Linear Algebra

As well as other useful operations, such as finding eigenvalues or eigenvectors:

julia> A = [-4. -17.; 2. 2.]
2×2 Matrix{Float64}:
 -4.0  -17.0
  2.0    2.0
julia> eigvals(A)
2-element Vector{ComplexF64}:
 -1.0 - 5.0im
 -1.0 + 5.0im
julia> eigvecs(A)
2×2 Matrix{ComplexF64}:
  0.945905-0.0im        0.945905+0.0im
 -0.166924+0.278207im  -0.166924-0.278207im

In addition, Julia provides many which can be used to speed up problems such as linear solve or matrix exponentiation by pre-factorizing a matrix into a form more amenable (for performance or memory reasons) to the problem. See the documentation on factorize for more information. As an example:

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

Since A is not Hermitian, symmetric, triangular, tridiagonal, or bidiagonal, an LU factorization may be the best we can do. Compare with:

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

Here, Julia was able to detect that B is in fact symmetric, and used a more appropriate factorization. Often it’s possible to write more efficient code for a matrix that is known to have certain properties e.g. it is symmetric, or tridiagonal. Julia provides some special types so that you can “tag” matrices as having these properties. For instance:

julia> B = [1.5 2 -4; 2 -1 -3; -4 -3 5]
3×3 Matrix{Float64}:
  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, Matrix{Float64}}:
  1.5   2.0  -4.0
  2.0  -1.0  -3.0
 -4.0  -3.0   5.0

sB has been tagged as a matrix that’s (real) symmetric, so for later operations we might perform on it, such as eigenfactorization or computing matrix-vector products, efficiencies can be found by only referencing half of it. For example:

julia> B = [1.5 2 -4; 2 -1 -3; -4 -3 5]
3×3 Matrix{Float64}:
  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, Matrix{Float64}}:
  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 Vector{Int64}:
 1
 2
 3
julia> sB\x
3-element Vector{Float64}:
 -1.7391304347826084
 -1.1086956521739126
 -1.4565217391304346

The \ operation here performs the linear solution. The left-division operator is pretty powerful and it’s easy to write compact, readable code that is flexible enough to solve all sorts of systems of linear equations.

arise often in linear algebra and are frequently associated with various matrix factorizations. Julia features a rich collection of special matrix types, which allow for fast computation with specialized routines that are specially developed for particular matrix types.

The following tables summarize the types of special matrices that have been implemented in Julia, as well as whether hooks to various optimized methods for them in LAPACK are available.

Legend:

Legend:

A UniformScaling operator represents a scalar times the identity operator, λ*I. The identity operator I is defined as a constant and is an instance of UniformScaling. The size of these operators are generic and match the other matrix in the binary operations , -, and \. For A+I and A-I this means that A must be square. Multiplication with the identity operator I is a noop (except for checking that the scaling factor is one) and therefore almost without overhead.

To see the UniformScaling operator in action:

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

If you need to solve many systems of the form (A+μI)x = b for the same A and different μ, it might be beneficial to first compute the Hessenberg factorization F of A via the function. Given F, Julia employs an efficient algorithm for (F+μ*I) \ b (equivalent to (A+μ*I)x \ b) and related operations like determinants.

Matrix factorizations (a.k.a. matrix decompositions) compute the factorization of a matrix into a product of matrices, and are one of the central concepts in linear algebra.

The following table summarizes the types of matrix factorizations that have been implemented in Julia. Details of their associated methods can be found in the section of the Linear Algebra documentation.

Linear algebra functions in Julia are largely implemented by calling functions from LAPACK. Sparse matrix factorizations call functions from . Other sparse solvers are available as Julia packages.

Base.:* — Method

*(A::AbstractMatrix, B::AbstractMatrix)

Matrix multiplication.

Examples

julia> [1 1; 0 1] * [1 0; 1 1]
2×2 Matrix{Int64}:
 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 Vector{Float64}:
 32.0
 18.0
julia> A * X == B
true

LinearAlgebra.SingularException — Type

SingularException

Exception thrown when the input matrix has one or more zero-valued eigenvalues, and is not invertible. A linear solve involving such a matrix cannot be computed. The info field indicates the location of (one of) the singular value(s).

LinearAlgebra.PosDefException — Type

PosDefException

Exception thrown when the input matrix was not . Some linear algebra functions and factorizations are only applicable to positive definite matrices. The info field indicates the location of (one of) the eigenvalue(s) which is (are) less than/equal to 0.

source

— Type

ZeroPivotException <: Exception

Exception thrown when a matrix factorization/solve encounters a zero in a pivot (diagonal) position and cannot proceed. This may not mean that the matrix is singular: it may be fruitful to switch to a diffent factorization such as pivoted LU that can re-order variables to eliminate spurious zero pivots. The info field indicates the location of (one of) the zero pivot(s).

source

— Function

dot(x, y)
x ⋅ y

Compute the dot product between two vectors. For complex vectors, the first vector is conjugated.

dot also works on arbitrary iterable objects, including arrays of any dimension, as long as dot is defined on the elements.

dot is semantically equivalent to sum(dot(vx,vy) for (vx,vy) in zip(x, y)), with the added restriction that the arguments must have equal lengths.

x ⋅ y (where ⋅ can be typed by tab-completing \cdot in the REPL) is a synonym for dot(x, y).

Examples

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

source

— Method

dot(x, A, y)

Compute the generalized dot product dot(x, A*y) between two vectors x and y, without storing the intermediate result of A*y. As for the two-argument dot(_,_), this acts recursively. Moreover, for complex vectors, the first vector is conjugated.

Julia 1.4

Three-argument dot requires at least Julia 1.4.

Examples

julia> dot([1; 1], [1 2; 3 4], [2; 3])
26
julia> dot(1:5, reshape(1:25, 5, 5), 2:6)
4850
julia> ⋅(1:5, reshape(1:25, 5, 5), 2:6) == dot(1:5, reshape(1:25, 5, 5), 2:6)
true

LinearAlgebra.cross — Function

cross(x, y)
×(x,y)

Compute the cross product of two 3-vectors.

Examples

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

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 Matrix{Float64}:
 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, Vector{Float64}}:
 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.

source

— 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 Matrix{Int64}:
 1  2  3
 4  5  6
 7  8  9
julia> Diagonal(A)
3×3 Diagonal{Int64, Vector{Int64}}:
 1  ⋅  ⋅
 ⋅  5  ⋅
 ⋅  ⋅  9

source

Diagonal(V::AbstractVector)

Construct a matrix with V as its diagonal.

Examples

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

source

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 Matrix{Int64}:
 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, Vector{Int64}}:
 1  1  ⋅  ⋅
 ⋅  2  2  ⋅
 ⋅  ⋅  3  3
 ⋅  ⋅  ⋅  4
julia> Bidiagonal(A, :L) # contains the main diagonal and first subdiagonal of A
4×4 Bidiagonal{Int64, Vector{Int64}}:
 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 (or Array(_) for short).

For SymTridiagonal block matrices, the elements of dv are symmetrized. The argument ev is interpreted as the superdiagonal. Blocks from the subdiagonal are (materialized) transpose of the corresponding superdiagonal blocks.

Examples

julia> dv = [1, 2, 3, 4]
4-element Vector{Int64}:
 1
 2
 3
 4
julia> ev = [7, 8, 9]
3-element Vector{Int64}:
 7
 8
 9
julia> SymTridiagonal(dv, ev)
4×4 SymTridiagonal{Int64, Vector{Int64}}:
 1  7  ⋅  ⋅
 7  2  8  ⋅
 ⋅  8  3  9
 ⋅  ⋅  9  4
julia> A = SymTridiagonal(fill([1 2; 3 4], 3), fill([1 2; 3 4], 2));
julia> A[1,1]
2×2 Symmetric{Int64, Matrix{Int64}}:
 1  2
 2  4
julia> A[1,2]
2×2 Matrix{Int64}:
 1  2
 3  4
julia> A[2,1]
2×2 Matrix{Int64}:
 1  3
 2  4

source

SymTridiagonal(A::AbstractMatrix)

Construct a symmetric tridiagonal matrix from the diagonal and first superdiagonal of the symmetric matrix A.

Examples

julia> A = [1 2 3; 2 4 5; 3 5 6]
3×3 Matrix{Int64}:
 1  2  3
 2  4  5
 3  5  6
julia> SymTridiagonal(A)
3×3 SymTridiagonal{Int64, Vector{Int64}}:
 1  2  ⋅
 2  4  5
 ⋅  5  6
julia> B = reshape([[1 2; 2 3], [1 2; 3 4], [1 3; 2 4], [1 2; 2 3]], 2, 2);
julia> SymTridiagonal(B)
2×2 SymTridiagonal{Matrix{Int64}, Vector{Matrix{Int64}}}:
 [1 2; 2 3]  [1 3; 2 4]
 [1 2; 3 4]  [1 2; 2 3]

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 (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, Vector{Int64}}:
 7  4  ⋅  ⋅
 1  8  5  ⋅
 ⋅  2  9  6
 ⋅  ⋅  3  0

source

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 Matrix{Int64}:
 1  2  3  4
 1  2  3  4
 1  2  3  4
 1  2  3  4
julia> Tridiagonal(A)
4×4 Tridiagonal{Int64, Vector{Int64}}:
 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 Matrix{Int64}:
 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, Matrix{Int64}}:
 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, Matrix{Int64}}:
 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}, Matrix{Complex{Int64}}}:
 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}, Matrix{Complex{Int64}}}:
 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 Matrix{Float64}:
 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, Matrix{Float64}}:
 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 Matrix{Float64}:
 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, Matrix{Float64}}:
 1.0  2.0  3.0
  ⋅   5.0  6.0
  ⋅    ⋅   9.0

LinearAlgebra.UnitLowerTriangular — Type

UnitLowerTriangular(A::AbstractMatrix)

Construct a UnitLowerTriangular view of the matrix A. Such a view has the of the eltype of A on its diagonal.

Examples

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

LinearAlgebra.UnitUpperTriangular — Type

UnitUpperTriangular(A::AbstractMatrix)

Construct an UnitUpperTriangular view of the matrix A. Such a view has the of the eltype of A on its diagonal.

Examples

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

LinearAlgebra.UpperHessenberg — Type

UpperHessenberg(A::AbstractMatrix)

Construct an UpperHessenberg view of the matrix A. Entries of A below the first subdiagonal are ignored.

Efficient algorithms are implemented for H \ b, det(H), and similar.

See also the function to factor any matrix into a similar upper-Hessenberg matrix.

If F::Hessenberg 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 = [1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16]
4×4 Matrix{Int64}:
  1   2   3   4
  5   6   7   8
  9  10  11  12
 13  14  15  16
julia> UpperHessenberg(A)
4×4 UpperHessenberg{Int64, Matrix{Int64}}:
 1   2   3   4
 5   6   7   8
 ⋅  10  11  12
 ⋅   ⋅  15  16

LinearAlgebra.UniformScaling — Type

UniformScaling{T<:Number}

Generically sized uniform scaling operator defined as a scalar times the identity operator, λ*I. Although without an explicit size, it acts similarly to a matrix in many cases and includes support for some indexing. See also .

Julia 1.6

Indexing using ranges is available as of Julia 1.6.

Examples

julia> J = UniformScaling(2.)
UniformScaling{Float64}
2.0*I
julia> A = [1. 2.; 3. 4.]
2×2 Matrix{Float64}:
 1.0  2.0
 3.0  4.0
julia> J*A
2×2 Matrix{Float64}:
 2.0  4.0
 6.0  8.0
julia> J[1:2, 1:2]
2×2 Matrix{Float64}:
 2.0  0.0
 0.0  2.0

source

— 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 Matrix{Complex{Int64}}:
 1+0im  0+2im  3+0im
 0+1im  2+0im  3+0im

LinearAlgebra.UniformScaling — Method

(I::UniformScaling)(n::Integer)

Construct a Diagonal matrix from a UniformScaling.

Julia 1.2

This method is available as of Julia 1.2.

Examples

julia> I(3)
3×3 Diagonal{Bool, Vector{Bool}}:
 1  ⋅  ⋅
 ⋅  1  ⋅
 ⋅  ⋅  1
julia> (0.7*I)(3)
3×3 Diagonal{Float64, Vector{Float64}}:
 0.7   ⋅    ⋅
  ⋅   0.7   ⋅
  ⋅    ⋅   0.7

LinearAlgebra.Factorization — Type

LinearAlgebra.Factorization

Abstract type for a.k.a. matrix decompositions. See online documentation for a list of available matrix factorizations.

LinearAlgebra.LU — Type

LU <: Factorization

Matrix factorization type of the LU factorization of a square matrix A. This is the return type of , the corresponding matrix factorization function.

The individual components of the factorization F::LU can be accessed via getproperty:

Iterating the factorization produces the components F.L, F.U, and F.p.

Examples

julia> A = [4 3; 6 3]
2×2 Matrix{Int64}:
 4  3
 6  3
julia> F = lu(A)
LU{Float64, Matrix{Float64}}
L factor:
2×2 Matrix{Float64}:
 1.0       0.0
 0.666667  1.0
U factor:
2×2 Matrix{Float64}:
 6.0  3.0
 0.0  1.0
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::SparseMatrixCSC; check = true) -> F::UmfpackLU

Compute the LU factorization of a sparse matrix A.

For sparse A with real or complex element type, the return type of F is UmfpackLU{Tv, Ti}, with Tv = or ComplexF64 respectively and Ti is an integer type (Int32 or ).

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.

The individual components of the factorization F can be accessed by indexing:

The relation between F and A is

F.L*F.U == (F.Rs .* A)[F.p, F.q]

F further supports the following functions:

• cond

Note

lu(A::SparseMatrixCSC) uses the UMFPACK library that is part of SuiteSparse. As this library only supports sparse matrices with Float64 or ComplexF64 elements, lu converts A into a copy that is of type SparseMatrixCSC{Float64} or SparseMatrixCSC{ComplexF64} as appropriate.

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 and <.

The individual components of the factorization F can be accessed via :

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:

Examples

julia> A = [4 3; 6 3]
2×2 Matrix{Int64}:
 4  3
 6  3
julia> F = lu(A)
LU{Float64, Matrix{Float64}}
L factor:
2×2 Matrix{Float64}:
 1.0       0.0
 0.666667  1.0
U factor:
2×2 Matrix{Float64}:
 6.0  3.0
 0.0  1.0
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

source

— Function

lu!(F::UmfpackLU, A::SparseMatrixCSC; check=true) -> F::UmfpackLU

Compute the LU factorization of a sparse matrix A, reusing the symbolic factorization of an already existing LU factorization stored in F. The sparse matrix A must have an identical nonzero pattern as the matrix used to create the LU factorization F, otherwise an error is thrown.

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.

Note

lu!(F::UmfpackLU, A::SparseMatrixCSC) uses the UMFPACK library that is part of SuiteSparse. As this library only supports sparse matrices with or ComplexF64 elements, lu! converts A into a copy that is of type SparseMatrixCSC{Float64} or SparseMatrixCSC{ComplexF64} as appropriate.

Julia 1.5

lu! for UmfpackLU requires at least Julia 1.5.

Examples

julia> A = sparse(Float64[1.0 2.0; 0.0 3.0]);
julia> F = lu(A);
julia> B = sparse(Float64[1.0 1.0; 0.0 1.0]);
julia> lu!(F, B);
julia> F \ ones(2)
2-element Vector{Float64}:
 0.0
 1.0

source

lu!(A, pivot=Val(true); check = true) -> LU

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

LinearAlgebra.Cholesky — Type

Cholesky <: Factorization

Matrix factorization type of the Cholesky factorization of a dense symmetric/Hermitian positive definite matrix A. This is the return type of , the corresponding matrix factorization function.

The triangular Cholesky factor can be obtained from the factorization F::Cholesky via F.L and F.U.

Iterating the decomposition produces the components L and U.

Examples

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

source

— Type

CholeskyPivoted

Matrix factorization type of the pivoted Cholesky factorization of a dense symmetric/Hermitian positive semi-definite matrix A. This is the return type of cholesky(_, Val(true)), the corresponding matrix factorization function.

The triangular Cholesky factor can be obtained from the factorization F::CholeskyPivoted via F.L and F.U.

Iterating the decomposition produces the components L and U.

Examples

julia> A = [4. 12. -16.; 12. 37. -43.; -16. -43. 98.]
3×3 Matrix{Float64}:
   4.0   12.0  -16.0
  12.0   37.0  -43.0
 -16.0  -43.0   98.0
julia> C = cholesky(A, Val(true))
CholeskyPivoted{Float64, Matrix{Float64}}
U factor with rank 3:
3×3 UpperTriangular{Float64, Matrix{Float64}}:
 9.89949  -4.34366  -1.61624
  ⋅        4.25825   1.1694
  ⋅         ⋅        0.142334
permutation:
3-element Vector{Int64}:
 3
 2
 1
julia> l, u = C; # destructuring via iteration
julia> l == C.L && u == C.U
true

LinearAlgebra.cholesky — Function

cholesky(A::SparseMatrixCSC; shift = 0.0, check = true, perm = nothing) -> CHOLMOD.Factor

Compute the Cholesky factorization of a sparse positive definite matrix A. A must be a or a Symmetric/ view of a SparseMatrixCSC. Note that even if A doesn’t have the type tag, it must still be symmetric or Hermitian. If perm is not given, a fill-reducing permutation is used. F = cholesky(A) is most frequently used to solve systems of equations with F\b, but also the methods diag, , and logdet are defined for F. You can also extract individual factors from F, using F.L. However, since pivoting is on by default, the factorization is internally represented as A == P'*L*L'*P with a permutation matrix P; using just L without accounting for P will give incorrect answers. To include the effects of permutation, it’s typically preferable to extract “combined” factors like PtL = F.PtL (the equivalent of P'*L) and LtP = F.UP (the equivalent of L'*P).

When check = true, an error is thrown if the decomposition fails. When check = false, responsibility for checking the decomposition’s validity (via ) lies with the user.

Setting the optional shift keyword argument computes the factorization of A+shift*I instead of A. If the perm argument is provided, it should be a permutation of 1:size(A,1) giving the ordering to use (instead of CHOLMOD’s default AMD ordering).

Examples

In the following example, the fill-reducing permutation used is [3, 2, 1]. If perm is set to 1:3 to enforce no permutation, the number of nonzero elements in the factor is 6.

julia> A = [2 1 1; 1 2 0; 1 0 2]
3×3 Matrix{Int64}:
 2  1  1
 1  2  0
 1  0  2
julia> C = cholesky(sparse(A))
SuiteSparse.CHOLMOD.Factor{Float64}
type:    LLt
method:  simplicial
maxnnz:  5
nnz:     5
success: true
julia> C.p
3-element Vector{Int64}:
 3
 2
 1
julia> L = sparse(C.L);
julia> Matrix(L)
3×3 Matrix{Float64}:
 1.41421   0.0       0.0
 0.0       1.41421   0.0
 0.707107  0.707107  1.0
julia> L * L' ≈ A[C.p, C.p]
true
julia> P = sparse(1:3, C.p, ones(3))
3×3 SparseMatrixCSC{Float64, Int64} with 3 stored entries:
  ⋅    ⋅   1.0
  ⋅   1.0   ⋅
 1.0   ⋅    ⋅
julia> P' * L * L' * P ≈ A
true
julia> C = cholesky(sparse(A), perm=1:3)
SuiteSparse.CHOLMOD.Factor{Float64}
type:    LLt
method:  simplicial
maxnnz:  6
nnz:     6
success: true
julia> L = sparse(C.L);
julia> Matrix(L)
3×3 Matrix{Float64}:
 1.41421    0.0       0.0
 0.707107   1.22474   0.0
 0.707107  -0.408248  1.1547
julia> L * L' ≈ A
true

Note

This method uses the CHOLMOD library from SuiteSparse, which only supports doubles or complex doubles. Input matrices not of those element types will be converted to SparseMatrixCSC{Float64} or SparseMatrixCSC{ComplexF64} as appropriate.

Many other functions from CHOLMOD are wrapped but not exported from the Base.SparseArrays.CHOLMOD module.

source

cholesky(A, Val(false); check = true) -> Cholesky

Compute the Cholesky factorization of a dense symmetric positive definite matrix A and return a factorization. The matrix A can either be a Symmetric or 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: , \, , det, and isposdef.

If you have a matrix A that is slightly non-Hermitian due to roundoff errors in its construction, wrap it in Hermitian(A) before passing it to cholesky in order to treat it as perfectly Hermitian.

When check = true, an error is thrown if the decomposition fails. When check = false, responsibility for checking the decomposition’s validity (via ) lies with the user.

Examples

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