# 线性代数

除了（且作为一部分）对多维数组的支持，Julia 还提供了许多常见和实用的线性代数运算的本地实现。基本的运算，比如 `tr`

，`det`

和 `inv`

都是支持的：

```
julia> A = [1 2 3; 4 1 6; 7 8 1]
3×3 Array{Int64,2}:
1 2 3
4 1 6
7 8 1
julia> tr(A)
3
julia> det(A)
104.0
julia> inv(A)
3×3 Array{Float64,2}:
-0.451923 0.211538 0.0865385
0.365385 -0.192308 0.0576923
0.240385 0.0576923 -0.0673077
```

还有其它实用的运算，比如寻找特征值或特征向量：

```
julia> A = [-4. -17.; 2. 2.]
2×2 Array{Float64,2}:
-4.0 -17.0
2.0 2.0
julia> eigvals(A)
2-element Array{Complex{Float64},1}:
-1.0 + 5.0im
-1.0 - 5.0im
julia> eigvecs(A)
2×2 Array{Complex{Float64},2}:
0.945905+0.0im 0.945905-0.0im
-0.166924-0.278207im -0.166924+0.278207im
```

此外，Julia 提供了多种矩阵分解，它们可用于加快问题的求解，比如线性求解或矩阵或矩阵求幂，这通过将矩阵预先分解成更适合问题的形式（出于性能或内存上的原因）。有关的更多信息，请参阅文档 `factorize`

。举个例子：

```
julia> A = [1.5 2 -4; 3 -1 -6; -10 2.3 4]
3×3 Array{Float64,2}:
1.5 2.0 -4.0
3.0 -1.0 -6.0
-10.0 2.3 4.0
julia> factorize(A)
LU{Float64,Array{Float64,2}}
L factor:
3×3 Array{Float64,2}:
1.0 0.0 0.0
-0.15 1.0 0.0
-0.3 -0.132196 1.0
U factor:
3×3 Array{Float64,2}:
-10.0 2.3 4.0
0.0 2.345 -3.4
0.0 0.0 -5.24947
```

因为 `A`

不是埃尔米特、对称、三角、三对角或双对角矩阵，LU 分解也许是我们能做的最好分解。与之相比：

```
julia> B = [1.5 2 -4; 2 -1 -3; -4 -3 5]
3×3 Array{Float64,2}:
1.5 2.0 -4.0
2.0 -1.0 -3.0
-4.0 -3.0 5.0
julia> factorize(B)
BunchKaufman{Float64,Array{Float64,2}}
D factor:
3×3 Tridiagonal{Float64,Array{Float64,1}}:
-1.64286 0.0 ⋅
0.0 -2.8 0.0
⋅ 0.0 5.0
U factor:
3×3 UnitUpperTriangular{Float64,Array{Float64,2}}:
1.0 0.142857 -0.8
⋅ 1.0 -0.6
⋅ ⋅ 1.0
permutation:
3-element Array{Int64,1}:
1
2
3
```

在这里，Julia 能够发现 `B`

确实是对称矩阵，并且使用一种更适当的分解。针对一个具有某些属性的矩阵，比如一个对称或三对角矩阵，往往有可能写出更高效的代码。Julia 提供了一些特殊的类型好让你可以根据矩阵所具有的属性「标记」它们。例如：

```
julia> B = [1.5 2 -4; 2 -1 -3; -4 -3 5]
3×3 Array{Float64,2}:
1.5 2.0 -4.0
2.0 -1.0 -3.0
-4.0 -3.0 5.0
julia> sB = Symmetric(B)
3×3 Symmetric{Float64,Array{Float64,2}}:
1.5 2.0 -4.0
2.0 -1.0 -3.0
-4.0 -3.0 5.0
```

`sB`

已经被标记成（实）对称矩阵，所以对于之后可能在它上面执行的操作，例如特征因子化或矩阵-向量乘积，只引用矩阵的一半可以提高效率。举个例子：

```
julia> B = [1.5 2 -4; 2 -1 -3; -4 -3 5]
3×3 Array{Float64,2}:
1.5 2.0 -4.0
2.0 -1.0 -3.0
-4.0 -3.0 5.0
julia> sB = Symmetric(B)
3×3 Symmetric{Float64,Array{Float64,2}}:
1.5 2.0 -4.0
2.0 -1.0 -3.0
-4.0 -3.0 5.0
julia> x = [1; 2; 3]
3-element Array{Int64,1}:
1
2
3
julia> sB\x
3-element Array{Float64,1}:
-1.7391304347826084
-1.1086956521739126
-1.4565217391304346
```

`\`

运算在这里执行线性求解。左除运算符相当强大，很容易写出紧凑、可读的代码，它足够灵活，可以求解各种线性方程组。

## 特殊矩阵

具有特殊对称性和结构的矩阵经常在线性代数中出现并且与各种矩阵分解相关。Julia 具有丰富的特殊矩阵类型，可以快速计算专门为特定矩阵类型开发的专用例程。

下表总结了在 Julia 中已经实现的特殊矩阵类型，以及为它们提供各种优化方法的钩子在 LAPACK 中是否可用。

类型 | 描述 |
---|---|

`Symmetric` | 对称矩阵 |

`Hermitian` | 埃尔米特矩阵 |

`UpperTriangular` | 上三角矩阵 |

`LowerTriangular` | 下三角矩阵 |

`Tridiagonal` | 三对角矩阵 |

`SymTridiagonal` | 对称三对角矩阵 |

`Bidiagonal` | 上/下双对角矩阵 |

`Diagonal` | 对角矩阵 |

`UniformScaling` | 等比缩放运算符 |

### 基本运算

矩阵类型 | `+` | `-` | `*` | `\` | 其它具有优化方法的函数 |
---|---|---|---|---|---|

`Symmetric` | MV | `inv` , `sqrt` , `exp` | |||

`Hermitian` | MV | `inv` , `sqrt` , `exp` | |||

`UpperTriangular` | MV | MV | `inv` , `det` | ||

`LowerTriangular` | MV | MV | `inv` , `det` | ||

`SymTridiagonal` | M | M | MS | MV | `eigmax` , `eigmin` |

`Tridiagonal` | M | M | MS | MV | |

`Bidiagonal` | M | M | MS | MV | |

`Diagonal` | M | M | MV | MV | `inv` , `det` , `logdet` , `/` |

`UniformScaling` | M | M | MVS | MVS | `/` |

图例：

键 | 描述 |
---|---|

M (matrix) | 针对矩阵与矩阵运算的优化方法可用 |

V (vector) | 针对矩阵与向量运算的优化方法可用 |

S (scalar) | 针对矩阵与标量运算的优化方法可用 |

### 矩阵分解

矩阵类型 | LAPACK | `eigen` | `eigvals` | `eigvecs` | `svd` | `svdvals` |
---|---|---|---|---|---|---|

`Symmetric` | SY | ARI | ||||

`Hermitian` | HE | ARI | ||||

`UpperTriangular` | TR | A | A | A | ||

`LowerTriangular` | TR | A | A | A | ||

`SymTridiagonal` | ST | A | ARI | AV | ||

`Tridiagonal` | GT | |||||

`Bidiagonal` | BD | A | A | |||

`Diagonal` | DI | A |

图例：

键 | 描述 | 例子 |
---|---|---|

A (all) | 找到所有特征值和/或特征向量的优化方法可用 | 例如，`eigvals(M)` |

R (range) | 通过第 `ih` 个特征值寻找第 `il` 个特征值的优化方法可用 | `eigvals(M, il, ih)` |

I (interval) | 寻找在区间 [`vl` , `vh` ] 内的特征值的优化方法可用 | `eigvals(M, vl, vh)` |

V (vectors) | 寻找对应于特征值 `x=[x1, x2,...]` 的特征向量的优化方法可用 | `eigvecs(M, x)` |

### 等比缩放运算符

`UniformScaling`

运算符代表一个标量乘以恒同运算符，`λ*I`

。恒同运算符 `I`

定义为常量和 `UniformScaling`

的实例。这些运算符是通用的，并且会在二元运算符 `+`

，`-`

，`*`

和 `\`

中与另一个矩阵相匹配。对于 `A+I`

和 `A-I`

，这意味着 `A`

必须是个方阵。与恒同运算符 `I`

相乘是一个空操作（除了检查比例因子是一），因此几乎没有开销。

来查看 `UniformScaling`

运算符的运行结果：

```
julia> U = UniformScaling(2);
julia> a = [1 2; 3 4]
2×2 Array{Int64,2}:
1 2
3 4
julia> a + U
2×2 Array{Int64,2}:
3 2
3 6
julia> a * U
2×2 Array{Int64,2}:
2 4
6 8
julia> [a U]
2×4 Array{Int64,2}:
1 2 2 0
3 4 0 2
julia> b = [1 2 3; 4 5 6]
2×3 Array{Int64,2}:
1 2 3
4 5 6
julia> b - U
ERROR: DimensionMismatch("matrix is not square: dimensions are (2, 3)")
Stacktrace:
[...]
```

## 矩阵分解

矩阵分解将矩阵分解成矩阵乘积，是线性代数的中心概念。

下表总结了在 Julia 中已经实现了的矩阵分解的类型。其相关方法的细节可以在线性代数文档中的标准函数这一节中找到。

类型 | 描述 |
---|---|

`Cholesky` | Cholesky 分解 |

`CholeskyPivoted` | Pivoted Cholesky 分解 |

`LU` | LU 分解 |

`LUTridiagonal` | 针对 `Tridiagonal` 矩阵的 LU 分解 |

`QR` | QR 分解 |

`QRCompactWY` | QR 分解的紧凑 WY 形式 |

`QRPivoted` | Pivoted QR 分解 |

`Hessenberg` | Hessenberg 分解 |

`Eigen` | 谱分解 |

`SVD` | 奇异值分解 |

`GeneralizedSVD` | 广义 SVD |

## 标准函数

Julia 中的线性代数函数主要通过调用 LAPACK 中的函数来实现。稀疏分解则调用 SuiteSparse 中的函数。

`Base.:*`

— Method.`*(A::AbstractMatrix, B::AbstractMatrix)`

Matrix multiplication.

**Examples**

```
julia> [1 1; 0 1] * [1 0; 1 1]
2×2 Array{Int64,2}:
2 1
1 1
```

`Base.:\`

— Method.`\(A, B)`

Matrix division using a polyalgorithm. For input matrices `A`

and `B`

, the result `X`

is such that `A*X == B`

when `A`

is square. The solver that is used depends upon the structure of `A`

. If `A`

is upper or lower triangular (or diagonal), no factorization of `A`

is required and the system is solved with either forward or backward substitution. For non-triangular square matrices, an LU factorization is used.

For rectangular `A`

the result is the minimum-norm least squares solution computed by a pivoted QR factorization of `A`

and a rank estimate of `A`

based on the R factor.

When `A`

is sparse, a similar polyalgorithm is used. For indefinite matrices, the `LDLt`

factorization does not use pivoting during the numerical factorization and therefore the procedure can fail even for invertible matrices.

**Examples**

```
julia> A = [1 0; 1 -2]; B = [32; -4];
julia> X = A \ B
2-element Array{Float64,1}:
32.0
18.0
julia> A * X == B
true
```

`LinearAlgebra.dot`

— Function.```
dot(x, y)
x ⋅ y
```

For any iterable containers `x`

and `y`

(including arrays of any dimension) of numbers (or any element type for which `dot`

is defined), compute the dot product (or inner product or scalar product), i.e. the sum of `dot(x[i],y[i])`

, as if they were vectors.

`x ⋅ y`

(where `⋅`

can be typed by tab-completing `\cdot`

in the REPL) is a synonym for `dot(x, y)`

.

**Examples**

```
julia> dot(1:5, 2:6)
70
julia> x = fill(2., (5,5));
julia> y = fill(3., (5,5));
julia> dot(x, y)
150.0
```

```
dot(x, y)
x ⋅ y
```

Compute the dot product between two vectors. For complex vectors, the first vector is conjugated. When the vectors have equal lengths, calling `dot`

is semantically equivalent to `sum(dot(vx,vy) for (vx,vy) in zip(x, y))`

.

**Examples**

```
julia> dot([1; 1], [2; 3])
5
julia> dot([im; im], [1; 1])
0 - 2im
```

`LinearAlgebra.cross`

— Function.```
cross(x, y)
×(x,y)
```

Compute the cross product of two 3-vectors.

**Examples**

```
julia> a = [0;1;0]
3-element Array{Int64,1}:
0
1
0
julia> b = [0;0;1]
3-element Array{Int64,1}:
0
0
1
julia> cross(a,b)
3-element Array{Int64,1}:
1
0
0
```

`LinearAlgebra.factorize`

— Function.`factorize(A)`

Compute a convenient factorization of `A`

, based upon the type of the input matrix. `factorize`

checks `A`

to see if it is symmetric/triangular/etc. if `A`

is passed as a generic matrix. `factorize`

checks every element of `A`

to verify/rule out each property. It will short-circuit as soon as it can rule out symmetry/triangular structure. The return value can be reused for efficient solving of multiple systems. For example: `A=factorize(A); x=A\b; y=A\C`

.

Properties of `A` | type of factorization |
---|---|

Positive-definite | Cholesky (see `cholesky` ) |

Dense Symmetric/Hermitian | Bunch-Kaufman (see `bunchkaufman` ) |

Sparse Symmetric/Hermitian | LDLt (see `ldlt` ) |

Triangular | Triangular |

Diagonal | Diagonal |

Bidiagonal | Bidiagonal |

Tridiagonal | LU (see `lu` ) |

Symmetric real tridiagonal | LDLt (see `ldlt` ) |

General square | LU (see `lu` ) |

General non-square | QR (see `qr` ) |

If `factorize`

is called on a Hermitian positive-definite matrix, for instance, then `factorize`

will return a Cholesky factorization.

**Examples**

```
julia> A = Array(Bidiagonal(fill(1.0, (5, 5)), :U))
5×5 Array{Float64,2}:
1.0 1.0 0.0 0.0 0.0
0.0 1.0 1.0 0.0 0.0
0.0 0.0 1.0 1.0 0.0
0.0 0.0 0.0 1.0 1.0
0.0 0.0 0.0 0.0 1.0
julia> factorize(A) # factorize will check to see that A is already factorized
5×5 Bidiagonal{Float64,Array{Float64,1}}:
1.0 1.0 ⋅ ⋅ ⋅
⋅ 1.0 1.0 ⋅ ⋅
⋅ ⋅ 1.0 1.0 ⋅
⋅ ⋅ ⋅ 1.0 1.0
⋅ ⋅ ⋅ ⋅ 1.0
```

This returns a `5×5 Bidiagonal{Float64}`

, which can now be passed to other linear algebra functions (e.g. eigensolvers) which will use specialized methods for `Bidiagonal`

types.

`LinearAlgebra.Diagonal`

— Type.`Diagonal(A::AbstractMatrix)`

Construct a matrix from the diagonal of `A`

.

**Examples**

```
julia> A = [1 2 3; 4 5 6; 7 8 9]
3×3 Array{Int64,2}:
1 2 3
4 5 6
7 8 9
julia> Diagonal(A)
3×3 Diagonal{Int64,Array{Int64,1}}:
1 ⋅ ⋅
⋅ 5 ⋅
⋅ ⋅ 9
```

`Diagonal(V::AbstractVector)`

Construct a matrix with `V`

as its diagonal.

**Examples**

```
julia> V = [1, 2]
2-element Array{Int64,1}:
1
2
julia> Diagonal(V)
2×2 Diagonal{Int64,Array{Int64,1}}:
1 ⋅
⋅ 2
```

`LinearAlgebra.Bidiagonal`

— Type.`Bidiagonal(dv::V, ev::V, uplo::Symbol) where V <: AbstractVector`

Constructs an upper (`uplo=:U`

) or lower (`uplo=:L`

) bidiagonal matrix using the given diagonal (`dv`

) and off-diagonal (`ev`

) vectors. The result is of type `Bidiagonal`

and provides efficient specialized linear solvers, but may be converted into a regular matrix with `convert(Array, _)`

(or `Array(_)`

for short). The length of `ev`

must be one less than the length of `dv`

.

**Examples**

```
julia> dv = [1, 2, 3, 4]
4-element Array{Int64,1}:
1
2
3
4
julia> ev = [7, 8, 9]
3-element Array{Int64,1}:
7
8
9
julia> Bu = Bidiagonal(dv, ev, :U) # ev is on the first superdiagonal
4×4 Bidiagonal{Int64,Array{Int64,1}}:
1 7 ⋅ ⋅
⋅ 2 8 ⋅
⋅ ⋅ 3 9
⋅ ⋅ ⋅ 4
julia> Bl = Bidiagonal(dv, ev, :L) # ev is on the first subdiagonal
4×4 Bidiagonal{Int64,Array{Int64,1}}:
1 ⋅ ⋅ ⋅
7 2 ⋅ ⋅
⋅ 8 3 ⋅
⋅ ⋅ 9 4
```

`Bidiagonal(A, uplo::Symbol)`

Construct a `Bidiagonal`

matrix from the main diagonal of `A`

and its first super- (if `uplo=:U`

) or sub-diagonal (if `uplo=:L`

).

**Examples**

```
julia> A = [1 1 1 1; 2 2 2 2; 3 3 3 3; 4 4 4 4]
4×4 Array{Int64,2}:
1 1 1 1
2 2 2 2
3 3 3 3
4 4 4 4
julia> Bidiagonal(A, :U) # contains the main diagonal and first superdiagonal of A
4×4 Bidiagonal{Int64,Array{Int64,1}}:
1 1 ⋅ ⋅
⋅ 2 2 ⋅
⋅ ⋅ 3 3
⋅ ⋅ ⋅ 4
julia> Bidiagonal(A, :L) # contains the main diagonal and first subdiagonal of A
4×4 Bidiagonal{Int64,Array{Int64,1}}:
1 ⋅ ⋅ ⋅
2 2 ⋅ ⋅
⋅ 3 3 ⋅
⋅ ⋅ 4 4
```

`LinearAlgebra.SymTridiagonal`

— Type.`SymTridiagonal(dv::V, ev::V) where V <: AbstractVector`

Construct a symmetric tridiagonal matrix from the diagonal (`dv`

) and first sub/super-diagonal (`ev`

), respectively. The result is of type `SymTridiagonal`

and provides efficient specialized eigensolvers, but may be converted into a regular matrix with `convert(Array, _)`

(or `Array(_)`

for short).

**Examples**

```
julia> dv = [1, 2, 3, 4]
4-element Array{Int64,1}:
1
2
3
4
julia> ev = [7, 8, 9]
3-element Array{Int64,1}:
7
8
9
julia> SymTridiagonal(dv, ev)
4×4 SymTridiagonal{Int64,Array{Int64,1}}:
1 7 ⋅ ⋅
7 2 8 ⋅
⋅ 8 3 9
⋅ ⋅ 9 4
```

`SymTridiagonal(A::AbstractMatrix)`

Construct a symmetric tridiagonal matrix from the diagonal and first sub/super-diagonal, of the symmetric matrix `A`

.

**Examples**

```
julia> A = [1 2 3; 2 4 5; 3 5 6]
3×3 Array{Int64,2}:
1 2 3
2 4 5
3 5 6
julia> SymTridiagonal(A)
3×3 SymTridiagonal{Int64,Array{Int64,1}}:
1 2 ⋅
2 4 5
⋅ 5 6
```

`LinearAlgebra.Tridiagonal`

— Type.`Tridiagonal(dl::V, d::V, du::V) where V <: AbstractVector`

Construct a tridiagonal matrix from the first subdiagonal, diagonal, and first superdiagonal, respectively. The result is of type `Tridiagonal`

and provides efficient specialized linear solvers, but may be converted into a regular matrix with `convert(Array, _)`

(or `Array(_)`

for short). The lengths of `dl`

and `du`

must be one less than the length of `d`

.

**Examples**

```
julia> dl = [1, 2, 3];
julia> du = [4, 5, 6];
julia> d = [7, 8, 9, 0];
julia> Tridiagonal(dl, d, du)
4×4 Tridiagonal{Int64,Array{Int64,1}}:
7 4 ⋅ ⋅
1 8 5 ⋅
⋅ 2 9 6
⋅ ⋅ 3 0
```

`Tridiagonal(A)`

Construct a tridiagonal matrix from the first sub-diagonal, diagonal and first super-diagonal of the matrix `A`

.

**Examples**

```
julia> A = [1 2 3 4; 1 2 3 4; 1 2 3 4; 1 2 3 4]
4×4 Array{Int64,2}:
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
julia> Tridiagonal(A)
4×4 Tridiagonal{Int64,Array{Int64,1}}:
1 2 ⋅ ⋅
1 2 3 ⋅
⋅ 2 3 4
⋅ ⋅ 3 4
```

`LinearAlgebra.Symmetric`

— Type.`Symmetric(A, uplo=:U)`

Construct a `Symmetric`

view of the upper (if `uplo = :U`

) or lower (if `uplo = :L`

) triangle of the matrix `A`

.

**Examples**

```
julia> A = [1 0 2 0 3; 0 4 0 5 0; 6 0 7 0 8; 0 9 0 1 0; 2 0 3 0 4]
5×5 Array{Int64,2}:
1 0 2 0 3
0 4 0 5 0
6 0 7 0 8
0 9 0 1 0
2 0 3 0 4
julia> Supper = Symmetric(A)
5×5 Symmetric{Int64,Array{Int64,2}}:
1 0 2 0 3
0 4 0 5 0
2 0 7 0 8
0 5 0 1 0
3 0 8 0 4
julia> Slower = Symmetric(A, :L)
5×5 Symmetric{Int64,Array{Int64,2}}:
1 0 6 0 2
0 4 0 9 0
6 0 7 0 3
0 9 0 1 0
2 0 3 0 4
```

Note that `Supper`

will not be equal to `Slower`

unless `A`

is itself symmetric (e.g. if `A == transpose(A)`

).

`LinearAlgebra.Hermitian`

— Type.`Hermitian(A, uplo=:U)`

Construct a `Hermitian`

view of the upper (if `uplo = :U`

) or lower (if `uplo = :L`

) triangle of the matrix `A`

.

**Examples**

```
julia> A = [1 0 2+2im 0 3-3im; 0 4 0 5 0; 6-6im 0 7 0 8+8im; 0 9 0 1 0; 2+2im 0 3-3im 0 4];
julia> Hupper = Hermitian(A)
5×5 Hermitian{Complex{Int64},Array{Complex{Int64},2}}:
1+0im 0+0im 2+2im 0+0im 3-3im
0+0im 4+0im 0+0im 5+0im 0+0im
2-2im 0+0im 7+0im 0+0im 8+8im
0+0im 5+0im 0+0im 1+0im 0+0im
3+3im 0+0im 8-8im 0+0im 4+0im
julia> Hlower = Hermitian(A, :L)
5×5 Hermitian{Complex{Int64},Array{Complex{Int64},2}}:
1+0im 0+0im 6+6im 0+0im 2-2im
0+0im 4+0im 0+0im 9+0im 0+0im
6-6im 0+0im 7+0im 0+0im 3+3im
0+0im 9+0im 0+0im 1+0im 0+0im
2+2im 0+0im 3-3im 0+0im 4+0im
```

Note that `Hupper`

will not be equal to `Hlower`

unless `A`

is itself Hermitian (e.g. if `A == adjoint(A)`

).

All non-real parts of the diagonal will be ignored.

`Hermitian(fill(complex(1,1), 1, 1)) == fill(1, 1, 1)`

`LinearAlgebra.LowerTriangular`

— Type.`LowerTriangular(A::AbstractMatrix)`

Construct a `LowerTriangular`

view of the matrix `A`

.

**Examples**

```
julia> A = [1.0 2.0 3.0; 4.0 5.0 6.0; 7.0 8.0 9.0]
3×3 Array{Float64,2}:
1.0 2.0 3.0
4.0 5.0 6.0
7.0 8.0 9.0
julia> LowerTriangular(A)
3×3 LowerTriangular{Float64,Array{Float64,2}}:
1.0 ⋅ ⋅
4.0 5.0 ⋅
7.0 8.0 9.0
```

`LinearAlgebra.UpperTriangular`

— Type.`UpperTriangular(A::AbstractMatrix)`

Construct an `UpperTriangular`

view of the matrix `A`

.

**Examples**

```
julia> A = [1.0 2.0 3.0; 4.0 5.0 6.0; 7.0 8.0 9.0]
3×3 Array{Float64,2}:
1.0 2.0 3.0
4.0 5.0 6.0
7.0 8.0 9.0
julia> UpperTriangular(A)
3×3 UpperTriangular{Float64,Array{Float64,2}}:
1.0 2.0 3.0
⋅ 5.0 6.0
⋅ ⋅ 9.0
```

`LinearAlgebra.UniformScaling`

— Type.`UniformScaling{T<:Number}`

Generically sized uniform scaling operator defined as a scalar times the identity operator, `λ*I`

. See also `I`

.

**Examples**

```
julia> J = UniformScaling(2.)
UniformScaling{Float64}
2.0*I
julia> A = [1. 2.; 3. 4.]
2×2 Array{Float64,2}:
1.0 2.0
3.0 4.0
julia> J*A
2×2 Array{Float64,2}:
2.0 4.0
6.0 8.0
```

`LinearAlgebra.lu`

— Function.`lu(A, pivot=Val(true); check = true) -> F::LU`

Compute the LU factorization of `A`

.

When `check = true`

, an error is thrown if the decomposition fails. When `check = false`

, responsibility for checking the decomposition's validity (via `issuccess`

) lies with the user.

In most cases, if `A`

is a subtype `S`

of `AbstractMatrix{T}`

with an element type `T`

supporting `+`

, `-`

, `*`

and `/`

, the return type is `LU{T,S{T}}`

. If pivoting is chosen (default) the element type should also support `abs`

and `<`

.

The individual components of the factorization `F`

can be accessed via `getproperty`

:

Component | Description |
---|---|

`F.L` | `L` (lower triangular) part of `LU` |

`F.U` | `U` (upper triangular) part of `LU` |

`F.p` | (right) permutation `Vector` |

`F.P` | (right) permutation `Matrix` |

Iterating the factorization produces the components `F.L`

, `F.U`

, and `F.p`

.

The relationship between `F`

and `A`

is

`F.L*F.U == A[F.p, :]`

`F`

further supports the following functions:

Supported function | `LU` | `LU{T,Tridiagonal{T}}` |
---|---|---|

`/` | ✓ | |

`\` | ✓ | ✓ |

`inv` | ✓ | ✓ |

`det` | ✓ | ✓ |

`logdet` | ✓ | ✓ |

`logabsdet` | ✓ | ✓ |

`size` | ✓ | ✓ |

**Examples**

```
julia> A = [4 3; 6 3]
2×2 Array{Int64,2}:
4 3
6 3
julia> F = lu(A)
LU{Float64,Array{Float64,2}}
L factor:
2×2 Array{Float64,2}:
1.0 0.0
1.5 1.0
U factor:
2×2 Array{Float64,2}:
4.0 3.0
0.0 -1.5
julia> F.L * F.U == A[F.p, :]
true
julia> l, u, p = lu(A); # destructuring via iteration
julia> l == F.L && u == F.U && p == F.p
true
```

`LinearAlgebra.lu!`

— Function.`lu!(A, pivot=Val(true); check = true) -> LU`

`lu!`

is the same as `lu`

, but saves space by overwriting the input `A`

, instead of creating a copy. An `InexactError`

exception is thrown if the factorization produces a number not representable by the element type of `A`

, e.g. for integer types.

**Examples**

```
julia> A = [4. 3.; 6. 3.]
2×2 Array{Float64,2}:
4.0 3.0
6.0 3.0
julia> F = lu!(A)
LU{Float64,Array{Float64,2}}
L factor:
2×2 Array{Float64,2}:
1.0 0.0
0.666667 1.0
U factor:
2×2 Array{Float64,2}:
6.0 3.0
0.0 1.0
julia> iA = [4 3; 6 3]
2×2 Array{Int64,2}:
4 3
6 3
julia> lu!(iA)
ERROR: InexactError: Int64(Int64, 0.6666666666666666)
Stacktrace:
[...]
```

`LinearAlgebra.cholesky`

— Function.`cholesky(A, Val(false); check = true) -> Cholesky`

Compute the Cholesky factorization of a dense symmetric positive definite matrix `A`

and return a `Cholesky`

factorization. The matrix `A`

can either be a `Symmetric`

or `Hermitian`

`StridedMatrix`

or a *perfectly* symmetric or Hermitian `StridedMatrix`

. The triangular Cholesky factor can be obtained from the factorization `F`

with: `F.L`

and `F.U`

. The following functions are available for `Cholesky`

objects: `size`

, `\`

, `inv`

, `det`

, `logdet`

and `isposdef`

.

When `check = true`

, an error is thrown if the decomposition fails. When `check = false`

, responsibility for checking the decomposition's validity (via `issuccess`

) lies with the user.

**Examples**

```
julia> A = [4. 12. -16.; 12. 37. -43.; -16. -43. 98.]
3×3 Array{Float64,2}:
4.0 12.0 -16.0
12.0 37.0 -43.0
-16.0 -43.0 98.0
julia> C = cholesky(A)
Cholesky{Float64,Array{Float64,2}}
U factor:
3×3 UpperTriangular{Float64,Array{Float64,2}}:
2.0 6.0 -8.0
⋅ 1.0 5.0
⋅ ⋅ 3.0
julia> C.U
3×3 UpperTriangular{Float64,Array{Float64,2}}:
2.0 6.0 -8.0
⋅ 1.0 5.0
⋅ ⋅ 3.0
julia> C.L
3×3 LowerTriangular{Float64,Array{Float64,2}}:
2.0 ⋅ ⋅
6.0 1.0 ⋅
-8.0 5.0 3.0
julia> C.L * C.U == A
true
```

`cholesky(A, Val(true); tol = 0.0, check = true) -> CholeskyPivoted`

Compute the pivoted Cholesky factorization of a dense symmetric positive semi-definite matrix `A`

and return a `CholeskyPivoted`

factorization. The matrix `A`

can either be a `Symmetric`

or `Hermitian`

`StridedMatrix`

or a *perfectly* symmetric or Hermitian `StridedMatrix`

. The triangular Cholesky factor can be obtained from the factorization `F`

with: `F.L`

and `F.U`

. The following functions are available for `PivotedCholesky`

objects: `size`

, `\`

, `inv`

, `det`

, and `rank`

. The argument `tol`

determines the tolerance for determining the rank. For negative values, the tolerance is the machine precision.

When `check = true`

, an error is thrown if the decomposition fails. When `check = false`

, responsibility for checking the decomposition's validity (via `issuccess`

) lies with the user.

`LinearAlgebra.cholesky!`

— Function.`cholesky!(A, Val(false); check = true) -> Cholesky`

The same as `cholesky`

, but saves space by overwriting the input `A`

, instead of creating a copy. An `InexactError`

exception is thrown if the factorization produces a number not representable by the element type of `A`

, e.g. for integer types.

**Examples**

```
julia> A = [1 2; 2 50]
2×2 Array{Int64,2}:
1 2
2 50
julia> cholesky!(A)
ERROR: InexactError: Int64(Int64, 6.782329983125268)
Stacktrace:
[...]
```

`cholesky!(A, Val(true); tol = 0.0, check = true) -> CholeskyPivoted`

The same as `cholesky`

, but saves space by overwriting the input `A`

, instead of creating a copy. An `InexactError`

exception is thrown if the factorization produces a number not representable by the element type of `A`

, e.g. for integer types.

`LinearAlgebra.lowrankupdate`

— Function.`lowrankupdate(C::Cholesky, v::StridedVector) -> CC::Cholesky`

Update a Cholesky factorization `C`

with the vector `v`

. If `A = C.U'C.U`

then `CC = cholesky(C.U'C.U + v*v')`

but the computation of `CC`

only uses `O(n^2)`

operations.

`LinearAlgebra.lowrankdowndate`

— Function.`lowrankdowndate(C::Cholesky, v::StridedVector) -> CC::Cholesky`

Downdate a Cholesky factorization `C`

with the vector `v`

. If `A = C.U'C.U`

then `CC = cholesky(C.U'C.U - v*v')`

but the computation of `CC`

only uses `O(n^2)`

operations.

`LinearAlgebra.lowrankupdate!`

— Function.`lowrankupdate!(C::Cholesky, v::StridedVector) -> CC::Cholesky`

Update a Cholesky factorization `C`

with the vector `v`

. If `A = C.U'C.U`

then `CC = cholesky(C.U'C.U + v*v')`

but the computation of `CC`

only uses `O(n^2)`

operations. The input factorization `C`

is updated in place such that on exit `C == CC`

. The vector `v`

is destroyed during the computation.

`LinearAlgebra.lowrankdowndate!`

— Function.`lowrankdowndate!(C::Cholesky, v::StridedVector) -> CC::Cholesky`

Downdate a Cholesky factorization `C`

with the vector `v`

. If `A = C.U'C.U`

then `CC = cholesky(C.U'C.U - v*v')`

but the computation of `CC`

only uses `O(n^2)`

operations. The input factorization `C`

is updated in place such that on exit `C == CC`

. The vector `v`

is destroyed during the computation.

`LinearAlgebra.ldlt`

— Function.`ldlt(S::SymTridiagonal) -> LDLt`

Compute an `LDLt`

factorization of the real symmetric tridiagonal matrix `S`

such that `S = L*Diagonal(d)*L'`

where `L`

is a unit lower triangular matrix and `d`

is a vector. The main use of an `LDLt`

factorization `F = ldlt(S)`

is to solve the linear system of equations `Sx = b`

with `F\b`

.

**Examples**

```
julia> S = SymTridiagonal([3., 4., 5.], [1., 2.])
3×3 SymTridiagonal{Float64,Array{Float64,1}}:
3.0 1.0 ⋅
1.0 4.0 2.0
⋅ 2.0 5.0
julia> ldltS = ldlt(S);
julia> b = [6., 7., 8.];
julia> ldltS \ b
3-element Array{Float64,1}:
1.7906976744186047
0.627906976744186
1.3488372093023255
julia> S \ b
3-element Array{Float64,1}:
1.7906976744186047
0.627906976744186
1.3488372093023255
```

`LinearAlgebra.ldlt!`

— Function.`ldlt!(S::SymTridiagonal) -> LDLt`

Same as `ldlt`

, but saves space by overwriting the input `S`

, instead of creating a copy.

**Examples**

```
julia> S = SymTridiagonal([3., 4., 5.], [1., 2.])
3×3 SymTridiagonal{Float64,Array{Float64,1}}:
3.0 1.0 ⋅
1.0 4.0 2.0
⋅ 2.0 5.0
julia> ldltS = ldlt!(S);
julia> ldltS === S
false
julia> S
3×3 SymTridiagonal{Float64,Array{Float64,1}}:
3.0 0.333333 ⋅
0.333333 3.66667 0.545455
⋅ 0.545455 3.90909
```

`LinearAlgebra.qr`

— Function.`qr(A, pivot=Val(false)) -> F`

Compute the QR factorization of the matrix `A`

: an orthogonal (or unitary if `A`

is complex-valued) matrix `Q`

, and an upper triangular matrix `R`

such that

The returned object `F`

stores the factorization in a packed format:

if

`pivot == Val(true)`

then`F`

is a`QRPivoted`

object,otherwise if the element type of

`A`

is a BLAS type (`Float32`

,`Float64`

,`ComplexF32`

or`ComplexF64`

), then`F`

is a`QRCompactWY`

object,otherwise

`F`

is a`QR`

object.

The individual components of the decomposition `F`

can be retrieved via property accessors:

`F.Q`

: the orthogonal/unitary matrix`Q`

`F.R`

: the upper triangular matrix`R`

`F.p`

: the permutation vector of the pivot (`QRPivoted`

only)`F.P`

: the permutation matrix of the pivot (`QRPivoted`

only)

Iterating the decomposition produces the components `Q`

, `R`

, and if extant `p`

.

The following functions are available for the `QR`

objects: `inv`

, `size`

, and `\`

. When `A`

is rectangular, `\`

will return a least squares solution and if the solution is not unique, the one with smallest norm is returned.

Multiplication with respect to either full/square or non-full/square `Q`

is allowed, i.e. both `F.Q*F.R`

and `F.Q*A`

are supported. A `Q`

matrix can be converted into a regular matrix with `Matrix`

. This operation returns the "thin" Q factor, i.e., if `A`

is `m`

×`n`

with `m>=n`

, then `Matrix(F.Q)`

yields an `m`

×`n`

matrix with orthonormal columns. To retrieve the "full" Q factor, an `m`

×`m`

orthogonal matrix, use `F.Q*Matrix(I,m,m)`

. If `m<=n`

, then `Matrix(F.Q)`

yields an `m`

×`m`

orthogonal matrix.

**Examples**

```
julia> A = [3.0 -6.0; 4.0 -8.0; 0.0 1.0]
3×2 Array{Float64,2}:
3.0 -6.0
4.0 -8.0
0.0 1.0
julia> F = qr(A)
LinearAlgebra.QRCompactWY{Float64,Array{Float64,2}}
Q factor:
3×3 LinearAlgebra.QRCompactWYQ{Float64,Array{Float64,2}}:
-0.6 0.0 0.8
-0.8 0.0 -0.6
0.0 -1.0 0.0
R factor:
2×2 Array{Float64,2}:
-5.0 10.0
0.0 -1.0
julia> F.Q * F.R == A
true
```

`qr`

returns multiple types because LAPACK uses several representations that minimize the memory storage requirements of products of Householder elementary reflectors, so that the `Q`

and `R`

matrices can be stored compactly rather as two separate dense matrices.

`LinearAlgebra.qr!`

— Function.`qr!(A, pivot=Val(false))`

`qr!`

is the same as `qr`

when `A`

is a subtype of `StridedMatrix`

, but saves space by overwriting the input `A`

, instead of creating a copy. An `InexactError`

exception is thrown if the factorization produces a number not representable by the element type of `A`

, e.g. for integer types.

**Examples**

```
julia> a = [1. 2.; 3. 4.]
2×2 Array{Float64,2}:
1.0 2.0
3.0 4.0
julia> qr!(a)
LinearAlgebra.QRCompactWY{Float64,Array{Float64,2}}
Q factor:
2×2 LinearAlgebra.QRCompactWYQ{Float64,Array{Float64,2}}:
-0.316228 -0.948683
-0.948683 0.316228
R factor:
2×2 Array{Float64,2}:
-3.16228 -4.42719
0.0 -0.632456
julia> a = [1 2; 3 4]
2×2 Array{Int64,2}:
1 2
3 4
julia> qr!(a)
ERROR: InexactError: Int64(Int64, -3.1622776601683795)
Stacktrace:
[...]
```

`LinearAlgebra.QR`

— Type.`QR <: Factorization`

A QR matrix factorization stored in a packed format, typically obtained from `qr`

. If $A$ is an `m`

×`n`

matrix, then

where $Q$ is an orthogonal/unitary matrix and $R$ is upper triangular. The matrix $Q$ is stored as a sequence of Householder reflectors $v_i$ and coefficients $\tau_i$ where:

Iterating the decomposition produces the components `Q`

and `R`

.

The object has two fields:

`factors`

is an`m`

×`n`

matrix.The upper triangular part contains the elements of $R$, that is

`R = triu(F.factors)`

for a`QR`

object`F`

.The subdiagonal part contains the reflectors $v_i$ stored in a packed format where $v_i$ is the $i$th column of the matrix

`V = I + tril(F.factors, -1)`

.

`τ`

is a vector of length`min(m,n)`

containing the coefficients $au_i$.

`LinearAlgebra.QRCompactWY`

— Type.`QRCompactWY <: Factorization`

A QR matrix factorization stored in a compact blocked format, typically obtained from `qr`

. If $A$ is an `m`

×`n`

matrix, then

where $Q$ is an orthogonal/unitary matrix and $R$ is upper triangular. It is similar to the `QR`

format except that the orthogonal/unitary matrix $Q$ is stored in *Compact WY* format [Schreiber1989], as a lower trapezoidal matrix $V$ and an upper triangular matrix $T$ where

such that $v_i$ is the $i$th column of $V$, and $au_i$ is the $i$th diagonal element of $T$.

Iterating the decomposition produces the components `Q`

and `R`

.

The object has two fields:

`factors`

, as in the`QR`

type, is an`m`

×`n`

matrix.The upper triangular part contains the elements of $R$, that is

`R = triu(F.factors)`

for a`QR`

object`F`

.The subdiagonal part contains the reflectors $v_i$ stored in a packed format such that

`V = I + tril(F.factors, -1)`

.

`T`

is a square matrix with`min(m,n)`

columns, whose upper triangular part gives the matrix $T$ above (the subdiagonal elements are ignored).

This format should not to be confused with the older *WY* representation [Bischof1987].

**[Bischof1987]**

C Bischof and C Van Loan, "The WY representation for products of Householder matrices", SIAM J Sci Stat Comput 8 (1987), s2-s13. doi:10.1137/0908009

**[Schreiber1989]**

R Schreiber and C Van Loan, "A storage-efficient WY representation for products of Householder transformations", SIAM J Sci Stat Comput 10 (1989), 53-57. doi:10.1137/0910005

`LinearAlgebra.QRPivoted`

— Type.`QRPivoted <: Factorization`

A QR matrix factorization with column pivoting in a packed format, typically obtained from `qr`

. If $A$ is an `m`

×`n`

matrix, then

where $P$ is a permutation matrix, $Q$ is an orthogonal/unitary matrix and $R$ is upper triangular. The matrix $Q$ is stored as a sequence of Householder reflectors:

Iterating the decomposition produces the components `Q`

, `R`

, and `p`

.

The object has three fields:

`factors`

is an`m`

×`n`

matrix.The upper triangular part contains the elements of $R$, that is

`R = triu(F.factors)`

for a`QR`

object`F`

.The subdiagonal part contains the reflectors $v_i$ stored in a packed format where $v_i$ is the $i$th column of the matrix

`V = I + tril(F.factors, -1)`

.

`τ`

is a vector of length`min(m,n)`

containing the coefficients $au_i$.`jpvt`

is an integer vector of length`n`

corresponding to the permutation $P$.

`LinearAlgebra.lq!`

— Function.`lq!(A) -> LQ`

Compute the LQ factorization of `A`

, using the input matrix as a workspace. See also `lq`

.

`LinearAlgebra.lq`

— Function.`lq(A) -> S::LQ`

Compute the LQ decomposition of `A`

. The decomposition's lower triangular component can be obtained from the `LQ`

object `S`

via `S.L`

, and the orthogonal/unitary component via `S.Q`

, such that `A ≈ S.L*S.Q`

.

Iterating the decomposition produces the components `S.L`

and `S.Q`

.

The LQ decomposition is the QR decomposition of `transpose(A)`

.

**Examples**

```
julia> A = [5. 7.; -2. -4.]
2×2 Array{Float64,2}:
5.0 7.0
-2.0 -4.0
julia> S = lq(A)
LQ{Float64,Array{Float64,2}} with factors L and Q:
[-8.60233 0.0; 4.41741 -0.697486]
[-0.581238 -0.813733; -0.813733 0.581238]
julia> S.L * S.Q
2×2 Array{Float64,2}:
5.0 7.0
-2.0 -4.0
julia> l, q = S; # destructuring via iteration
julia> l == S.L && q == S.Q
true
```

`LinearAlgebra.bunchkaufman`

— Function.`bunchkaufman(A, rook::Bool=false; check = true) -> S::BunchKaufman`

Compute the Bunch-Kaufman [Bunch1977] factorization of a `Symmetric`

or `Hermitian`

matrix `A`

as $P'*U*D*U'*P$ or $P'*L*D*L'*P$, depending on which triangle is stored in `A`

, and return a `BunchKaufman`

object. Note that if `A`

is complex symmetric then `U'`

and `L'`

denote the unconjugated transposes, i.e. `transpose(U)`

and `transpose(L)`

.

Iterating the decomposition produces the components `S.D`

, `S.U`

or `S.L`

as appropriate given `S.uplo`

, and `S.p`

.

If `rook`

is `true`

, rook pivoting is used. If `rook`

is false, rook pivoting is not used.

`check = true`

, an error is thrown if the decomposition fails. When `check = false`

, responsibility for checking the decomposition's validity (via `issuccess`

) lies with the user.

The following functions are available for `BunchKaufman`

objects: `size`

, `\`

, `inv`

, `issymmetric`

, `ishermitian`

, `getindex`

.

**[Bunch1977]**

J R Bunch and L Kaufman, Some stable methods for calculating inertia

and solving symmetric linear systems, Mathematics of Computation 31:137 (1977), 163-179. url.

**Examples**

```
julia> A = [1 2; 2 3]
2×2 Array{Int64,2}:
1 2
2 3
julia> S = bunchkaufman(A)
BunchKaufman{Float64,Array{Float64,2}}
D factor:
2×2 Tridiagonal{Float64,Array{Float64,1}}:
-0.333333 0.0
0.0 3.0
U factor:
2×2 UnitUpperTriangular{Float64,Array{Float64,2}}:
1.0 0.666667
⋅ 1.0
permutation:
2-element Array{Int64,1}:
1
2
julia> d, u, p = S; # destructuring via iteration
julia> d == S.D && u == S.U && p == S.p
true
```

`LinearAlgebra.bunchkaufman!`

— Function.`bunchkaufman!(A, rook::Bool=false; check = true) -> BunchKaufman`

`bunchkaufman!`

is the same as `bunchkaufman`

, but saves space by overwriting the input `A`

, instead of creating a copy.

`LinearAlgebra.eigvals`

— Function.`eigvals(A; permute::Bool=true, scale::Bool=true) -> values`

Return the eigenvalues of `A`

.

For general non-symmetric matrices it is possible to specify how the matrix is balanced before the eigenvalue calculation. The option `permute=true`

permutes the matrix to become closer to upper triangular, and `scale=true`

scales the matrix by its diagonal elements to make rows and columns more equal in norm. The default is `true`

for both options.

**Examples**

```
julia> diag_matrix = [1 0; 0 4]
2×2 Array{Int64,2}:
1 0
0 4
julia> eigvals(diag_matrix)
2-element Array{Float64,1}:
1.0
4.0
```

For a scalar input, `eigvals`

will return a scalar.

**Example**

```
julia> eigvals(-2)
-2
```

`eigvals(A, B) -> values`

Computes the generalized eigenvalues of `A`

and `B`

.

**Examples**

```
julia> A = [1 0; 0 -1]
2×2 Array{Int64,2}:
1 0
0 -1
julia> B = [0 1; 1 0]
2×2 Array{Int64,2}:
0 1
1 0
julia> eigvals(A,B)
2-element Array{Complex{Float64},1}:
0.0 + 1.0im
0.0 - 1.0im
```

`eigvals(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> values`

Returns the eigenvalues of `A`

. It is possible to calculate only a subset of the eigenvalues by specifying a `UnitRange`

`irange`

covering indices of the sorted eigenvalues, e.g. the 2nd to 8th eigenvalues.

```
julia> A = SymTridiagonal([1.; 2.; 1.], [2.; 3.])
3×3 SymTridiagonal{Float64,Array{Float64,1}}:
1.0 2.0 ⋅
2.0 2.0 3.0
⋅ 3.0 1.0
julia> eigvals(A, 2:2)
1-element Array{Float64,1}:
0.9999999999999996
julia> eigvals(A)
3-element Array{Float64,1}:
-2.1400549446402604
1.0000000000000002
5.140054944640259
```

`eigvals(A::Union{SymTridiagonal, Hermitian, Symmetric}, vl::Real, vu::Real) -> values`

Returns the eigenvalues of `A`

. It is possible to calculate only a subset of the eigenvalues by specifying a pair `vl`

and `vu`

for the lower and upper boundaries of the eigenvalues.

```
julia> A = SymTridiagonal([1.; 2.; 1.], [2.; 3.])
3×3 SymTridiagonal{Float64,Array{Float64,1}}:
1.0 2.0 ⋅
2.0 2.0 3.0
⋅ 3.0 1.0
julia> eigvals(A, -1, 2)
1-element Array{Float64,1}:
1.0000000000000009
julia> eigvals(A)
3-element Array{Float64,1}:
-2.1400549446402604
1.0000000000000002
5.140054944640259
```

`LinearAlgebra.eigvals!`

— Function.`eigvals!(A; permute::Bool=true, scale::Bool=true) -> values`

Same as `eigvals`

, but saves space by overwriting the input `A`

, instead of creating a copy. The option `permute=true`

permutes the matrix to become closer to upper triangular, and `scale=true`

scales the matrix by its diagonal elements to make rows and columns more equal in norm.

The input matrix `A`

will not contain its eigenvalues after `eigvals!`

is called on it - `A`

is used as a workspace.

**Examples**

```
julia> A = [1. 2.; 3. 4.]
2×2 Array{Float64,2}:
1.0 2.0
3.0 4.0
julia> eigvals!(A)
2-element Array{Float64,1}:
-0.3722813232690143
5.372281323269014
julia> A
2×2 Array{Float64,2}:
-0.372281 -1.0
0.0 5.37228
```

`eigvals!(A, B) -> values`

Same as `eigvals`

, but saves space by overwriting the input `A`

(and `B`

), instead of creating copies.

The input matrices `A`

and `B`

will not contain their eigenvalues after `eigvals!`

is called. They are used as workspaces.

**Examples**

```
julia> A = [1. 0.; 0. -1.]
2×2 Array{Float64,2}:
1.0 0.0
0.0 -1.0
julia> B = [0. 1.; 1. 0.]
2×2 Array{Float64,2}:
0.0 1.0
1.0 0.0
julia> eigvals!(A, B)
2-element Array{Complex{Float64},1}:
0.0 + 1.0im
0.0 - 1.0im
julia> A
2×2 Array{Float64,2}:
-0.0 -1.0
1.0 -0.0
julia> B
2×2 Array{Float64,2}:
1.0 0.0
0.0 1.0
```

`eigvals!(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> values`

Same as `eigvals`

, but saves space by overwriting the input `A`

, instead of creating a copy. `irange`

is a range of eigenvalue *indices* to search for - for instance, the 2nd to 8th eigenvalues.

`eigvals!(A::Union{SymTridiagonal, Hermitian, Symmetric}, vl::Real, vu::Real) -> values`

Same as `eigvals`

, but saves space by overwriting the input `A`

, instead of creating a copy. `vl`

is the lower bound of the interval to search for eigenvalues, and `vu`

is the upper bound.

`LinearAlgebra.eigmax`

— Function.`eigmax(A; permute::Bool=true, scale::Bool=true)`

Return the largest eigenvalue of `A`

. The option `permute=true`

permutes the matrix to become closer to upper triangular, and `scale=true`

scales the matrix by its diagonal elements to make rows and columns more equal in norm. Note that if the eigenvalues of `A`

are complex, this method will fail, since complex numbers cannot be sorted.

**Examples**

```
julia> A = [0 im; -im 0]
2×2 Array{Complex{Int64},2}:
0+0im 0+1im
0-1im 0+0im
julia> eigmax(A)
1.0
julia> A = [0 im; -1 0]
2×2 Array{Complex{Int64},2}:
0+0im 0+1im
-1+0im 0+0im
julia> eigmax(A)
ERROR: DomainError with Complex{Int64}[0+0im 0+1im; -1+0im 0+0im]:
`A` cannot have complex eigenvalues.
Stacktrace:
[...]
```

`LinearAlgebra.eigmin`

— Function.`eigmin(A; permute::Bool=true, scale::Bool=true)`

Return the smallest eigenvalue of `A`

. The option `permute=true`

permutes the matrix to become closer to upper triangular, and `scale=true`

scales the matrix by its diagonal elements to make rows and columns more equal in norm. Note that if the eigenvalues of `A`

are complex, this method will fail, since complex numbers cannot be sorted.

**Examples**

```
julia> A = [0 im; -im 0]
2×2 Array{Complex{Int64},2}:
0+0im 0+1im
0-1im 0+0im
julia> eigmin(A)
-1.0
julia> A = [0 im; -1 0]
2×2 Array{Complex{Int64},2}:
0+0im 0+1im
-1+0im 0+0im
julia> eigmin(A)
ERROR: DomainError with Complex{Int64}[0+0im 0+1im; -1+0im 0+0im]:
`A` cannot have complex eigenvalues.
Stacktrace:
[...]
```

`LinearAlgebra.eigvecs`

— Function.`eigvecs(A::SymTridiagonal[, eigvals]) -> Matrix`

Return a matrix `M`

whose columns are the eigenvectors of `A`

. (The `k`

th eigenvector can be obtained from the slice `M[:, k]`

.)

If the optional vector of eigenvalues `eigvals`

is specified, `eigvecs`

returns the specific corresponding eigenvectors.

**Examples**

```
julia> A = SymTridiagonal([1.; 2.; 1.], [2.; 3.])
3×3 SymTridiagonal{Float64,Array{Float64,1}}:
1.0 2.0 ⋅
2.0 2.0 3.0
⋅ 3.0 1.0
julia> eigvals(A)
3-element Array{Float64,1}:
-2.1400549446402604
1.0000000000000002
5.140054944640259
julia> eigvecs(A)
3×3 Array{Float64,2}:
0.418304 -0.83205 0.364299
-0.656749 -7.39009e-16 0.754109
0.627457 0.5547 0.546448
julia> eigvecs(A, [1.])
3×1 Array{Float64,2}:
0.8320502943378438
4.263514128092366e-17
-0.5547001962252291
```

`eigvecs(A; permute::Bool=true, scale::Bool=true) -> Matrix`

Return a matrix `M`

whose columns are the eigenvectors of `A`

. (The `k`

th eigenvector can be obtained from the slice `M[:, k]`

.) The `permute`

and `scale`

keywords are the same as for `eigen`

.

**Examples**

```
julia> eigvecs([1.0 0.0 0.0; 0.0 3.0 0.0; 0.0 0.0 18.0])
3×3 Array{Float64,2}:
1.0 0.0 0.0
0.0 1.0 0.0
0.0 0.0 1.0
```

`eigvecs(A, B) -> Matrix`

Return a matrix `M`

whose columns are the generalized eigenvectors of `A`

and `B`

. (The `k`

th eigenvector can be obtained from the slice `M[:, k]`

.)

**Examples**

```
julia> A = [1 0; 0 -1]
2×2 Array{Int64,2}:
1 0
0 -1
julia> B = [0 1; 1 0]
2×2 Array{Int64,2}:
0 1
1 0
julia> eigvecs(A, B)
2×2 Array{Complex{Float64},2}:
0.0-1.0im 0.0+1.0im
-1.0-0.0im -1.0+0.0im
```

`LinearAlgebra.eigen`

— Function.`eigen(A; permute::Bool=true, scale::Bool=true) -> Eigen`

Computes the eigenvalue decomposition of `A`

, returning an `Eigen`

factorization object `F`

which contains the eigenvalues in `F.values`

and the eigenvectors in the columns of the matrix `F.vectors`

. (The `k`

th eigenvector can be obtained from the slice `F.vectors[:, k]`

.)

Iterating the decomposition produces the components `F.values`

and `F.vectors`

.

The following functions are available for `Eigen`

objects: `inv`

, `det`

, and `isposdef`

.

For general nonsymmetric matrices it is possible to specify how the matrix is balanced before the eigenvector calculation. The option `permute=true`

permutes the matrix to become closer to upper triangular, and `scale=true`

scales the matrix by its diagonal elements to make rows and columns more equal in norm. The default is `true`

for both options.

**Examples**

```
julia> F = eigen([1.0 0.0 0.0; 0.0 3.0 0.0; 0.0 0.0 18.0])
Eigen{Float64,Float64,Array{Float64,2},Array{Float64,1}}
eigenvalues:
3-element Array{Float64,1}:
1.0
3.0
18.0
eigenvectors:
3×3 Array{Float64,2}:
1.0 0.0 0.0
0.0 1.0 0.0
0.0 0.0 1.0
julia> F.values
3-element Array{Float64,1}:
1.0
3.0
18.0
julia> F.vectors
3×3 Array{Float64,2}:
1.0 0.0 0.0
0.0 1.0 0.0
0.0 0.0 1.0
julia> vals, vecs = F; # destructuring via iteration
julia> vals == F.values && vecs == F.vectors
true
```

`eigen(A, B) -> GeneralizedEigen`

Computes the generalized eigenvalue decomposition of `A`

and `B`

, returning a `GeneralizedEigen`

factorization object `F`

which contains the generalized eigenvalues in `F.values`

and the generalized eigenvectors in the columns of the matrix `F.vectors`

. (The `k`

th generalized eigenvector can be obtained from the slice `F.vectors[:, k]`

.)

Iterating the decomposition produces the components `F.values`

and `F.vectors`

.

**Examples**

```
julia> A = [1 0; 0 -1]
2×2 Array{Int64,2}:
1 0
0 -1
julia> B = [0 1; 1 0]
2×2 Array{Int64,2}:
0 1
1 0
julia> F = eigen(A, B);
julia> F.values
2-element Array{Complex{Float64},1}:
0.0 + 1.0im
0.0 - 1.0im
julia> F.vectors
2×2 Array{Complex{Float64},2}:
0.0-1.0im 0.0+1.0im
-1.0-0.0im -1.0+0.0im
julia> vals, vecs = F; # destructuring via iteration
julia> vals == F.values && vecs == F.vectors
true
```

`eigen(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> Eigen`

Computes the eigenvalue decomposition of `A`

, returning an `Eigen`

factorization object `F`

which contains the eigenvalues in `F.values`

and the eigenvectors in the columns of the matrix `F.vectors`

. (The `k`

th eigenvector can be obtained from the slice `F.vectors[:, k]`

.)

Iterating the decomposition produces the components `F.values`

and `F.vectors`

.

The following functions are available for `Eigen`

objects: `inv`

, `det`

, and `isposdef`

.

The `UnitRange`

`irange`

specifies indices of the sorted eigenvalues to search for.

If `irange`

is not `1:n`

, where `n`

is the dimension of `A`

, then the returned factorization will be a *truncated* factorization.

`eigen(A::Union{SymTridiagonal, Hermitian, Symmetric}, vl::Real, vu::Real) -> Eigen`

Computes the eigenvalue decomposition of `A`

, returning an `Eigen`

factorization object `F`

which contains the eigenvalues in `F.values`

and the eigenvectors in the columns of the matrix `F.vectors`

. (The `k`

th eigenvector can be obtained from the slice `F.vectors[:, k]`

.)

Iterating the decomposition produces the components `F.values`

and `F.vectors`

.

The following functions are available for `Eigen`

objects: `inv`

, `det`

, and `isposdef`

.

`vl`

is the lower bound of the window of eigenvalues to search for, and `vu`

is the upper bound.

If [`vl`

, `vu`

] does not contain all eigenvalues of `A`

, then the returned factorization will be a *truncated* factorization.

`LinearAlgebra.eigen!`

— Function.`eigen!(A, [B])`

Same as `eigen`

, but saves space by overwriting the input `A`

(and `B`

), instead of creating a copy.

`LinearAlgebra.hessenberg`

— Function.`hessenberg(A) -> Hessenberg`

Compute the Hessenberg decomposition of `A`

and return a `Hessenberg`

object. If `F`

is the factorization object, the unitary matrix can be accessed with `F.Q`

and the Hessenberg matrix with `F.H`

. When `Q`

is extracted, the resulting type is the `HessenbergQ`

object, and may be converted to a regular matrix with `convert(Array, _)`

(or `Array(_)`

for short).

Iterating the decomposition produces the factors `F.Q`

and `F.H`

.

**Examples**

```
julia> A = [4. 9. 7.; 4. 4. 1.; 4. 3. 2.]
3×3 Array{Float64,2}:
4.0 9.0 7.0
4.0 4.0 1.0
4.0 3.0 2.0
julia> F = hessenberg(A);
julia> F.Q * F.H * F.Q'
3×3 Array{Float64,2}:
4.0 9.0 7.0
4.0 4.0 1.0
4.0 3.0 2.0
julia> q, h = F; # destructuring via iteration
julia> q == F.Q && h == F.H
true
```

`LinearAlgebra.hessenberg!`

— Function.`hessenberg!(A) -> Hessenberg`

`hessenberg!`

is the same as `hessenberg`

, but saves space by overwriting the input `A`

, instead of creating a copy.

`LinearAlgebra.schur!`

— Function.`schur!(A::StridedMatrix) -> F::Schur`

Same as `schur`

but uses the input argument `A`

as workspace.

**Examples**

```
julia> A = [5. 7.; -2. -4.]
2×2 Array{Float64,2}:
5.0 7.0
-2.0 -4.0
julia> F = schur!(A)
Schur{Float64,Array{Float64,2}}
T factor:
2×2 Array{Float64,2}:
3.0 9.0
0.0 -2.0
Z factor:
2×2 Array{Float64,2}:
0.961524 0.274721
-0.274721 0.961524
eigenvalues:
2-element Array{Float64,1}:
3.0
-2.0
julia> A
2×2 Array{Float64,2}:
3.0 9.0
0.0 -2.0
```

`schur!(A::StridedMatrix, B::StridedMatrix) -> F::GeneralizedSchur`

Same as `schur`

but uses the input matrices `A`

and `B`

as workspace.

`LinearAlgebra.schur`

— Function.`schur(A::StridedMatrix) -> F::Schur`

Computes the Schur factorization of the matrix `A`

. The (quasi) triangular Schur factor can be obtained from the `Schur`

object `F`

with either `F.Schur`

or `F.T`

and the orthogonal/unitary Schur vectors can be obtained with `F.vectors`

or `F.Z`

such that `A = F.vectors * F.Schur * F.vectors'`

. The eigenvalues of `A`

can be obtained with `F.values`

.

Iterating the decomposition produces the components `F.T`

, `F.Z`

, and `F.values`

.

**Examples**

```
julia> A = [5. 7.; -2. -4.]
2×2 Array{Float64,2}:
5.0 7.0
-2.0 -4.0
julia> F = schur(A)
Schur{Float64,Array{Float64,2}}
T factor:
2×2 Array{Float64,2}:
3.0 9.0
0.0 -2.0
Z factor:
2×2 Array{Float64,2}:
0.961524 0.274721
-0.274721 0.961524
eigenvalues:
2-element Array{Float64,1}:
3.0
-2.0
julia> F.vectors * F.Schur * F.vectors'
2×2 Array{Float64,2}:
5.0 7.0
-2.0 -4.0
julia> t, z, vals = F; # destructuring via iteration
julia> t == F.T && z == F.Z && vals == F.values
true
```

`schur(A::StridedMatrix, B::StridedMatrix) -> F::GeneralizedSchur`

Computes the Generalized Schur (or QZ) factorization of the matrices `A`

and `B`

. The (quasi) triangular Schur factors can be obtained from the `Schur`

object `F`

with `F.S`

and `F.T`

, the left unitary/orthogonal Schur vectors can be obtained with `F.left`

or `F.Q`

and the right unitary/orthogonal Schur vectors can be obtained with `F.right`

or `F.Z`

such that `A=F.left*F.S*F.right'`

and `B=F.left*F.T*F.right'`

. The generalized eigenvalues of `A`

and `B`

can be obtained with `F.α./F.β`

.

Iterating the decomposition produces the components `F.S`

, `F.T`

, `F.Q`

, `F.Z`

, `F.α`

, and `F.β`

.

`LinearAlgebra.ordschur`

— Function.`ordschur(F::Schur, select::Union{Vector{Bool},BitVector}) -> F::Schur`

Reorders the Schur factorization `F`

of a matrix `A = Z*T*Z'`

according to the logical array `select`

returning the reordered factorization `F`

object. The selected eigenvalues appear in the leading diagonal of `F.Schur`

and the corresponding leading columns of `F.vectors`

form an orthogonal/unitary basis of the corresponding right invariant subspace. In the real case, a complex conjugate pair of eigenvalues must be either both included or both excluded via `select`

.

`ordschur(F::GeneralizedSchur, select::Union{Vector{Bool},BitVector}) -> F::GeneralizedSchur`

Reorders the Generalized Schur factorization `F`

of a matrix pair `(A, B) = (Q*S*Z', Q*T*Z')`

according to the logical array `select`

and returns a GeneralizedSchur object `F`

. The selected eigenvalues appear in the leading diagonal of both `F.S`

and `F.T`

, and the left and right orthogonal/unitary Schur vectors are also reordered such that `(A, B) = F.Q*(F.S, F.T)*F.Z'`

still holds and the generalized eigenvalues of `A`

and `B`

can still be obtained with `F.α./F.β`

.

`LinearAlgebra.ordschur!`

— Function.`ordschur!(F::Schur, select::Union{Vector{Bool},BitVector}) -> F::Schur`

Same as `ordschur`

but overwrites the factorization `F`

.

`ordschur!(F::GeneralizedSchur, select::Union{Vector{Bool},BitVector}) -> F::GeneralizedSchur`

Same as `ordschur`

but overwrites the factorization `F`

.

`LinearAlgebra.svd`

— Function.`svd(A; full::Bool = false) -> SVD`

Compute the singular value decomposition (SVD) of `A`

and return an `SVD`

object.

`U`

, `S`

, `V`

and `Vt`

can be obtained from the factorization `F`

with `F.U`

, `F.S`

, `F.V`

and `F.Vt`

, such that `A = U * Diagonal(S) * Vt`

. The algorithm produces `Vt`

and hence `Vt`

is more efficient to extract than `V`

. The singular values in `S`

are sorted in descending order.

Iterating the decomposition produces the components `U`

, `S`

, and `V`

.

If `full = false`

(default), a "thin" SVD is returned. For a $M \times N$ matrix `A`

, in the full factorization `U`

is `M \times M`

and `V`

is `N \times N`

, while in the thin factorization `U`

is `M \times K`

and `V`

is `N \times K`

, where `K = \min(M,N)`

is the number of singular values.

**Examples**

```
julia> A = [1. 0. 0. 0. 2.; 0. 0. 3. 0. 0.; 0. 0. 0. 0. 0.; 0. 2. 0. 0. 0.]
4×5 Array{Float64,2}:
1.0 0.0 0.0 0.0 2.0
0.0 0.0 3.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 2.0 0.0 0.0 0.0
julia> F = svd(A);
julia> F.U * Diagonal(F.S) * F.Vt
4×5 Array{Float64,2}:
1.0 0.0 0.0 0.0 2.0
0.0 0.0 3.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 2.0 0.0 0.0 0.0
```

`svd(A, B) -> GeneralizedSVD`

Compute the generalized SVD of `A`

and `B`

, returning a `GeneralizedSVD`

factorization object `F`

, such that `A = F.U*F.D1*F.R0*F.Q'`

and `B = F.V*F.D2*F.R0*F.Q'`

.

For an M-by-N matrix `A`

and P-by-N matrix `B`

,

`U`

is a M-by-M orthogonal matrix,`V`

is a P-by-P orthogonal matrix,`Q`

is a N-by-N orthogonal matrix,`D1`

is a M-by-(K+L) diagonal matrix with 1s in the first K entries,`D2`

is a P-by-(K+L) matrix whose top right L-by-L block is diagonal,`R0`

is a (K+L)-by-N matrix whose rightmost (K+L)-by-(K+L) block is nonsingular upper block triangular,

`K+L`

is the effective numerical rank of the matrix `[A; B]`

.

Iterating the decomposition produces the components `U`

, `V`

, `Q`

, `D1`

, `D2`

, and `R0`

.

The entries of `F.D1`

and `F.D2`

are related, as explained in the LAPACK documentation for the generalized SVD and the xGGSVD3 routine which is called underneath (in LAPACK 3.6.0 and newer).

**Examples**

```
julia> A = [1. 0.; 0. -1.]
2×2 Array{Float64,2}:
1.0 0.0
0.0 -1.0
julia> B = [0. 1.; 1. 0.]
2×2 Array{Float64,2}:
0.0 1.0
1.0 0.0
julia> F = svd(A, B);
julia> F.U*F.D1*F.R0*F.Q'
2×2 Array{Float64,2}:
1.0 0.0
0.0 -1.0
julia> F.V*F.D2*F.R0*F.Q'
2×2 Array{Float64,2}:
0.0 1.0
1.0 0.0
```

`LinearAlgebra.svd!`

— Function.`svd!(A; full::Bool = false) -> SVD`

`svd!`

is the same as `svd`

, but saves space by overwriting the input `A`

, instead of creating a copy.

**Examples**

```
julia> A = [1. 0. 0. 0. 2.; 0. 0. 3. 0. 0.; 0. 0. 0. 0. 0.; 0. 2. 0. 0. 0.]
4×5 Array{Float64,2}:
1.0 0.0 0.0 0.0 2.0
0.0 0.0 3.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 2.0 0.0 0.0 0.0
julia> F = svd!(A);
julia> F.U * Diagonal(F.S) * F.Vt
4×5 Array{Float64,2}:
1.0 0.0 0.0 0.0 2.0
0.0 0.0 3.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 2.0 0.0 0.0 0.0
julia> A
4×5 Array{Float64,2}:
-2.23607 0.0 0.0 0.0 0.618034
0.0 -3.0 1.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 -2.0 0.0 0.0
```

`svd!(A, B) -> GeneralizedSVD`

`svd!`

is the same as `svd`

, but modifies the arguments `A`

and `B`

in-place, instead of making copies.

**Examples**

```
julia> A = [1. 0.; 0. -1.]
2×2 Array{Float64,2}:
1.0 0.0
0.0 -1.0
julia> B = [0. 1.; 1. 0.]
2×2 Array{Float64,2}:
0.0 1.0
1.0 0.0
julia> F = svd!(A, B);
julia> F.U*F.D1*F.R0*F.Q'
2×2 Array{Float64,2}:
1.0 0.0
0.0 -1.0
julia> F.V*F.D2*F.R0*F.Q'
2×2 Array{Float64,2}:
0.0 1.0
1.0 0.0
julia> A
2×2 Array{Float64,2}:
1.41421 0.0
0.0 -1.41421
julia> B
2×2 Array{Float64,2}:
1.0 -0.0
0.0 -1.0
```

`LinearAlgebra.svdvals`

— Function.`svdvals(A)`

Return the singular values of `A`

in descending order.

**Examples**

```
julia> A = [1. 0. 0. 0. 2.; 0. 0. 3. 0. 0.; 0. 0. 0. 0. 0.; 0. 2. 0. 0. 0.]
4×5 Array{Float64,2}:
1.0 0.0 0.0 0.0 2.0
0.0 0.0 3.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 2.0 0.0 0.0 0.0
julia> svdvals(A)
4-element Array{Float64,1}:
3.0
2.23606797749979
2.0
0.0
```

`svdvals(A, B)`

Return the generalized singular values from the generalized singular value decomposition of `A`

and `B`

. See also `svd`

.

**Examples**

```
julia> A = [1. 0.; 0. -1.]
2×2 Array{Float64,2}:
1.0 0.0
0.0 -1.0
julia> B = [0. 1.; 1. 0.]
2×2 Array{Float64,2}:
0.0 1.0
1.0 0.0
julia> svdvals(A, B)
2-element Array{Float64,1}:
1.0
1.0
```

`LinearAlgebra.svdvals!`

— Function.`svdvals!(A)`

Return the singular values of `A`

, saving space by overwriting the input. See also `svdvals`

and `svd`

.

**Examples**

```
julia> A = [1. 0. 0. 0. 2.; 0. 0. 3. 0. 0.; 0. 0. 0. 0. 0.; 0. 2. 0. 0. 0.]
4×5 Array{Float64,2}:
1.0 0.0 0.0 0.0 2.0
0.0 0.0 3.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 2.0 0.0 0.0 0.0
julia> svdvals!(A)
4-element Array{Float64,1}:
3.0
2.23606797749979
2.0
0.0
julia> A
4×5 Array{Float64,2}:
-2.23607 0.0 0.0 0.0 0.618034
0.0 -3.0 1.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 -2.0 0.0 0.0
```

`svdvals!(A, B)`

Return the generalized singular values from the generalized singular value decomposition of `A`

and `B`

, saving space by overwriting `A`

and `B`

. See also `svd`

and `svdvals`

.

**Examples**

```
julia> A = [1. 0.; 0. -1.]
2×2 Array{Float64,2}:
1.0 0.0
0.0 -1.0
julia> B = [0. 1.; 1. 0.]
2×2 Array{Float64,2}:
0.0 1.0
1.0 0.0
julia> svdvals!(A, B)
2-element Array{Float64,1}:
1.0
1.0
julia> A
2×2 Array{Float64,2}:
1.41421 0.0
0.0 -1.41421
julia> B
2×2 Array{Float64,2}:
1.0 -0.0
0.0 -1.0
```

`LinearAlgebra.Givens`

— Type.`LinearAlgebra.Givens(i1,i2,c,s) -> G`

A Givens rotation linear operator. The fields `c`

and `s`

represent the cosine and sine of the rotation angle, respectively. The `Givens`

type supports left multiplication `G*A`

and conjugated transpose right multiplication `A*G'`

. The type doesn't have a `size`

and can therefore be multiplied with matrices of arbitrary size as long as `i2<=size(A,2)`

for `G*A`

or `i2<=size(A,1)`

for `A*G'`

.

See also: `givens`

`LinearAlgebra.givens`

— Function.`givens(f::T, g::T, i1::Integer, i2::Integer) where {T} -> (G::Givens, r::T)`

Computes the Givens rotation `G`

and scalar `r`

such that for any vector `x`

where

```
x[i1] = f
x[i2] = g
```

the result of the multiplication

`y = G*x`

has the property that

```
y[i1] = r
y[i2] = 0
```

See also: `LinearAlgebra.Givens`

`givens(A::AbstractArray, i1::Integer, i2::Integer, j::Integer) -> (G::Givens, r)`

Computes the Givens rotation `G`

and scalar `r`

such that the result of the multiplication

`B = G*A`

has the property that

```
B[i1,j] = r
B[i2,j] = 0
```

See also: `LinearAlgebra.Givens`

`givens(x::AbstractVector, i1::Integer, i2::Integer) -> (G::Givens, r)`

Computes the Givens rotation `G`

and scalar `r`

such that the result of the multiplication

`B = G*x`

has the property that

```
B[i1] = r
B[i2] = 0
```

See also: `LinearAlgebra.Givens`

`LinearAlgebra.triu`

— Function.`triu(M)`

Upper triangle of a matrix.

**Examples**

```
julia> a = fill(1.0, (4,4))
4×4 Array{Float64,2}:
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
julia> triu(a)
4×4 Array{Float64,2}:
1.0 1.0 1.0 1.0
0.0 1.0 1.0 1.0
0.0 0.0 1.0 1.0
0.0 0.0 0.0 1.0
```

`triu(M, k::Integer)`

Returns the upper triangle of `M`

starting from the `k`

th superdiagonal.

**Examples**

```
julia> a = fill(1.0, (4,4))
4×4 Array{Float64,2}:
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
julia> triu(a,3)
4×4 Array{Float64,2}:
0.0 0.0 0.0 1.0
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
julia> triu(a,-3)
4×4 Array{Float64,2}:
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
```

`LinearAlgebra.triu!`

— Function.`triu!(M)`

Upper triangle of a matrix, overwriting `M`

in the process. See also `triu`

.

`triu!(M, k::Integer)`

Return the upper triangle of `M`

starting from the `k`

th superdiagonal, overwriting `M`

in the process.

**Examples**

```
julia> M = [1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5]
5×5 Array{Int64,2}:
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
julia> triu!(M, 1)
5×5 Array{Int64,2}:
0 2 3 4 5
0 0 3 4 5
0 0 0 4 5
0 0 0 0 5
0 0 0 0 0
```

`LinearAlgebra.tril`

— Function.`tril(M)`

Lower triangle of a matrix.

**Examples**

```
julia> a = fill(1.0, (4,4))
4×4 Array{Float64,2}:
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
julia> tril(a)
4×4 Array{Float64,2}:
1.0 0.0 0.0 0.0
1.0 1.0 0.0 0.0
1.0 1.0 1.0 0.0
1.0 1.0 1.0 1.0
```

`tril(M, k::Integer)`

Returns the lower triangle of `M`

starting from the `k`

th superdiagonal.

**Examples**

```
julia> a = fill(1.0, (4,4))
4×4 Array{Float64,2}:
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
julia> tril(a,3)
4×4 Array{Float64,2}:
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0
julia> tril(a,-3)
4×4 Array{Float64,2}:
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
1.0 0.0 0.0 0.0
```

`LinearAlgebra.tril!`

— Function.`tril!(M)`

Lower triangle of a matrix, overwriting `M`

in the process. See also `tril`

.

`tril!(M, k::Integer)`

Return the lower triangle of `M`

starting from the `k`

th superdiagonal, overwriting `M`

in the process.

**Examples**

```
julia> M = [1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5]
5×5 Array{Int64,2}:
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
julia> tril!(M, 2)
5×5 Array{Int64,2}:
1 2 3 0 0
1 2 3 4 0
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
```

`LinearAlgebra.diagind`

— Function.`diagind(M, k::Integer=0)`

An `AbstractRange`

giving the indices of the `k`

th diagonal of the matrix `M`

.

**Examples**

```
julia> A = [1 2 3; 4 5 6; 7 8 9]
3×3 Array{Int64,2}:
1 2 3
4 5 6
7 8 9
julia> diagind(A,-1)
2:4:6
```

`LinearAlgebra.diag`

— Function.`diag(M, k::Integer=0)`

The `k`

th diagonal of a matrix, as a vector.

See also: `diagm`

**Examples**

```
julia> A = [1 2 3; 4 5 6; 7 8 9]
3×3 Array{Int64,2}:
1 2 3
4 5 6
7 8 9
julia> diag(A,1)
2-element Array{Int64,1}:
2
6
```

`LinearAlgebra.diagm`

— Function.`diagm(kv::Pair{<:Integer,<:AbstractVector}...)`

Construct a square matrix from `Pair`

s of diagonals and vectors. Vector `kv.second`

will be placed on the `kv.first`

diagonal. `diagm`

constructs a full matrix; if you want storage-efficient versions with fast arithmetic, see `Diagonal`

, `Bidiagonal`

`Tridiagonal`

and `SymTridiagonal`

.

**Examples**

```
julia> diagm(1 => [1,2,3])
4×4 Array{Int64,2}:
0 1 0 0
0 0 2 0
0 0 0 3
0 0 0 0
julia> diagm(1 => [1,2,3], -1 => [4,5])
4×4 Array{Int64,2}:
0 1 0 0
4 0 2 0
0 5 0 3
0 0 0 0
```

`LinearAlgebra.rank`

— Function.`rank(A[, tol::Real])`

Compute the rank of a matrix by counting how many singular values of `A`

have magnitude greater than `tol*σ₁`

where `σ₁`

is `A`

's largest singular values. By default, the value of `tol`

is the smallest dimension of `A`

multiplied by the `eps`

of the `eltype`

of `A`

.

**Examples**

```
julia> rank(Matrix(I, 3, 3))
3
julia> rank(diagm(0 => [1, 0, 2]))
2
julia> rank(diagm(0 => [1, 0.001, 2]), 0.1)
2
julia> rank(diagm(0 => [1, 0.001, 2]), 0.00001)
3
```

`LinearAlgebra.norm`

— Function.`norm(A, p::Real=2)`

For any iterable container `A`

(including arrays of any dimension) of numbers (or any element type for which `norm`

is defined), compute the `p`

-norm (defaulting to `p=2`

) as if `A`

were a vector of the corresponding length.

The `p`

-norm is defined as

with $a_i$ the entries of $A$, $| a_i |$ the `norm`

of $a_i$, and $n$ the length of $A$. Since the `p`

-norm is computed using the `norm`

s of the entries of `A`

, the `p`

-norm of a vector of vectors is not compatible with the interpretation of it as a block vector in general if `p != 2`

.

`p`

can assume any numeric value (even though not all values produce a mathematically valid vector norm). In particular, `norm(A, Inf)`

returns the largest value in `abs.(A)`

, whereas `norm(A, -Inf)`

returns the smallest. If `A`

is a matrix and `p=2`

, then this is equivalent to the Frobenius norm.

The second argument `p`

is not necessarily a part of the interface for `norm`

, i.e. a custom type may only implement `norm(A)`

without second argument.

Use `opnorm`

to compute the operator norm of a matrix.

**Examples**

```
julia> v = [3, -2, 6]
3-element Array{Int64,1}:
3
-2
6
julia> norm(v)
7.0
julia> norm(v, 1)
11.0
julia> norm(v, Inf)
6.0
julia> norm([1 2 3; 4 5 6; 7 8 9])
16.881943016134134
julia> norm([1 2 3 4 5 6 7 8 9])
16.881943016134134
julia> norm(1:9)
16.881943016134134
julia> norm(hcat(v,v), 1) == norm(vcat(v,v), 1) != norm([v,v], 1)
true
julia> norm(hcat(v,v), 2) == norm(vcat(v,v), 2) == norm([v,v], 2)
true
julia> norm(hcat(v,v), Inf) == norm(vcat(v,v), Inf) != norm([v,v], Inf)
true
```

`norm(x::Number, p::Real=2)`

For numbers, return $\left( |x|^p \right)^{1/p}$.

**Examples**

```
julia> norm(2, 1)
2
julia> norm(-2, 1)
2
julia> norm(2, 2)
2
julia> norm(-2, 2)
2
julia> norm(2, Inf)
2
julia> norm(-2, Inf)
2
```

`LinearAlgebra.opnorm`

— Function.`opnorm(A::AbstractMatrix, p::Real=2)`

Compute the operator norm (or matrix norm) induced by the vector `p`

-norm, where valid values of `p`

are `1`

, `2`

, or `Inf`

. (Note that for sparse matrices, `p=2`

is currently not implemented.) Use `norm`

to compute the Frobenius norm.

When `p=1`

, the operator norm is the maximum absolute column sum of `A`

:

with $a_{ij}$ the entries of $A$, and $m$ and $n$ its dimensions.

When `p=2`

, the operator norm is the spectral norm, equal to the largest singular value of `A`

.

When `p=Inf`

, the operator norm is the maximum absolute row sum of `A`

:

**Examples**

```
julia> A = [1 -2 -3; 2 3 -1]
2×3 Array{Int64,2}:
1 -2 -3
2 3 -1
julia> opnorm(A, Inf)
6.0
julia> opnorm(A, 1)
5.0
```

`opnorm(x::Number, p::Real=2)`

For numbers, return $\left( |x|^p \right)^{1/p}$. This is equivalent to `norm`

.

```
opnorm(A::Adjoint{<:Any,<:AbstracVector}, q::Real=2)
opnorm(A::Transpose{<:Any,<:AbstracVector}, q::Real=2)
```

For Adjoint/Transpose-wrapped vectors, return the operator $q$-norm of `A`

, which is equivalent to the `p`

-norm with value `p = q/(q-1)`

. They coincide at `p = q = 2`

. Use `norm`

to compute the `p`

norm of `A`

as a vector.

The difference in norm between a vector space and its dual arises to preserve the relationship between duality and the dot product, and the result is consistent with the operator `p`

-norm of a `1 × n`

matrix.

**Examples**

```
julia> v = [1; im];
julia> vc = v';
julia> opnorm(vc, 1)
1.0
julia> norm(vc, 1)
2.0
julia> norm(v, 1)
2.0
julia> opnorm(vc, 2)
1.4142135623730951
julia> norm(vc, 2)
1.4142135623730951
julia> norm(v, 2)
1.4142135623730951
julia> opnorm(vc, Inf)
2.0
julia> norm(vc, Inf)
1.0
julia> norm(v, Inf)
1.0
```

`LinearAlgebra.normalize!`

— Function.`LinearAlgebra.normalize`

— Function.`normalize(v::AbstractVector, p::Real=2)`

Normalize the vector `v`

so that its `p`

-norm equals unity, i.e. `norm(v, p) == vecnorm(v, p) == 1`

. See also `normalize!`

and `norm`

.

**Examples**

```
julia> a = [1,2,4];
julia> b = normalize(a)
3-element Array{Float64,1}:
0.2182178902359924
0.4364357804719848
0.8728715609439696
julia> norm(b)
1.0
julia> c = normalize(a, 1)
3-element Array{Float64,1}:
0.14285714285714285
0.2857142857142857
0.5714285714285714
julia> norm(c, 1)
1.0
```

`LinearAlgebra.cond`

— Function.`cond(M, p::Real=2)`

Condition number of the matrix `M`

, computed using the operator `p`

-norm. Valid values for `p`

are `1`

, `2`

(default), or `Inf`

.

`LinearAlgebra.condskeel`

— Function.`condskeel(M, [x, p::Real=Inf])`

Skeel condition number $\kappa_S$ of the matrix `M`

, optionally with respect to the vector `x`

, as computed using the operator `p`

-norm. $\left\vert M \right\vert$ denotes the matrix of (entry wise) absolute values of $M$; $\left\vert M \right\vert_{ij} = \left\vert M_{ij} \right\vert$. Valid values for `p`

are `1`

, `2`

and `Inf`

(default).

This quantity is also known in the literature as the Bauer condition number, relative condition number, or componentwise relative condition number.

`LinearAlgebra.tr`

— Function.`tr(M)`

Matrix trace. Sums the diagonal elements of `M`

.

**Examples**

```
julia> A = [1 2; 3 4]
2×2 Array{Int64,2}:
1 2
3 4
julia> tr(A)
5
```

`LinearAlgebra.det`

— Function.`det(M)`

Matrix determinant.

**Examples**

```
julia> M = [1 0; 2 2]
2×2 Array{Int64,2}:
1 0
2 2
julia> det(M)
2.0
```

`LinearAlgebra.logdet`

— Function.`logdet(M)`

Log of matrix determinant. Equivalent to `log(det(M))`

, but may provide increased accuracy and/or speed.

**Examples**

```
julia> M = [1 0; 2 2]
2×2 Array{Int64,2}:
1 0
2 2
julia> logdet(M)
0.6931471805599453
julia> logdet(Matrix(I, 3, 3))
0.0
```

`LinearAlgebra.logabsdet`

— Function.`logabsdet(M)`

Log of absolute value of matrix determinant. Equivalent to `(log(abs(det(M))), sign(det(M)))`

, but may provide increased accuracy and/or speed.

**Examples**

```
julia> A = [-1. 0.; 0. 1.]
2×2 Array{Float64,2}:
-1.0 0.0
0.0 1.0
julia> det(A)
-1.0
julia> logabsdet(A)
(0.0, -1.0)
julia> B = [2. 0.; 0. 1.]
2×2 Array{Float64,2}:
2.0 0.0
0.0 1.0
julia> det(B)
2.0
julia> logabsdet(B)
(0.6931471805599453, 1.0)
```

`Base.inv`

— Method.`inv(M)`

Matrix inverse. Computes matrix `N`

such that `M * N = I`

, where `I`

is the identity matrix. Computed by solving the left-division `N = M \ I`

.

**Examples**

```
julia> M = [2 5; 1 3]
2×2 Array{Int64,2}:
2 5
1 3
julia> N = inv(M)
2×2 Array{Float64,2}:
3.0 -5.0
-1.0 2.0
julia> M*N == N*M == Matrix(I, 2, 2)
true
```

`LinearAlgebra.pinv`

— Function.`pinv(M[, rtol::Real])`

Computes the Moore-Penrose pseudoinverse.

For matrices `M`

with floating point elements, it is convenient to compute the pseudoinverse by inverting only singular values greater than `rtol * maximum(svdvals(M))`

.

The optimal choice of `rtol`

varies both with the value of `M`

and the intended application of the pseudoinverse. The default value of `rtol`

is `eps(real(float(one(eltype(M)))))*minimum(size(M))`

, which is essentially machine epsilon for the real part of a matrix element multiplied by the larger matrix dimension. For inverting dense ill-conditioned matrices in a least-squares sense, `rtol = sqrt(eps(real(float(one(eltype(M))))))`

is recommended.

For more information, see [issue8859], [B96], [S84], [KY88].

**Examples**

```
julia> M = [1.5 1.3; 1.2 1.9]
2×2 Array{Float64,2}:
1.5 1.3
1.2 1.9
julia> N = pinv(M)
2×2 Array{Float64,2}:
1.47287 -1.00775
-0.930233 1.16279
julia> M * N
2×2 Array{Float64,2}:
1.0 -2.22045e-16
4.44089e-16 1.0
```

**[issue8859]**

Issue 8859, "Fix least squares", https://github.com/JuliaLang/julia/pull/8859

**[B96]**

Åke Björck, "Numerical Methods for Least Squares Problems", SIAM Press, Philadelphia, 1996, "Other Titles in Applied Mathematics", Vol. 51. doi:10.1137/1.9781611971484

**[S84]**

G. W. Stewart, "Rank Degeneracy", SIAM Journal on Scientific and Statistical Computing, 5(2), 1984, 403-413. doi:10.1137/0905030

**[KY88]**

Konstantinos Konstantinides and Kung Yao, "Statistical analysis of effective singular values in matrix rank determination", IEEE Transactions on Acoustics, Speech and Signal Processing, 36(5), 1988, 757-763. doi:10.1109/29.1585

`LinearAlgebra.nullspace`

— Function.`nullspace(M[, rtol::Real])`

Computes a basis for the nullspace of `M`

by including the singular vectors of A whose singular have magnitude are greater than `rtol*σ₁`

, where `σ₁`

is `A`

's largest singular values. By default, the value of `rtol`

is the smallest dimension of `A`

multiplied by the `eps`

of the `eltype`

of `A`

.

**Examples**

```
julia> M = [1 0 0; 0 1 0; 0 0 0]
3×3 Array{Int64,2}:
1 0 0
0 1 0
0 0 0
julia> nullspace(M)
3×1 Array{Float64,2}:
0.0
0.0
1.0
julia> nullspace(M, 2)
3×3 Array{Float64,2}:
0.0 1.0 0.0
1.0 0.0 0.0
0.0 0.0 1.0
```

`Base.kron`

— Function.`kron(A, B)`

Kronecker tensor product of two vectors or two matrices.

**Examples**

```
julia> A = [1 2; 3 4]
2×2 Array{Int64,2}:
1 2
3 4
julia> B = [im 1; 1 -im]
2×2 Array{Complex{Int64},2}:
0+1im 1+0im
1+0im 0-1im
julia> kron(A, B)
4×4 Array{Complex{Int64},2}:
0+1im 1+0im 0+2im 2+0im
1+0im 0-1im 2+0im 0-2im
0+3im 3+0im 0+4im 4+0im
3+0im 0-3im 4+0im 0-4im
```

`Base.exp`

— Method.`exp(A::AbstractMatrix)`

Compute the matrix exponential of `A`

, defined by

For symmetric or Hermitian `A`

, an eigendecomposition (`eigen`

) is used, otherwise the scaling and squaring algorithm (see [H05]) is chosen.

**[H05]**

Nicholas J. Higham, "The squaring and scaling method for the matrix exponential revisited", SIAM Journal on Matrix Analysis and Applications, 26(4), 2005, 1179-1193. doi:10.1137/090768539

**Examples**

```
julia> A = Matrix(1.0I, 2, 2)
2×2 Array{Float64,2}:
1.0 0.0
0.0 1.0
julia> exp(A)
2×2 Array{Float64,2}:
2.71828 0.0
0.0 2.71828
```

`Base.log`

— Method.`log(A{T}::StridedMatrix{T})`

If `A`

has no negative real eigenvalue, compute the principal matrix logarithm of `A`

, i.e. the unique matrix $X$ such that $e^X = A$ and $-\pi < Im(\lambda) < \pi$ for all the eigenvalues $\lambda$ of $X$. If `A`

has nonpositive eigenvalues, a nonprincipal matrix function is returned whenever possible.

If `A`

is symmetric or Hermitian, its eigendecomposition (`eigen`

) is used, if `A`

is triangular an improved version of the inverse scaling and squaring method is employed (see [AH12] and [AHR13]). For general matrices, the complex Schur form (`schur`

) is computed and the triangular algorithm is used on the triangular factor.

**[AH12]**

Awad H. Al-Mohy and Nicholas J. Higham, "Improved inverse scaling and squaring algorithms for the matrix logarithm", SIAM Journal on Scientific Computing, 34(4), 2012, C153-C169. doi:10.1137/110852553

**[AHR13]**

Awad H. Al-Mohy, Nicholas J. Higham and Samuel D. Relton, "Computing the Fréchet derivative of the matrix logarithm and estimating the condition number", SIAM Journal on Scientific Computing, 35(4), 2013, C394-C410. doi:10.1137/120885991

**Examples**

```
julia> A = Matrix(2.7182818*I, 2, 2)
2×2 Array{Float64,2}:
2.71828 0.0
0.0 2.71828
julia> log(A)
2×2 Array{Float64,2}:
1.0 0.0
0.0 1.0
```

`Base.sqrt`

— Method.`sqrt(A::AbstractMatrix)`

If `A`

has no negative real eigenvalues, compute the principal matrix square root of `A`

, that is the unique matrix $X$ with eigenvalues having positive real part such that $X^2 = A$. Otherwise, a nonprincipal square root is returned.

If `A`

is symmetric or Hermitian, its eigendecomposition (`eigen`

) is used to compute the square root. Otherwise, the square root is determined by means of the Björck-Hammarling method [BH83], which computes the complex Schur form (`schur`

) and then the complex square root of the triangular factor.

**[BH83]**

Åke Björck and Sven Hammarling, "A Schur method for the square root of a matrix", Linear Algebra and its Applications, 52-53, 1983, 127-140. doi:10.1016/0024-3795(83)80010-X

**Examples**

```
julia> A = [4 0; 0 4]
2×2 Array{Int64,2}:
4 0
0 4
julia> sqrt(A)
2×2 Array{Float64,2}:
2.0 0.0
0.0 2.0
```

`Base.cos`

— Method.`cos(A::AbstractMatrix)`

Compute the matrix cosine of a square matrix `A`

.

If `A`

is symmetric or Hermitian, its eigendecomposition (`eigen`

) is used to compute the cosine. Otherwise, the cosine is determined by calling `exp`

.

**Examples**

```
julia> cos(fill(1.0, (2,2)))
2×2 Array{Float64,2}:
0.291927 -0.708073
-0.708073 0.291927
```

`Base.sin`

— Method.`sin(A::AbstractMatrix)`

Compute the matrix sine of a square matrix `A`

.

If `A`

is symmetric or Hermitian, its eigendecomposition (`eigen`

) is used to compute the sine. Otherwise, the sine is determined by calling `exp`

.

**Examples**

```
julia> sin(fill(1.0, (2,2)))
2×2 Array{Float64,2}:
0.454649 0.454649
0.454649 0.454649
```

`Base.Math.sincos`

— Method.`sincos(A::AbstractMatrix)`

Compute the matrix sine and cosine of a square matrix `A`

.

**Examples**

```
julia> S, C = sincos(fill(1.0, (2,2)));
julia> S
2×2 Array{Float64,2}:
0.454649 0.454649
0.454649 0.454649
julia> C
2×2 Array{Float64,2}:
0.291927 -0.708073
-0.708073 0.291927
```

`Base.tan`

— Method.`tan(A::AbstractMatrix)`

Compute the matrix tangent of a square matrix `A`

.

If `A`

is symmetric or Hermitian, its eigendecomposition (`eigen`

) is used to compute the tangent. Otherwise, the tangent is determined by calling `exp`

.

**Examples**

```
julia> tan(fill(1.0, (2,2)))
2×2 Array{Float64,2}:
-1.09252 -1.09252
-1.09252 -1.09252
```

`Base.Math.sec`

— Method.`sec(A::AbstractMatrix)`

Compute the matrix secant of a square matrix `A`

.

`Base.Math.csc`

— Method.`csc(A::AbstractMatrix)`

Compute the matrix cosecant of a square matrix `A`

.

`Base.Math.cot`

— Method.`cot(A::AbstractMatrix)`

Compute the matrix cotangent of a square matrix `A`

.

`Base.cosh`

— Method.`cosh(A::AbstractMatrix)`

Compute the matrix hyperbolic cosine of a square matrix `A`

.

`Base.sinh`

— Method.`sinh(A::AbstractMatrix)`

Compute the matrix hyperbolic sine of a square matrix `A`

.

`Base.tanh`

— Method.`tanh(A::AbstractMatrix)`

Compute the matrix hyperbolic tangent of a square matrix `A`

.

`Base.Math.sech`

— Method.`sech(A::AbstractMatrix)`

Compute the matrix hyperbolic secant of square matrix `A`

.

`Base.Math.csch`

— Method.`csch(A::AbstractMatrix)`

Compute the matrix hyperbolic cosecant of square matrix `A`

.

`Base.Math.coth`

— Method.`coth(A::AbstractMatrix)`

Compute the matrix hyperbolic cotangent of square matrix `A`

.

`Base.acos`

— Method.`acos(A::AbstractMatrix)`

Compute the inverse matrix cosine of a square matrix `A`

.

If `A`

is symmetric or Hermitian, its eigendecomposition (`eigen`

) is used to compute the inverse cosine. Otherwise, the inverse cosine is determined by using `log`

and `sqrt`

. For the theory and logarithmic formulas used to compute this function, see [AH16_1].

**[AH16_1]**

Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577

**Examples**

```
julia> acos(cos([0.5 0.1; -0.2 0.3]))
2×2 Array{Complex{Float64},2}:
0.5-5.55112e-17im 0.1-2.77556e-17im
-0.2+2.498e-16im 0.3-3.46945e-16im
```

`Base.asin`

— Method.`asin(A::AbstractMatrix)`

Compute the inverse matrix sine of a square matrix `A`

.

If `A`

is symmetric or Hermitian, its eigendecomposition (`eigen`

) is used to compute the inverse sine. Otherwise, the inverse sine is determined by using `log`

and `sqrt`

. For the theory and logarithmic formulas used to compute this function, see [AH16_2].

**[AH16_2]**

Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577

**Examples**

```
julia> asin(sin([0.5 0.1; -0.2 0.3]))
2×2 Array{Complex{Float64},2}:
0.5-4.16334e-17im 0.1-5.55112e-17im
-0.2+9.71445e-17im 0.3-1.249e-16im
```

`Base.atan`

— Method.`atan(A::AbstractMatrix)`

Compute the inverse matrix tangent of a square matrix `A`

.

If `A`

is symmetric or Hermitian, its eigendecomposition (`eigen`

) is used to compute the inverse tangent. Otherwise, the inverse tangent is determined by using `log`

. For the theory and logarithmic formulas used to compute this function, see [AH16_3].

**[AH16_3]**

Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577

**Examples**

```
julia> atan(tan([0.5 0.1; -0.2 0.3]))
2×2 Array{Complex{Float64},2}:
0.5+1.38778e-17im 0.1-2.77556e-17im
-0.2+6.93889e-17im 0.3-4.16334e-17im
```

`Base.Math.asec`

— Method.`asec(A::AbstractMatrix)`

Compute the inverse matrix secant of `A`

.

`Base.Math.acsc`

— Method.`acsc(A::AbstractMatrix)`

Compute the inverse matrix cosecant of `A`

.

`Base.Math.acot`

— Method.`acot(A::AbstractMatrix)`

Compute the inverse matrix cotangent of `A`

.

`Base.acosh`

— Method.`acosh(A::AbstractMatrix)`

Compute the inverse hyperbolic matrix cosine of a square matrix `A`

. For the theory and logarithmic formulas used to compute this function, see [AH16_4].

**[AH16_4]**

`Base.asinh`

— Method.`asinh(A::AbstractMatrix)`

Compute the inverse hyperbolic matrix sine of a square matrix `A`

. For the theory and logarithmic formulas used to compute this function, see [AH16_5].

**[AH16_5]**

`Base.atanh`

— Method.`atanh(A::AbstractMatrix)`

Compute the inverse hyperbolic matrix tangent of a square matrix `A`

. For the theory and logarithmic formulas used to compute this function, see [AH16_6].

**[AH16_6]**

`Base.Math.asech`

— Method.`asech(A::AbstractMatrix)`

Compute the inverse matrix hyperbolic secant of `A`

.

`Base.Math.acsch`

— Method.`acsch(A::AbstractMatrix)`

Compute the inverse matrix hyperbolic cosecant of `A`

.

`Base.Math.acoth`

— Method.`acoth(A::AbstractMatrix)`

Compute the inverse matrix hyperbolic cotangent of `A`

.

`LinearAlgebra.lyap`

— Function.`lyap(A, C)`

Computes the solution `X`

to the continuous Lyapunov equation `AX + XA' + C = 0`

, where no eigenvalue of `A`

has a zero real part and no two eigenvalues are negative complex conjugates of each other.

**Examples**

```
julia> A = [3. 4.; 5. 6]
2×2 Array{Float64,2}:
3.0 4.0
5.0 6.0
julia> B = [1. 1.; 1. 2.]
2×2 Array{Float64,2}:
1.0 1.0
1.0 2.0
julia> X = lyap(A, B)
2×2 Array{Float64,2}:
0.5 -0.5
-0.5 0.25
julia> A*X + X*A' + B
2×2 Array{Float64,2}:
0.0 6.66134e-16
6.66134e-16 8.88178e-16
```

`LinearAlgebra.sylvester`

— Function.`sylvester(A, B, C)`

Computes the solution `X`

to the Sylvester equation `AX + XB + C = 0`

, where `A`

, `B`

and `C`

have compatible dimensions and `A`

and `-B`

have no eigenvalues with equal real part.

**Examples**

```
julia> A = [3. 4.; 5. 6]
2×2 Array{Float64,2}:
3.0 4.0
5.0 6.0
julia> B = [1. 1.; 1. 2.]
2×2 Array{Float64,2}:
1.0 1.0
1.0 2.0
julia> C = [1. 2.; -2. 1]
2×2 Array{Float64,2}:
1.0 2.0
-2.0 1.0
julia> X = sylvester(A, B, C)
2×2 Array{Float64,2}:
-4.46667 1.93333
3.73333 -1.8
julia> A*X + X*B + C
2×2 Array{Float64,2}:
2.66454e-15 1.77636e-15
-3.77476e-15 4.44089e-16
```

`LinearAlgebra.issuccess`

— Function.`issuccess(F::Factorization)`

Test that a factorization of a matrix succeeded.

```
julia> F = cholesky([1 0; 0 1]);
julia> LinearAlgebra.issuccess(F)
true
julia> F = lu([1 0; 0 0]; check = false);
julia> LinearAlgebra.issuccess(F)
false
```

`LinearAlgebra.issymmetric`

— Function.`issymmetric(A) -> Bool`

Test whether a matrix is symmetric.

**Examples**

```
julia> a = [1 2; 2 -1]
2×2 Array{Int64,2}:
1 2
2 -1
julia> issymmetric(a)
true
julia> b = [1 im; -im 1]
2×2 Array{Complex{Int64},2}:
1+0im 0+1im
0-1im 1+0im
julia> issymmetric(b)
false
```

`LinearAlgebra.isposdef`

— Function.`isposdef(A) -> Bool`

Test whether a matrix is positive definite (and Hermitian) by trying to perform a Cholesky factorization of `A`

. See also `isposdef!`

**Examples**

```
julia> A = [1 2; 2 50]
2×2 Array{Int64,2}:
1 2
2 50
julia> isposdef(A)
true
```

`LinearAlgebra.isposdef!`

— Function.`isposdef!(A) -> Bool`

Test whether a matrix is positive definite (and Hermitian) by trying to perform a Cholesky factorization of `A`

, overwriting `A`

in the process. See also `isposdef`

.

**Examples**

```
julia> A = [1. 2.; 2. 50.];
julia> isposdef!(A)
true
julia> A
2×2 Array{Float64,2}:
1.0 2.0
2.0 6.78233
```

`LinearAlgebra.istril`

— Function.`istril(A::AbstractMatrix, k::Integer = 0) -> Bool`

Test whether `A`

is lower triangular starting from the `k`

th superdiagonal.

**Examples**

```
julia> a = [1 2; 2 -1]
2×2 Array{Int64,2}:
1 2
2 -1
julia> istril(a)
false
julia> istril(a, 1)
true
julia> b = [1 0; -im -1]
2×2 Array{Complex{Int64},2}:
1+0im 0+0im
0-1im -1+0im
julia> istril(b)
true
julia> istril(b, -1)
false
```

`LinearAlgebra.istriu`

— Function.`istriu(A::AbstractMatrix, k::Integer = 0) -> Bool`

Test whether `A`

is upper triangular starting from the `k`

th superdiagonal.

**Examples**

```
julia> a = [1 2; 2 -1]
2×2 Array{Int64,2}:
1 2
2 -1
julia> istriu(a)
false
julia> istriu(a, -1)
true
julia> b = [1 im; 0 -1]
2×2 Array{Complex{Int64},2}:
1+0im 0+1im
0+0im -1+0im
julia> istriu(b)
true
julia> istriu(b, 1)
false
```

`LinearAlgebra.isdiag`

— Function.`isdiag(A) -> Bool`

Test whether a matrix is diagonal.

**Examples**

```
julia> a = [1 2; 2 -1]
2×2 Array{Int64,2}:
1 2
2 -1
julia> isdiag(a)
false
julia> b = [im 0; 0 -im]
2×2 Array{Complex{Int64},2}:
0+1im 0+0im
0+0im 0-1im
julia> isdiag(b)
true
```

`LinearAlgebra.ishermitian`

— Function.`ishermitian(A) -> Bool`

Test whether a matrix is Hermitian.

**Examples**

```
julia> a = [1 2; 2 -1]
2×2 Array{Int64,2}:
1 2
2 -1
julia> ishermitian(a)
true
julia> b = [1 im; -im 1]
2×2 Array{Complex{Int64},2}:
1+0im 0+1im
0-1im 1+0im
julia> ishermitian(b)
true
```

`Base.transpose`

— Function.`transpose(A)`

Lazy transpose. Mutating the returned object should appropriately mutate `A`

. Often, but not always, yields `Transpose(A)`

, where `Transpose`

is a lazy transpose wrapper. Note that this operation is recursive.

This operation is intended for linear algebra usage - for general data manipulation see `permutedims`

, which is non-recursive.

**Examples**

```
julia> A = [3+2im 9+2im; 8+7im 4+6im]
2×2 Array{Complex{Int64},2}:
3+2im 9+2im
8+7im 4+6im
julia> transpose(A)
2×2 Transpose{Complex{Int64},Array{Complex{Int64},2}}:
3+2im 8+7im
9+2im 4+6im
```

`LinearAlgebra.transpose!`

— Function.`transpose!(dest,src)`

Transpose array `src`

and store the result in the preallocated array `dest`

, which should have a size corresponding to `(size(src,2),size(src,1))`

. No in-place transposition is supported and unexpected results will happen if `src`

and `dest`

have overlapping memory regions.

**Examples**

```
julia> A = [3+2im 9+2im; 8+7im 4+6im]
2×2 Array{Complex{Int64},2}:
3+2im 9+2im
8+7im 4+6im
julia> B = zeros(Complex{Int64}, 2, 2)
2×2 Array{Complex{Int64},2}:
0+0im 0+0im
0+0im 0+0im
julia> transpose!(B, A);
julia> B
2×2 Array{Complex{Int64},2}:
3+2im 8+7im
9+2im 4+6im
julia> A
2×2 Array{Complex{Int64},2}:
3+2im 9+2im
8+7im 4+6im
```

`Base.adjoint`

— Function.`adjoint(A)`

Lazy adjoint (conjugate transposition) (also postfix `'`

). Note that `adjoint`

is applied recursively to elements.

This operation is intended for linear algebra usage - for general data manipulation see `permutedims`

.

**Examples**

```
julia> A = [3+2im 9+2im; 8+7im 4+6im]
2×2 Array{Complex{Int64},2}:
3+2im 9+2im
8+7im 4+6im
julia> adjoint(A)
2×2 Adjoint{Complex{Int64},Array{Complex{Int64},2}}:
3-2im 8-7im
9-2im 4-6im
```

`LinearAlgebra.adjoint!`

— Function.`adjoint!(dest,src)`

Conjugate transpose array `src`

and store the result in the preallocated array `dest`

, which should have a size corresponding to `(size(src,2),size(src,1))`

. No in-place transposition is supported and unexpected results will happen if `src`

and `dest`

have overlapping memory regions.

**Examples**

```
julia> A = [3+2im 9+2im; 8+7im 4+6im]
2×2 Array{Complex{Int64},2}:
3+2im 9+2im
8+7im 4+6im
julia> B = zeros(Complex{Int64}, 2, 2)
2×2 Array{Complex{Int64},2}:
0+0im 0+0im
0+0im 0+0im
julia> adjoint!(B, A);
julia> B
2×2 Array{Complex{Int64},2}:
3-2im 8-7im
9-2im 4-6im
julia> A
2×2 Array{Complex{Int64},2}:
3+2im 9+2im
8+7im 4+6im
```

`Base.copy`

— Method.```
copy(A::Transpose)
copy(A::Adjoint)
```

Eagerly evaluate the lazy matrix transpose/adjoint. Note that the transposition is applied recursively to elements.

This operation is intended for linear algebra usage - for general data manipulation see `permutedims`

, which is non-recursive.

**Examples**

```
julia> A = [1 2im; -3im 4]
2×2 Array{Complex{Int64},2}:
1+0im 0+2im
0-3im 4+0im
julia> T = transpose(A)
2×2 Transpose{Complex{Int64},Array{Complex{Int64},2}}:
1+0im 0-3im
0+2im 4+0im
julia> copy(T)
2×2 Array{Complex{Int64},2}:
1+0im 0-3im
0+2im 4+0im
```

`LinearAlgebra.stride1`

— Function.`stride1(A) -> Int`

Return the distance between successive array elements in dimension 1 in units of element size.

**Examples**

```
julia> A = [1,2,3,4]
4-element Array{Int64,1}:
1
2
3
4
julia> LinearAlgebra.stride1(A)
1
julia> B = view(A, 2:2:4)
2-element view(::Array{Int64,1}, 2:2:4) with eltype Int64:
2
4
julia> LinearAlgebra.stride1(B)
2
```

`LinearAlgebra.checksquare`

— Function.`LinearAlgebra.checksquare(A)`

Check that a matrix is square, then return its common dimension. For multiple arguments, return a vector.

**Examples**

```
julia> A = fill(1, (4,4)); B = fill(1, (5,5));
julia> LinearAlgebra.checksquare(A, B)
2-element Array{Int64,1}:
4
5
```

## 底层矩阵运算

在许多情况下，矩阵运算存在 in-place 版本，这允许你使用预先分配的输出向量或矩阵。在优化关键代码是这很实用，可以避免重复分配的开销。根据 Julia 的通常惯例，这些 in-place 运算后面带有 `!`

（例如，`mul!`

）。

`LinearAlgebra.mul!`

— Function.`mul!(Y, A, B) -> Y`

Calculates the matrix-matrix or matrix-vector product $AB$ and stores the result in `Y`

, overwriting the existing value of `Y`

. Note that `Y`

must not be aliased with either `A`

or `B`

.

**Examples**

```
julia> A=[1.0 2.0; 3.0 4.0]; B=[1.0 1.0; 1.0 1.0]; Y = similar(B); mul!(Y, A, B);
julia> Y
2×2 Array{Float64,2}:
3.0 3.0
7.0 7.0
```

`LinearAlgebra.lmul!`

— Function.`lmul!(a::Number, B::AbstractArray)`

Scale an array `B`

by a scalar `a`

overwriting `B`

in-place.

**Examples**

```
julia> B = [1 2; 3 4]
2×2 Array{Int64,2}:
1 2
3 4
julia> lmul!(2, B)
2×2 Array{Int64,2}:
2 4
6 8
```

`lmul!(A, B)`

Calculate the matrix-matrix product $AB$, overwriting `B`

, and return the result.

**Examples**

```
julia> B = [0 1; 1 0];
julia> A = LinearAlgebra.UpperTriangular([1 2; 0 3]);
julia> LinearAlgebra.lmul!(A, B);
julia> B
2×2 Array{Int64,2}:
2 1
3 0
```

`LinearAlgebra.rmul!`

— Function.`rmul!(A::AbstractArray, b::Number)`

Scale an array `A`

by a scalar `b`

overwriting `A`

in-place.

**Examples**

```
julia> A = [1 2; 3 4]
2×2 Array{Int64,2}:
1 2
3 4
julia> rmul!(A, 2)
2×2 Array{Int64,2}:
2 4
6 8
```

`rmul!(A, B)`

Calculate the matrix-matrix product $AB$, overwriting `A`

, and return the result.

**Examples**

```
julia> A = [0 1; 1 0];
julia> B = LinearAlgebra.UpperTriangular([1 2; 0 3]);
julia> LinearAlgebra.rmul!(A, B);
julia> A
2×2 Array{Int64,2}:
0 3
1 2
```

`LinearAlgebra.ldiv!`

— Function.`ldiv!(Y, A, B) -> Y`

Compute `A \ B`

in-place and store the result in `Y`

, returning the result.

The argument `A`

should *not* be a matrix. Rather, instead of matrices it should be a factorization object (e.g. produced by `factorize`

or `cholesky`

). The reason for this is that factorization itself is both expensive and typically allocates memory (although it can also be done in-place via, e.g., `lu!`

), and performance-critical situations requiring `ldiv!`

usually also require fine-grained control over the factorization of `A`

.

**Examples**

```
julia> A = [1 2.2 4; 3.1 0.2 3; 4 1 2];
julia> X = [1; 2.5; 3];
julia> Y = zero(X);
julia> ldiv!(Y, qr(A), X);
julia> Y
3-element Array{Float64,1}:
0.7128099173553719
-0.051652892561983674
0.10020661157024757
julia> A\X
3-element Array{Float64,1}:
0.7128099173553719
-0.05165289256198333
0.10020661157024785
```

`ldiv!(A, B)`

Compute `A \ B`

in-place and overwriting `B`

to store the result.

The argument `A`

should *not* be a matrix. Rather, instead of matrices it should be a factorization object (e.g. produced by `factorize`

or `cholesky`

). The reason for this is that factorization itself is both expensive and typically allocates memory (although it can also be done in-place via, e.g., `lu!`

), and performance-critical situations requiring `ldiv!`

usually also require fine-grained control over the factorization of `A`

.

**Examples**

```
julia> A = [1 2.2 4; 3.1 0.2 3; 4 1 2];
julia> X = [1; 2.5; 3];
julia> Y = copy(X);
julia> ldiv!(qr(A), X);
julia> X
3-element Array{Float64,1}:
0.7128099173553719
-0.051652892561983674
0.10020661157024757
julia> A\Y
3-element Array{Float64,1}:
0.7128099173553719
-0.05165289256198333
0.10020661157024785
```

`LinearAlgebra.rdiv!`

— Function.`rdiv!(A, B)`

Compute `A / B`

in-place and overwriting `A`

to store the result.

The argument `B`

should *not* be a matrix. Rather, instead of matrices it should be a factorization object (e.g. produced by `factorize`

or `cholesky`

). The reason for this is that factorization itself is both expensive and typically allocates memory (although it can also be done in-place via, e.g., `lu!`

), and performance-critical situations requiring `rdiv!`

usually also require fine-grained control over the factorization of `B`

.

## BLAS 函数

在 Julia 中（就像许多科学计算一样），密集线性代数运算是基于 LAPACK 库，它反过来建立在被称为 BLAS 的基本线性代数构建模块之上。高度优化的 BLAS 实现在每个计算机架构上可用，并且有时在高性能线性代数例程中直接调用 BLAS 函数很有用。

`LinearAlgebra.BLAS`

提供了一些 BLAS 函数的封装。那些改写了某个输入数组的 BLAS 函数的名称以 `'!'`

结尾。通常，一个 BLAS 函数定义了四个方法，分别针对 `Float64`

，`Float32`

，`ComplexF64`

和 `ComplexF32`

数组。

### BLAS 字符参数

许多 BLAS 函数接受的参数可以决定是否转置某个参数的（`trans`

），要引用矩阵的哪一个三角（`uplo`

或 `ul`

），是否可以假设三角矩阵的对角线上全为一（`dA`

），或者输入参数属于矩阵乘法中的哪一边（`side`

）。可能是：

#### 乘法顺序

`side` | 含义 |
---|---|

`'L'` | 参数位于矩阵与矩阵运算的左边。 |

`'R'` | 参数位于矩阵与矩阵运算的右边。 |

#### 三角引用

`uplo` /`ul` | 含义 |
---|---|

`'U'` | 只会使用矩阵的上三角部分。 |

`'L'` | 只会使用矩阵的下三角部分。 |

#### 转置运算

`trans` /`tX` | 含义 |
---|---|

`'N'` | 输入矩阵 `X` 不被转置或共轭。 |

`'T'` | 输入矩阵 `X` 会被转置。 |

`'C'` | 输入矩阵 `X` 会被共轭转置。 |

#### 单位对角线

`diag` /`dX` | 含义 |
---|---|

`'N'` | 矩阵 `X` 对角线上的值会被读取。 |

`'U'` | 矩阵 `X` 对角线上假设全为一。 |

`LinearAlgebra.BLAS`

— Module.Interface to BLAS subroutines.

`LinearAlgebra.BLAS.dotu`

— Function.`dotu(n, X, incx, Y, incy)`

Dot function for two complex vectors consisting of `n`

elements of array `X`

with stride `incx`

and `n`

elements of array `Y`

with stride `incy`

.

**Examples**

```
julia> BLAS.dotu(10, fill(1.0im, 10), 1, fill(1.0+im, 20), 2)
-10.0 + 10.0im
```

`LinearAlgebra.BLAS.dotc`

— Function.`dotc(n, X, incx, U, incy)`

Dot function for two complex vectors, consisting of `n`

elements of array `X`

with stride `incx`

and `n`

elements of array `U`

with stride `incy`

, conjugating the first vector.

**Examples**

```
julia> BLAS.dotc(10, fill(1.0im, 10), 1, fill(1.0+im, 20), 2)
10.0 - 10.0im
```

`LinearAlgebra.BLAS.blascopy!`

— Function.`blascopy!(n, X, incx, Y, incy)`

Copy `n`

elements of array `X`

with stride `incx`

to array `Y`

with stride `incy`

. Returns `Y`

.

`LinearAlgebra.BLAS.nrm2`

— Function.`nrm2(n, X, incx)`

2-norm of a vector consisting of `n`

elements of array `X`

with stride `incx`

.

**Examples**

```
julia> BLAS.nrm2(4, fill(1.0, 8), 2)
2.0
julia> BLAS.nrm2(1, fill(1.0, 8), 2)
1.0
```

`LinearAlgebra.BLAS.asum`

— Function.`asum(n, X, incx)`

Sum of the absolute values of the first `n`

elements of array `X`

with stride `incx`

.

**Examples**

```
julia> BLAS.asum(5, fill(1.0im, 10), 2)
5.0
julia> BLAS.asum(2, fill(1.0im, 10), 5)
2.0
```

`LinearAlgebra.axpy!`

— Function.`axpy!(a, X, Y)`

Overwrite `Y`

with `a*X + Y`

, where `a`

is a scalar. Return `Y`

.

**Examples**

```
julia> x = [1; 2; 3];
julia> y = [4; 5; 6];
julia> BLAS.axpy!(2, x, y)
3-element Array{Int64,1}:
6
9
12
```

`LinearAlgebra.BLAS.scal!`

— Function.`scal!(n, a, X, incx)`

Overwrite `X`

with `a*X`

for the first `n`

elements of array `X`

with stride `incx`

. Returns `X`

.

`LinearAlgebra.BLAS.scal`

— Function.`scal(n, a, X, incx)`

Return `X`

scaled by `a`

for the first `n`

elements of array `X`

with stride `incx`

.

`LinearAlgebra.BLAS.ger!`

— Function.`ger!(alpha, x, y, A)`

Rank-1 update of the matrix `A`

with vectors `x`

and `y`

as `alpha*x*y' + A`

.

`LinearAlgebra.BLAS.syr!`

— Function.`syr!(uplo, alpha, x, A)`

Rank-1 update of the symmetric matrix `A`

with vector `x`

as `alpha*x*transpose(x) + A`

. `uplo`

controls which triangle of `A`

is updated. Returns `A`

.

`LinearAlgebra.BLAS.syrk!`

— Function.`LinearAlgebra.BLAS.syrk`

— Function.`LinearAlgebra.BLAS.her!`

— Function.`her!(uplo, alpha, x, A)`

Methods for complex arrays only. Rank-1 update of the Hermitian matrix `A`

with vector `x`

as `alpha*x*x' + A`

. `uplo`

controls which triangle of `A`

is updated. Returns `A`

.

`LinearAlgebra.BLAS.herk!`

— Function.`LinearAlgebra.BLAS.herk`

— Function.`LinearAlgebra.BLAS.gbmv!`

— Function.`gbmv!(trans, m, kl, ku, alpha, A, x, beta, y)`

Update vector `y`

as `alpha*A*x + beta*y`

or `alpha*A'*x + beta*y`

according to `trans`

. The matrix `A`

is a general band matrix of dimension `m`

by `size(A,2)`

with `kl`

sub-diagonals and `ku`

super-diagonals. `alpha`

and `beta`

are scalars. Return the updated `y`

.

`LinearAlgebra.BLAS.gbmv`

— Function.`gbmv(trans, m, kl, ku, alpha, A, x)`

Return `alpha*A*x`

or `alpha*A'*x`

according to `trans`

. The matrix `A`

is a general band matrix of dimension `m`

by `size(A,2)`

with `kl`

sub-diagonals and `ku`

super-diagonals, and `alpha`

is a scalar.

`LinearAlgebra.BLAS.sbmv!`

— Function.`sbmv!(uplo, k, alpha, A, x, beta, y)`

Update vector `y`

as `alpha*A*x + beta*y`

where `A`

is a a symmetric band matrix of order `size(A,2)`

with `k`

super-diagonals stored in the argument `A`

. The storage layout for `A`

is described the reference BLAS module, level-2 BLAS at http://www.netlib.org/lapack/explore-html/. Only the `uplo`

triangle of `A`

is used.

Return the updated `y`

.

`LinearAlgebra.BLAS.sbmv`

— Method.`sbmv(uplo, k, alpha, A, x)`

Return `alpha*A*x`

where `A`

is a symmetric band matrix of order `size(A,2)`

with `k`

super-diagonals stored in the argument `A`

. Only the `uplo`

triangle of `A`

is used.

`LinearAlgebra.BLAS.sbmv`

— Method.`sbmv(uplo, k, A, x)`

Return `A*x`

where `A`

is a symmetric band matrix of order `size(A,2)`

with `k`

super-diagonals stored in the argument `A`

. Only the `uplo`

triangle of `A`

is used.

`LinearAlgebra.BLAS.gemm!`

— Function.`gemm!(tA, tB, alpha, A, B, beta, C)`

Update `C`

as `alpha*A*B + beta*C`

or the other three variants according to `tA`

and `tB`

. Return the updated `C`

.

`LinearAlgebra.BLAS.gemm`

— Method.`gemm(tA, tB, alpha, A, B)`

Return `alpha*A*B`

or the other three variants according to `tA`

and `tB`

.

`LinearAlgebra.BLAS.gemm`

— Method.`gemm(tA, tB, A, B)`

Return `A*B`

or the other three variants according to `tA`

and `tB`

.

`LinearAlgebra.BLAS.gemv!`

— Function.`gemv!(tA, alpha, A, x, beta, y)`

Update the vector `y`

as `alpha*A*x + beta*y`

or `alpha*A'x + beta*y`

according to `tA`

. `alpha`

and `beta`

are scalars. Return the updated `y`

.

`LinearAlgebra.BLAS.gemv`

— Method.`gemv(tA, alpha, A, x)`

Return `alpha*A*x`

or `alpha*A'x`

according to `tA`

. `alpha`

is a scalar.

`LinearAlgebra.BLAS.gemv`

— Method.`gemv(tA, A, x)`

Return `A*x`

or `A'x`

according to `tA`

.

`LinearAlgebra.BLAS.symm!`

— Function.`LinearAlgebra.BLAS.symm`

— Method.`LinearAlgebra.BLAS.symm`

— Method.`LinearAlgebra.BLAS.symv!`

— Function.`symv!(ul, alpha, A, x, beta, y)`

Update the vector `y`

as `alpha*A*x + beta*y`

. `A`

is assumed to be symmetric. Only the `ul`

triangle of `A`

is used. `alpha`

and `beta`

are scalars. Return the updated `y`

.

`LinearAlgebra.BLAS.symv`

— Method.`symv(ul, alpha, A, x)`

Return `alpha*A*x`

. `A`

is assumed to be symmetric. Only the `ul`

triangle of `A`

is used. `alpha`

is a scalar.

`LinearAlgebra.BLAS.symv`

— Method.`symv(ul, A, x)`

Return `A*x`

. `A`

is assumed to be symmetric. Only the `ul`

triangle of `A`

is used.

`LinearAlgebra.BLAS.trmm!`

— Function.`LinearAlgebra.BLAS.trmm`

— Function.`LinearAlgebra.BLAS.trsm!`

— Function.`LinearAlgebra.BLAS.trsm`

— Function.`LinearAlgebra.BLAS.trmv!`

— Function.`LinearAlgebra.BLAS.trmv`

— Function.`LinearAlgebra.BLAS.trsv!`

— Function.`LinearAlgebra.BLAS.trsv`

— Function.`LinearAlgebra.BLAS.set_num_threads`

— Function.`set_num_threads(n)`

Set the number of threads the BLAS library should use.

`LinearAlgebra.I`

— Constant.`I`

An object of type `UniformScaling`

, representing an identity matrix of any size.

**Examples**

```
julia> fill(1, (5,6)) * I == fill(1, (5,6))
true
julia> [1 2im 3; 1im 2 3] * I
2×3 Array{Complex{Int64},2}:
1+0im 0+2im 3+0im
0+1im 2+0im 3+0im
```

## LAPACK 函数

`LinearAlgebra.LAPACK`

提供了一些针对线性代数的 LAPACK 函数的封装。那些改写了输入数组的函数的名称以 `'!'`

结尾。

一个函数通常定义了 4 个方法，分别针对 `Float64`

，`Float32`

，`ComplexF64`

和 `ComplexF32`

数组。

请注意，由 Julia 提供的 LAPACK API 可以并且将来会改变。因此，这个 API 不是面向用户的，也没有承诺在将来的版本中支持/弃用这个特殊的函数集。

`LinearAlgebra.LAPACK`

— Module.Interfaces to LAPACK subroutines.

`LinearAlgebra.LAPACK.gbtrf!`

— Function.`gbtrf!(kl, ku, m, AB) -> (AB, ipiv)`

Compute the LU factorization of a banded matrix `AB`

. `kl`

is the first subdiagonal containing a nonzero band, `ku`

is the last superdiagonal containing one, and `m`

is the first dimension of the matrix `AB`

. Returns the LU factorization in-place and `ipiv`

, the vector of pivots used.

`LinearAlgebra.LAPACK.gbtrs!`

— Function.`gbtrs!(trans, kl, ku, m, AB, ipiv, B)`

Solve the equation `AB * X = B`

. `trans`

determines the orientation of `AB`

. It may be `N`

(no transpose), `T`

(transpose), or `C`

(conjugate transpose). `kl`

is the first subdiagonal containing a nonzero band, `ku`

is the last superdiagonal containing one, and `m`

is the first dimension of the matrix `AB`

. `ipiv`

is the vector of pivots returned from `gbtrf!`

. Returns the vector or matrix `X`

, overwriting `B`

in-place.

`LinearAlgebra.LAPACK.gebal!`

— Function.`gebal!(job, A) -> (ilo, ihi, scale)`

Balance the matrix `A`

before computing its eigensystem or Schur factorization. `job`

can be one of `N`

(`A`

will not be permuted or scaled), `P`

(`A`

will only be permuted), `S`

(`A`

will only be scaled), or `B`

(`A`

will be both permuted and scaled). Modifies `A`

in-place and returns `ilo`

, `ihi`

, and `scale`

. If permuting was turned on, `A[i,j] = 0`

if `j > i`

and `1 < j < ilo`

or `j > ihi`

. `scale`

contains information about the scaling/permutations performed.

`LinearAlgebra.LAPACK.gebak!`

— Function.`gebak!(job, side, ilo, ihi, scale, V)`

Transform the eigenvectors `V`

of a matrix balanced using `gebal!`

to the unscaled/unpermuted eigenvectors of the original matrix. Modifies `V`

in-place. `side`

can be `L`

(left eigenvectors are transformed) or `R`

(right eigenvectors are transformed).

`LinearAlgebra.LAPACK.gebrd!`

— Function.`gebrd!(A) -> (A, d, e, tauq, taup)`

Reduce `A`

in-place to bidiagonal form `A = QBP'`

. Returns `A`

, containing the bidiagonal matrix `B`

; `d`

, containing the diagonal elements of `B`

; `e`

, containing the off-diagonal elements of `B`

; `tauq`

, containing the elementary reflectors representing `Q`

; and `taup`

, containing the elementary reflectors representing `P`

.

`LinearAlgebra.LAPACK.gelqf!`

— Function.`gelqf!(A, tau)`

Compute the `LQ`

factorization of `A`

, `A = LQ`

. `tau`

contains scalars which parameterize the elementary reflectors of the factorization. `tau`

must have length greater than or equal to the smallest dimension of `A`

.

Returns `A`

and `tau`

modified in-place.

`gelqf!(A) -> (A, tau)`

Compute the `LQ`

factorization of `A`

, `A = LQ`

.

Returns `A`

, modified in-place, and `tau`

, which contains scalars which parameterize the elementary reflectors of the factorization.

`LinearAlgebra.LAPACK.geqlf!`

— Function.`geqlf!(A, tau)`

Compute the `QL`

factorization of `A`

, `A = QL`

. `tau`

contains scalars which parameterize the elementary reflectors of the factorization. `tau`

must have length greater than or equal to the smallest dimension of `A`

.

Returns `A`

and `tau`

modified in-place.

`geqlf!(A) -> (A, tau)`

Compute the `QL`

factorization of `A`

, `A = QL`

.

Returns `A`

, modified in-place, and `tau`

, which contains scalars which parameterize the elementary reflectors of the factorization.

`LinearAlgebra.LAPACK.geqrf!`

— Function.`geqrf!(A, tau)`

Compute the `QR`

factorization of `A`

, `A = QR`

. `tau`

contains scalars which parameterize the elementary reflectors of the factorization. `tau`

must have length greater than or equal to the smallest dimension of `A`

.

Returns `A`

and `tau`

modified in-place.

`geqrf!(A) -> (A, tau)`

Compute the `QR`

factorization of `A`

, `A = QR`

.

Returns `A`

, modified in-place, and `tau`

, which contains scalars which parameterize the elementary reflectors of the factorization.

`LinearAlgebra.LAPACK.geqp3!`

— Function.`geqp3!(A, jpvt, tau)`

Compute the pivoted `QR`

factorization of `A`

, `AP = QR`

using BLAS level 3. `P`

is a pivoting matrix, represented by `jpvt`

. `tau`

stores the elementary reflectors. `jpvt`

must have length length greater than or equal to `n`

if `A`

is an `(m x n)`

matrix. `tau`

must have length greater than or equal to the smallest dimension of `A`

.

`A`

, `jpvt`

, and `tau`

are modified in-place.

`geqp3!(A, jpvt) -> (A, jpvt, tau)`

Compute the pivoted `QR`

factorization of `A`

, `AP = QR`

using BLAS level 3. `P`

is a pivoting matrix, represented by `jpvt`

. `jpvt`

must have length greater than or equal to `n`

if `A`

is an `(m x n)`

matrix.

Returns `A`

and `jpvt`

, modified in-place, and `tau`

, which stores the elementary reflectors.

`geqp3!(A) -> (A, jpvt, tau)`

Compute the pivoted `QR`

factorization of `A`

, `AP = QR`

using BLAS level 3.

Returns `A`

, modified in-place, `jpvt`

, which represents the pivoting matrix `P`

, and `tau`

, which stores the elementary reflectors.

`LinearAlgebra.LAPACK.gerqf!`

— Function.`gerqf!(A, tau)`

Compute the `RQ`

factorization of `A`

, `A = RQ`

. `tau`

contains scalars which parameterize the elementary reflectors of the factorization. `tau`

must have length greater than or equal to the smallest dimension of `A`

.

Returns `A`

and `tau`

modified in-place.

`gerqf!(A) -> (A, tau)`

Compute the `RQ`

factorization of `A`

, `A = RQ`

.

`A`

, modified in-place, and `tau`

, which contains scalars which parameterize the elementary reflectors of the factorization.

`LinearAlgebra.LAPACK.geqrt!`

— Function.`geqrt!(A, T)`

Compute the blocked `QR`

factorization of `A`

, `A = QR`

. `T`

contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization. The first dimension of `T`

sets the block size and it must be between 1 and `n`

. The second dimension of `T`

must equal the smallest dimension of `A`

.

Returns `A`

and `T`

modified in-place.

`geqrt!(A, nb) -> (A, T)`

Compute the blocked `QR`

factorization of `A`

, `A = QR`

. `nb`

sets the block size and it must be between 1 and `n`

, the second dimension of `A`

.

Returns `A`

, modified in-place, and `T`

, which contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization.

`LinearAlgebra.LAPACK.geqrt3!`

— Function.`geqrt3!(A, T)`

Recursively computes the blocked `QR`

factorization of `A`

, `A = QR`

. `T`

contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization. The first dimension of `T`

sets the block size and it must be between 1 and `n`

. The second dimension of `T`

must equal the smallest dimension of `A`

.

Returns `A`

and `T`

modified in-place.

`geqrt3!(A) -> (A, T)`

Recursively computes the blocked `QR`

factorization of `A`

, `A = QR`

.

Returns `A`

, modified in-place, and `T`

, which contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization.

`LinearAlgebra.LAPACK.getrf!`

— Function.`getrf!(A) -> (A, ipiv, info)`

Compute the pivoted `LU`

factorization of `A`

, `A = LU`

.

Returns `A`

, modified in-place, `ipiv`

, the pivoting information, and an `info`

code which indicates success (`info = 0`

), a singular value in `U`

(`info = i`

, in which case `U[i,i]`

is singular), or an error code (`info < 0`

).

`LinearAlgebra.LAPACK.tzrzf!`

— Function.`tzrzf!(A) -> (A, tau)`

Transforms the upper trapezoidal matrix `A`

to upper triangular form in-place. Returns `A`

and `tau`

, the scalar parameters for the elementary reflectors of the transformation.

`LinearAlgebra.LAPACK.ormrz!`

— Function.`ormrz!(side, trans, A, tau, C)`

Multiplies the matrix `C`

by `Q`

from the transformation supplied by `tzrzf!`

. Depending on `side`

or `trans`

the multiplication can be left-sided (`side = L, Q*C`

) or right-sided (`side = R, C*Q`

) and `Q`

can be unmodified (`trans = N`

), transposed (`trans = T`

), or conjugate transposed (`trans = C`

). Returns matrix `C`

which is modified in-place with the result of the multiplication.

`LinearAlgebra.LAPACK.gels!`

— Function.`gels!(trans, A, B) -> (F, B, ssr)`

Solves the linear equation `A * X = B`

, `transpose(A) * X = B`

, or `adjoint(A) * X = B`

using a QR or LQ factorization. Modifies the matrix/vector `B`

in place with the solution. `A`

is overwritten with its `QR`

or `LQ`

factorization. `trans`

may be one of `N`

(no modification), `T`

(transpose), or `C`

(conjugate transpose). `gels!`

searches for the minimum norm/least squares solution. `A`

may be under or over determined. The solution is returned in `B`

.

`LinearAlgebra.LAPACK.gesv!`

— Function.`gesv!(A, B) -> (B, A, ipiv)`

Solves the linear equation `A * X = B`

where `A`

is a square matrix using the `LU`

factorization of `A`

. `A`

is overwritten with its `LU`

factorization and `B`

is overwritten with the solution `X`

. `ipiv`

contains the pivoting information for the `LU`

factorization of `A`

.

`LinearAlgebra.LAPACK.getrs!`

— Function.`getrs!(trans, A, ipiv, B)`

Solves the linear equation `A * X = B`

, `transpose(A) * X = B`

, or `adjoint(A) * X = B`

for square `A`

. Modifies the matrix/vector `B`

in place with the solution. `A`

is the `LU`

factorization from `getrf!`

, with `ipiv`

the pivoting information. `trans`

may be one of `N`

(no modification), `T`

(transpose), or `C`

(conjugate transpose).

`LinearAlgebra.LAPACK.getri!`

— Function.`getri!(A, ipiv)`

Computes the inverse of `A`

, using its `LU`

factorization found by `getrf!`

. `ipiv`

is the pivot information output and `A`

contains the `LU`

factorization of `getrf!`

. `A`

is overwritten with its inverse.

`LinearAlgebra.LAPACK.gesvx!`

— Function.`gesvx!(fact, trans, A, AF, ipiv, equed, R, C, B) -> (X, equed, R, C, B, rcond, ferr, berr, work)`

Solves the linear equation `A * X = B`

(`trans = N`

), `transpose(A) * X = B`

(`trans = T`

), or `adjoint(A) * X = B`

(`trans = C`

) using the `LU`

factorization of `A`

. `fact`

may be `E`

, in which case `A`

will be equilibrated and copied to `AF`

; `F`

, in which case `AF`

and `ipiv`

from a previous `LU`

factorization are inputs; or `N`

, in which case `A`

will be copied to `AF`

and then factored. If `fact = F`

, `equed`

may be `N`

, meaning `A`

has not been equilibrated; `R`

, meaning `A`

was multiplied by `Diagonal(R)`

from the left; `C`

, meaning `A`

was multiplied by `Diagonal(C)`

from the right; or `B`

, meaning `A`

was multiplied by `Diagonal(R)`

from the left and `Diagonal(C)`

from the right. If `fact = F`

and `equed = R`

or `B`

the elements of `R`

must all be positive. If `fact = F`

and `equed = C`

or `B`

the elements of `C`

must all be positive.

Returns the solution `X`

; `equed`

, which is an output if `fact`

is not `N`

, and describes the equilibration that was performed; `R`

, the row equilibration diagonal; `C`

, the column equilibration diagonal; `B`

, which may be overwritten with its equilibrated form `Diagonal(R)*B`

(if `trans = N`

and `equed = R,B`

) or `Diagonal(C)*B`

(if `trans = T,C`

and `equed = C,B`

); `rcond`

, the reciprocal condition number of `A`

after equilbrating; `ferr`

, the forward error bound for each solution vector in `X`

; `berr`

, the forward error bound for each solution vector in `X`

; and `work`

, the reciprocal pivot growth factor.

`gesvx!(A, B)`

The no-equilibration, no-transpose simplification of `gesvx!`

.

`LinearAlgebra.LAPACK.gelsd!`

— Function.`gelsd!(A, B, rcond) -> (B, rnk)`

Computes the least norm solution of `A * X = B`

by finding the `SVD`

factorization of `A`

, then dividing-and-conquering the problem. `B`

is overwritten with the solution `X`

. Singular values below `rcond`

will be treated as zero. Returns the solution in `B`

and the effective rank of `A`

in `rnk`

.

`LinearAlgebra.LAPACK.gelsy!`

— Function.`gelsy!(A, B, rcond) -> (B, rnk)`

Computes the least norm solution of `A * X = B`

by finding the full `QR`

factorization of `A`

, then dividing-and-conquering the problem. `B`

is overwritten with the solution `X`

. Singular values below `rcond`

will be treated as zero. Returns the solution in `B`

and the effective rank of `A`

in `rnk`

.

`LinearAlgebra.LAPACK.gglse!`

— Function.`gglse!(A, c, B, d) -> (X,res)`

Solves the equation `A * x = c`

where `x`

is subject to the equality constraint `B * x = d`

. Uses the formula `||c - A*x||^2 = 0`

to solve. Returns `X`

and the residual sum-of-squares.

`LinearAlgebra.LAPACK.geev!`

— Function.`geev!(jobvl, jobvr, A) -> (W, VL, VR)`

Finds the eigensystem of `A`

. If `jobvl = N`

, the left eigenvectors of `A`

aren't computed. If `jobvr = N`

, the right eigenvectors of `A`

aren't computed. If `jobvl = V`

or `jobvr = V`

, the corresponding eigenvectors are computed. Returns the eigenvalues in `W`

, the right eigenvectors in `VR`

, and the left eigenvectors in `VL`

.

`LinearAlgebra.LAPACK.gesdd!`

— Function.`gesdd!(job, A) -> (U, S, VT)`

Finds the singular value decomposition of `A`

, `A = U * S * V'`

, using a divide and conquer approach. If `job = A`

, all the columns of `U`

and the rows of `V'`

are computed. If `job = N`

, no columns of `U`

or rows of `V'`

are computed. If `job = O`

, `A`

is overwritten with the columns of (thin) `U`

and the rows of (thin) `V'`

. If `job = S`

, the columns of (thin) `U`

and the rows of (thin) `V'`

are computed and returned separately.

`LinearAlgebra.LAPACK.gesvd!`

— Function.`gesvd!(jobu, jobvt, A) -> (U, S, VT)`

Finds the singular value decomposition of `A`

, `A = U * S * V'`

. If `jobu = A`

, all the columns of `U`

are computed. If `jobvt = A`

all the rows of `V'`

are computed. If `jobu = N`

, no columns of `U`

are computed. If `jobvt = N`

no rows of `V'`

are computed. If `jobu = O`

, `A`

is overwritten with the columns of (thin) `U`

. If `jobvt = O`

, `A`

is overwritten with the rows of (thin) `V'`

. If `jobu = S`

, the columns of (thin) `U`

are computed and returned separately. If `jobvt = S`

the rows of (thin) `V'`

are computed and returned separately. `jobu`

and `jobvt`

can't both be `O`

.

Returns `U`

, `S`

, and `Vt`

, where `S`

are the singular values of `A`

.

`LinearAlgebra.LAPACK.ggsvd!`

— Function.`ggsvd!(jobu, jobv, jobq, A, B) -> (U, V, Q, alpha, beta, k, l, R)`

Finds the generalized singular value decomposition of `A`

and `B`

, `U'*A*Q = D1*R`

and `V'*B*Q = D2*R`

. `D1`

has `alpha`

on its diagonal and `D2`

has `beta`

on its diagonal. If `jobu = U`

, the orthogonal/unitary matrix `U`

is computed. If `jobv = V`

the orthogonal/unitary matrix `V`

is computed. If `jobq = Q`

, the orthogonal/unitary matrix `Q`

is computed. If `jobu`

, `jobv`

or `jobq`

is `N`

, that matrix is not computed. This function is only available in LAPACK versions prior to 3.6.0.

`LinearAlgebra.LAPACK.ggsvd3!`

— Function.`ggsvd3!(jobu, jobv, jobq, A, B) -> (U, V, Q, alpha, beta, k, l, R)`

Finds the generalized singular value decomposition of `A`

and `B`

, `U'*A*Q = D1*R`

and `V'*B*Q = D2*R`

. `D1`

has `alpha`

on its diagonal and `D2`

has `beta`

on its diagonal. If `jobu = U`

, the orthogonal/unitary matrix `U`

is computed. If `jobv = V`

the orthogonal/unitary matrix `V`

is computed. If `jobq = Q`

, the orthogonal/unitary matrix `Q`

is computed. If `jobu`

, `jobv`

, or `jobq`

is `N`

, that matrix is not computed. This function requires LAPACK 3.6.0.

`LinearAlgebra.LAPACK.geevx!`

— Function.`geevx!(balanc, jobvl, jobvr, sense, A) -> (A, w, VL, VR, ilo, ihi, scale, abnrm, rconde, rcondv)`

Finds the eigensystem of `A`

with matrix balancing. If `jobvl = N`

, the left eigenvectors of `A`

aren't computed. If `jobvr = N`

, the right eigenvectors of `A`

aren't computed. If `jobvl = V`

or `jobvr = V`

, the corresponding eigenvectors are computed. If `balanc = N`

, no balancing is performed. If `balanc = P`

, `A`

is permuted but not scaled. If `balanc = S`

, `A`

is scaled but not permuted. If `balanc = B`

, `A`

is permuted and scaled. If `sense = N`

, no reciprocal condition numbers are computed. If `sense = E`

, reciprocal condition numbers are computed for the eigenvalues only. If `sense = V`

, reciprocal condition numbers are computed for the right eigenvectors only. If `sense = B`

, reciprocal condition numbers are computed for the right eigenvectors and the eigenvectors. If `sense = E,B`

, the right and left eigenvectors must be computed.

`LinearAlgebra.LAPACK.ggev!`

— Function.`ggev!(jobvl, jobvr, A, B) -> (alpha, beta, vl, vr)`

Finds the generalized eigendecomposition of `A`

and `B`

. If `jobvl = N`

, the left eigenvectors aren't computed. If `jobvr = N`

, the right eigenvectors aren't computed. If `jobvl = V`

or `jobvr = V`

, the corresponding eigenvectors are computed.

`LinearAlgebra.LAPACK.gtsv!`

— Function.`gtsv!(dl, d, du, B)`

Solves the equation `A * X = B`

where `A`

is a tridiagonal matrix with `dl`

on the subdiagonal, `d`

on the diagonal, and `du`

on the superdiagonal.

Overwrites `B`

with the solution `X`

and returns it.

`LinearAlgebra.LAPACK.gttrf!`

— Function.`gttrf!(dl, d, du) -> (dl, d, du, du2, ipiv)`

Finds the `LU`

factorization of a tridiagonal matrix with `dl`

on the subdiagonal, `d`

on the diagonal, and `du`

on the superdiagonal.

Modifies `dl`

, `d`

, and `du`

in-place and returns them and the second superdiagonal `du2`

and the pivoting vector `ipiv`

.

`LinearAlgebra.LAPACK.gttrs!`

— Function.`gttrs!(trans, dl, d, du, du2, ipiv, B)`

Solves the equation `A * X = B`

(`trans = N`

), `transpose(A) * X = B`

(`trans = T`

), or `adjoint(A) * X = B`

(`trans = C`

) using the `LU`

factorization computed by `gttrf!`

. `B`

is overwritten with the solution `X`

.

`LinearAlgebra.LAPACK.orglq!`

— Function.`orglq!(A, tau, k = length(tau))`

Explicitly finds the matrix `Q`

of a `LQ`

factorization after calling `gelqf!`

on `A`

. Uses the output of `gelqf!`

. `A`

is overwritten by `Q`

.

`LinearAlgebra.LAPACK.orgqr!`

— Function.`orgqr!(A, tau, k = length(tau))`

Explicitly finds the matrix `Q`

of a `QR`

factorization after calling `geqrf!`

on `A`

. Uses the output of `geqrf!`

. `A`

is overwritten by `Q`

.

`LinearAlgebra.LAPACK.orgql!`

— Function.`orgql!(A, tau, k = length(tau))`

Explicitly finds the matrix `Q`

of a `QL`

factorization after calling `geqlf!`

on `A`

. Uses the output of `geqlf!`

. `A`

is overwritten by `Q`

.

`LinearAlgebra.LAPACK.orgrq!`

— Function.`orgrq!(A, tau, k = length(tau))`

Explicitly finds the matrix `Q`

of a `RQ`

factorization after calling `gerqf!`

on `A`

. Uses the output of `gerqf!`

. `A`

is overwritten by `Q`

.

`LinearAlgebra.LAPACK.ormlq!`

— Function.`ormlq!(side, trans, A, tau, C)`

Computes `Q * C`

(`trans = N`

), `transpose(Q) * C`

(`trans = T`

), `adjoint(Q) * C`

(`trans = C`

) for `side = L`

or the equivalent right-sided multiplication for `side = R`

using `Q`

from a `LQ`

factorization of `A`

computed using `gelqf!`

. `C`

is overwritten.

`LinearAlgebra.LAPACK.ormqr!`

— Function.`ormqr!(side, trans, A, tau, C)`

Computes `Q * C`

(`trans = N`

), `transpose(Q) * C`

(`trans = T`

), `adjoint(Q) * C`

(`trans = C`

) for `side = L`

or the equivalent right-sided multiplication for `side = R`

using `Q`

from a `QR`

factorization of `A`

computed using `geqrf!`

. `C`

is overwritten.

`LinearAlgebra.LAPACK.ormql!`

— Function.`ormql!(side, trans, A, tau, C)`

Computes `Q * C`

(`trans = N`

), `transpose(Q) * C`

(`trans = T`

), `adjoint(Q) * C`

(`trans = C`

) for `side = L`

or the equivalent right-sided multiplication for `side = R`

using `Q`

from a `QL`

factorization of `A`

computed using `geqlf!`

. `C`

is overwritten.

`LinearAlgebra.LAPACK.ormrq!`

— Function.`ormrq!(side, trans, A, tau, C)`

Computes `Q * C`

(`trans = N`

), `transpose(Q) * C`

(`trans = T`

), `adjoint(Q) * C`

(`trans = C`

) for `side = L`

or the equivalent right-sided multiplication for `side = R`

using `Q`

from a `RQ`

factorization of `A`

computed using `gerqf!`

. `C`

is overwritten.

`LinearAlgebra.LAPACK.gemqrt!`

— Function.`gemqrt!(side, trans, V, T, C)`

Computes `Q * C`

(`trans = N`

), `transpose(Q) * C`

(`trans = T`

), `adjoint(Q) * C`

(`trans = C`

) for `side = L`

or the equivalent right-sided multiplication for `side = R`

using `Q`

from a `QR`

factorization of `A`

computed using `geqrt!`

. `C`

is overwritten.

`LinearAlgebra.LAPACK.posv!`

— Function.`posv!(uplo, A, B) -> (A, B)`

Finds the solution to `A * X = B`

where `A`

is a symmetric or Hermitian positive definite matrix. If `uplo = U`

the upper Cholesky decomposition of `A`

is computed. If `uplo = L`

the lower Cholesky decomposition of `A`

is computed. `A`

is overwritten by its Cholesky decomposition. `B`

is overwritten with the solution `X`

.

`LinearAlgebra.LAPACK.potrf!`

— Function.`potrf!(uplo, A)`

Computes the Cholesky (upper if `uplo = U`

, lower if `uplo = L`

) decomposition of positive-definite matrix `A`

. `A`

is overwritten and returned with an info code.

`LinearAlgebra.LAPACK.potri!`

— Function.`potri!(uplo, A)`

Computes the inverse of positive-definite matrix `A`

after calling `potrf!`

to find its (upper if `uplo = U`

, lower if `uplo = L`

) Cholesky decomposition.

`A`

is overwritten by its inverse and returned.

`LinearAlgebra.LAPACK.potrs!`

— Function.`potrs!(uplo, A, B)`

Finds the solution to `A * X = B`

where `A`

is a symmetric or Hermitian positive definite matrix whose Cholesky decomposition was computed by `potrf!`

. If `uplo = U`

the upper Cholesky decomposition of `A`

was computed. If `uplo = L`

the lower Cholesky decomposition of `A`

was computed. `B`

is overwritten with the solution `X`

.

`LinearAlgebra.LAPACK.pstrf!`

— Function.`pstrf!(uplo, A, tol) -> (A, piv, rank, info)`

Computes the (upper if `uplo = U`

, lower if `uplo = L`

) pivoted Cholesky decomposition of positive-definite matrix `A`

with a user-set tolerance `tol`

. `A`

is overwritten by its Cholesky decomposition.

Returns `A`

, the pivots `piv`

, the rank of `A`

, and an `info`

code. If `info = 0`

, the factorization succeeded. If `info = i > 0`

, then `A`

is indefinite or rank-deficient.

`LinearAlgebra.LAPACK.ptsv!`

— Function.`ptsv!(D, E, B)`

Solves `A * X = B`

for positive-definite tridiagonal `A`

. `D`

is the diagonal of `A`

and `E`

is the off-diagonal. `B`

is overwritten with the solution `X`

and returned.

`LinearAlgebra.LAPACK.pttrf!`

— Function.`pttrf!(D, E)`

Computes the LDLt factorization of a positive-definite tridiagonal matrix with `D`

as diagonal and `E`

as off-diagonal. `D`

and `E`

are overwritten and returned.

`LinearAlgebra.LAPACK.pttrs!`

— Function.`pttrs!(D, E, B)`

Solves `A * X = B`

for positive-definite tridiagonal `A`

with diagonal `D`

and off-diagonal `E`

after computing `A`

's LDLt factorization using `pttrf!`

. `B`

is overwritten with the solution `X`

.

`LinearAlgebra.LAPACK.trtri!`

— Function.`trtri!(uplo, diag, A)`

Finds the inverse of (upper if `uplo = U`

, lower if `uplo = L`

) triangular matrix `A`

. If `diag = N`

, `A`

has non-unit diagonal elements. If `diag = U`

, all diagonal elements of `A`

are one. `A`

is overwritten with its inverse.

`LinearAlgebra.LAPACK.trtrs!`

— Function.`trtrs!(uplo, trans, diag, A, B)`

Solves `A * X = B`

(`trans = N`

), `transpose(A) * X = B`

(`trans = T`

), or `adjoint(A) * X = B`

(`trans = C`

) for (upper if `uplo = U`

, lower if `uplo = L`

) triangular matrix `A`

. If `diag = N`

, `A`

has non-unit diagonal elements. If `diag = U`

, all diagonal elements of `A`

are one. `B`

is overwritten with the solution `X`

.

`LinearAlgebra.LAPACK.trcon!`

— Function.`trcon!(norm, uplo, diag, A)`

Finds the reciprocal condition number of (upper if `uplo = U`

, lower if `uplo = L`

) triangular matrix `A`

. If `diag = N`

, `A`

has non-unit diagonal elements. If `diag = U`

, all diagonal elements of `A`

are one. If `norm = I`

, the condition number is found in the infinity norm. If `norm = O`

or `1`

, the condition number is found in the one norm.

`LinearAlgebra.LAPACK.trevc!`

— Function.`trevc!(side, howmny, select, T, VL = similar(T), VR = similar(T))`

Finds the eigensystem of an upper triangular matrix `T`

. If `side = R`

, the right eigenvectors are computed. If `side = L`

, the left eigenvectors are computed. If `side = B`

, both sets are computed. If `howmny = A`

, all eigenvectors are found. If `howmny = B`

, all eigenvectors are found and backtransformed using `VL`

and `VR`

. If `howmny = S`

, only the eigenvectors corresponding to the values in `select`

are computed.

`LinearAlgebra.LAPACK.trrfs!`

— Function.`trrfs!(uplo, trans, diag, A, B, X, Ferr, Berr) -> (Ferr, Berr)`

Estimates the error in the solution to `A * X = B`

(`trans = N`

), `transpose(A) * X = B`

(`trans = T`

), `adjoint(A) * X = B`

(`trans = C`

) for `side = L`

, or the equivalent equations a right-handed `side = R`

`X * A`

after computing `X`

using `trtrs!`

. If `uplo = U`

, `A`

is upper triangular. If `uplo = L`

, `A`

is lower triangular. If `diag = N`

, `A`

has non-unit diagonal elements. If `diag = U`

, all diagonal elements of `A`

are one. `Ferr`

and `Berr`

are optional inputs. `Ferr`

is the forward error and `Berr`

is the backward error, each component-wise.

`LinearAlgebra.LAPACK.stev!`

— Function.`stev!(job, dv, ev) -> (dv, Zmat)`

Computes the eigensystem for a symmetric tridiagonal matrix with `dv`

as diagonal and `ev`

as off-diagonal. If `job = N`

only the eigenvalues are found and returned in `dv`

. If `job = V`

then the eigenvectors are also found and returned in `Zmat`

.

`LinearAlgebra.LAPACK.stebz!`

— Function.`stebz!(range, order, vl, vu, il, iu, abstol, dv, ev) -> (dv, iblock, isplit)`

Computes the eigenvalues for a symmetric tridiagonal matrix with `dv`

as diagonal and `ev`

as off-diagonal. If `range = A`

, all the eigenvalues are found. If `range = V`

, the eigenvalues in the half-open interval `(vl, vu]`

are found. If `range = I`

, the eigenvalues with indices between `il`

and `iu`

are found. If `order = B`

, eigvalues are ordered within a block. If `order = E`

, they are ordered across all the blocks. `abstol`

can be set as a tolerance for convergence.

`LinearAlgebra.LAPACK.stegr!`

— Function.`stegr!(jobz, range, dv, ev, vl, vu, il, iu) -> (w, Z)`

Computes the eigenvalues (`jobz = N`

) or eigenvalues and eigenvectors (`jobz = V`

) for a symmetric tridiagonal matrix with `dv`

as diagonal and `ev`

as off-diagonal. If `range = A`

, all the eigenvalues are found. If `range = V`

, the eigenvalues in the half-open interval `(vl, vu]`

are found. If `range = I`

, the eigenvalues with indices between `il`

and `iu`

are found. The eigenvalues are returned in `w`

and the eigenvectors in `Z`

.

`LinearAlgebra.LAPACK.stein!`

— Function.`stein!(dv, ev_in, w_in, iblock_in, isplit_in)`

Computes the eigenvectors for a symmetric tridiagonal matrix with `dv`

as diagonal and `ev_in`

as off-diagonal. `w_in`

specifies the input eigenvalues for which to find corresponding eigenvectors. `iblock_in`

specifies the submatrices corresponding to the eigenvalues in `w_in`

. `isplit_in`

specifies the splitting points between the submatrix blocks.

`LinearAlgebra.LAPACK.syconv!`

— Function.`syconv!(uplo, A, ipiv) -> (A, work)`

Converts a symmetric matrix `A`

(which has been factorized into a triangular matrix) into two matrices `L`

and `D`

. If `uplo = U`

, `A`

is upper triangular. If `uplo = L`

, it is lower triangular. `ipiv`

is the pivot vector from the triangular factorization. `A`

is overwritten by `L`

and `D`

.

`LinearAlgebra.LAPACK.sysv!`

— Function.`sysv!(uplo, A, B) -> (B, A, ipiv)`

Finds the solution to `A * X = B`

for symmetric matrix `A`

. If `uplo = U`

, the upper half of `A`

is stored. If `uplo = L`

, the lower half is stored. `B`

is overwritten by the solution `X`

. `A`

is overwritten by its Bunch-Kaufman factorization. `ipiv`

contains pivoting information about the factorization.

`LinearAlgebra.LAPACK.sytrf!`

— Function.`sytrf!(uplo, A) -> (A, ipiv, info)`

Computes the Bunch-Kaufman factorization of a symmetric matrix `A`

. If `uplo = U`

, the upper half of `A`

is stored. If `uplo = L`

, the lower half is stored.

Returns `A`

, overwritten by the factorization, a pivot vector `ipiv`

, and the error code `info`

which is a non-negative integer. If `info`

is positive the matrix is singular and the diagonal part of the factorization is exactly zero at position `info`

.

`LinearAlgebra.LAPACK.sytri!`

— Function.`sytri!(uplo, A, ipiv)`

Computes the inverse of a symmetric matrix `A`

using the results of `sytrf!`

. If `uplo = U`

, the upper half of `A`

is stored. If `uplo = L`

, the lower half is stored. `A`

is overwritten by its inverse.

`LinearAlgebra.LAPACK.sytrs!`

— Function.`sytrs!(uplo, A, ipiv, B)`

Solves the equation `A * X = B`

for a symmetric matrix `A`

using the results of `sytrf!`

. If `uplo = U`

, the upper half of `A`

is stored. If `uplo = L`

, the lower half is stored. `B`

is overwritten by the solution `X`

.

`LinearAlgebra.LAPACK.hesv!`

— Function.`hesv!(uplo, A, B) -> (B, A, ipiv)`

Finds the solution to `A * X = B`

for Hermitian matrix `A`

. If `uplo = U`

, the upper half of `A`

is stored. If `uplo = L`

, the lower half is stored. `B`

is overwritten by the solution `X`

. `A`

is overwritten by its Bunch-Kaufman factorization. `ipiv`

contains pivoting information about the factorization.

`LinearAlgebra.LAPACK.hetrf!`

— Function.`hetrf!(uplo, A) -> (A, ipiv, info)`

Computes the Bunch-Kaufman factorization of a Hermitian matrix `A`

. If `uplo = U`

, the upper half of `A`

is stored. If `uplo = L`

, the lower half is stored.

Returns `A`

, overwritten by the factorization, a pivot vector `ipiv`

, and the error code `info`

which is a non-negative integer. If `info`

is positive the matrix is singular and the diagonal part of the factorization is exactly zero at position `info`

.

`LinearAlgebra.LAPACK.hetri!`

— Function.`hetri!(uplo, A, ipiv)`

Computes the inverse of a Hermitian matrix `A`

using the results of `sytrf!`

. If `uplo = U`

, the upper half of `A`

is stored. If `uplo = L`

, the lower half is stored. `A`

is overwritten by its inverse.

`LinearAlgebra.LAPACK.hetrs!`

— Function.`hetrs!(uplo, A, ipiv, B)`

Solves the equation `A * X = B`

for a Hermitian matrix `A`

using the results of `sytrf!`

. If `uplo = U`

, the upper half of `A`

is stored. If `uplo = L`

, the lower half is stored. `B`

is overwritten by the solution `X`

.

`LinearAlgebra.LAPACK.syev!`

— Function.`syev!(jobz, uplo, A)`

Finds the eigenvalues (`jobz = N`

) or eigenvalues and eigenvectors (`jobz = V`

) of a symmetric matrix `A`

. If `uplo = U`

, the upper triangle of `A`

is used. If `uplo = L`

, the lower triangle of `A`

is used.

`LinearAlgebra.LAPACK.syevr!`

— Function.`syevr!(jobz, range, uplo, A, vl, vu, il, iu, abstol) -> (W, Z)`

Finds the eigenvalues (`jobz = N`

) or eigenvalues and eigenvectors (`jobz = V`

) of a symmetric matrix `A`

. If `uplo = U`

, the upper triangle of `A`

is used. If `uplo = L`

, the lower triangle of `A`

is used. If `range = A`

, all the eigenvalues are found. If `range = V`

, the eigenvalues in the half-open interval `(vl, vu]`

are found. If `range = I`

, the eigenvalues with indices between `il`

and `iu`

are found. `abstol`

can be set as a tolerance for convergence.

The eigenvalues are returned in `W`

and the eigenvectors in `Z`

.

`LinearAlgebra.LAPACK.sygvd!`

— Function.`sygvd!(itype, jobz, uplo, A, B) -> (w, A, B)`

Finds the generalized eigenvalues (`jobz = N`

) or eigenvalues and eigenvectors (`jobz = V`

) of a symmetric matrix `A`

and symmetric positive-definite matrix `B`

. If `uplo = U`

, the upper triangles of `A`

and `B`

are used. If `uplo = L`

, the lower triangles of `A`

and `B`

are used. If `itype = 1`

, the problem to solve is `A * x = lambda * B * x`

. If `itype = 2`

, the problem to solve is `A * B * x = lambda * x`

. If `itype = 3`

, the problem to solve is `B * A * x = lambda * x`

.

`LinearAlgebra.LAPACK.bdsqr!`

— Function.`bdsqr!(uplo, d, e_, Vt, U, C) -> (d, Vt, U, C)`

Computes the singular value decomposition of a bidiagonal matrix with `d`

on the diagonal and `e_`

on the off-diagonal. If `uplo = U`

, `e_`

is the superdiagonal. If `uplo = L`

, `e_`

is the subdiagonal. Can optionally also compute the product `Q' * C`

.

Returns the singular values in `d`

, and the matrix `C`

overwritten with `Q' * C`

.

`LinearAlgebra.LAPACK.bdsdc!`

— Function.`bdsdc!(uplo, compq, d, e_) -> (d, e, u, vt, q, iq)`

Computes the singular value decomposition of a bidiagonal matrix with `d`

on the diagonal and `e_`

on the off-diagonal using a divide and conqueq method. If `uplo = U`

, `e_`

is the superdiagonal. If `uplo = L`

, `e_`

is the subdiagonal. If `compq = N`

, only the singular values are found. If `compq = I`

, the singular values and vectors are found. If `compq = P`

, the singular values and vectors are found in compact form. Only works for real types.

Returns the singular values in `d`

, and if `compq = P`

, the compact singular vectors in `iq`

.

`LinearAlgebra.LAPACK.gecon!`

— Function.`gecon!(normtype, A, anorm)`

Finds the reciprocal condition number of matrix `A`

. If `normtype = I`

, the condition number is found in the infinity norm. If `normtype = O`

or `1`

, the condition number is found in the one norm. `A`

must be the result of `getrf!`

and `anorm`

is the norm of `A`

in the relevant norm.

`LinearAlgebra.LAPACK.gehrd!`

— Function.`gehrd!(ilo, ihi, A) -> (A, tau)`

Converts a matrix `A`

to Hessenberg form. If `A`

is balanced with `gebal!`

then `ilo`

and `ihi`

are the outputs of `gebal!`

. Otherwise they should be `ilo = 1`

and `ihi = size(A,2)`

. `tau`

contains the elementary reflectors of the factorization.

`LinearAlgebra.LAPACK.orghr!`

— Function.`orghr!(ilo, ihi, A, tau)`

Explicitly finds `Q`

, the orthogonal/unitary matrix from `gehrd!`

. `ilo`

, `ihi`

, `A`

, and `tau`

must correspond to the input/output to `gehrd!`

.

`LinearAlgebra.LAPACK.gees!`

— Function.`gees!(jobvs, A) -> (A, vs, w)`

Computes the eigenvalues (`jobvs = N`

) or the eigenvalues and Schur vectors (`jobvs = V`

) of matrix `A`

. `A`

is overwritten by its Schur form.

Returns `A`

, `vs`

containing the Schur vectors, and `w`

, containing the eigenvalues.

`LinearAlgebra.LAPACK.gges!`

— Function.`gges!(jobvsl, jobvsr, A, B) -> (A, B, alpha, beta, vsl, vsr)`

Computes the generalized eigenvalues, generalized Schur form, left Schur vectors (`jobsvl = V`

), or right Schur vectors (`jobvsr = V`

) of `A`

and `B`

.

The generalized eigenvalues are returned in `alpha`

and `beta`

. The left Schur vectors are returned in `vsl`

and the right Schur vectors are returned in `vsr`

.

`LinearAlgebra.LAPACK.trexc!`

— Function.`trexc!(compq, ifst, ilst, T, Q) -> (T, Q)`

Reorder the Schur factorization of a matrix. If `compq = V`

, the Schur vectors `Q`

are reordered. If `compq = N`

they are not modified. `ifst`

and `ilst`

specify the reordering of the vectors.

`LinearAlgebra.LAPACK.trsen!`

— Function.`trsen!(compq, job, select, T, Q) -> (T, Q, w, s, sep)`

Reorder the Schur factorization of a matrix and optionally finds reciprocal condition numbers. If `job = N`

, no condition numbers are found. If `job = E`

, only the condition number for this cluster of eigenvalues is found. If `job = V`

, only the condition number for the invariant subspace is found. If `job = B`

then the condition numbers for the cluster and subspace are found. If `compq = V`

the Schur vectors `Q`

are updated. If `compq = N`

the Schur vectors are not modified. `select`

determines which eigenvalues are in the cluster.

Returns `T`

, `Q`

, reordered eigenvalues in `w`

, the condition number of the cluster of eigenvalues `s`

, and the condition number of the invariant subspace `sep`

.

`LinearAlgebra.LAPACK.tgsen!`

— Function.`tgsen!(select, S, T, Q, Z) -> (S, T, alpha, beta, Q, Z)`

Reorders the vectors of a generalized Schur decomposition. `select`

specifies the eigenvalues in each cluster.

`LinearAlgebra.LAPACK.trsyl!`

— Function.`trsyl!(transa, transb, A, B, C, isgn=1) -> (C, scale)`

Solves the Sylvester matrix equation `A * X +/- X * B = scale*C`

where `A`

and `B`

are both quasi-upper triangular. If `transa = N`

, `A`

is not modified. If `transa = T`

, `A`

is transposed. If `transa = C`

, `A`

is conjugate transposed. Similarly for `transb`

and `B`

. If `isgn = 1`

, the equation `A * X + X * B = scale * C`

is solved. If `isgn = -1`

, the equation `A * X - X * B = scale * C`

is solved.

Returns `X`

(overwriting `C`

) and `scale`

.