Matrix functionality

Basic matrix functionality

As well as the Ring and Matrix interfaces, the following functions are provided to manipulate matrices and to set and retrieve entries and other basic data associated with the matrices.

It is possible to create matrices directly, without first creating a corresponding matrix space. The following constructors are necessary, because unfortunately, Julia's matrices and linear algebra cannot be made to work in our context due to two independent problems:

In empty matrices (0 rows or columns) all that is known is the type of the matrix entries,

however for the complex types used in AbstractAlgebra, this information is not sufficient to create elements, hence zero(T) or friends cannot work

Many functions (e.g. det) assume that all types used embed into the real or complex numbers,

in Julia det(ones(Int, (1,1))) == 1.0, so the fact that this is exactly the integer 1 is lost. Furthermore, more general rings cannot be embedded into the reals at all.

matrix(R::Ring, arr::Matrix{T}) where T <: RingElement

Given an $m\times n$ Julia matrix of entries, construct the corresponding AbstractAlgebra.jl matrix over the given ring R, assuming all the entries can be coerced into R.

matrix(R::Ring, r::Int, c::Int, A::Vector{T}) where T <: RingElement

Construct the given $r\times c$ AbstractAlgebra.jl matrix over the ring R whose $(i, j)$ entry is given by A[c*(i - 1) + j], assuming that all the entries can be coerced into R.

zero_matrix(R::Ring, r::Int, c::Int)

Construct the $r\times c$ AbstractAlgebra.jl zero matrix over the ring R.

Examples

julia> M = matrix(ZZ, BigInt[3 1 2; 2 0 1])
[3   1   2]
[2   0   1]

julia> N = matrix(ZZ, 3, 2, BigInt[3, 1, 2, 2, 0, 1])
[3   1]
[2   2]
[0   1]

julia> P = zero_matrix(ZZ, 3, 2)
[0   0]
[0   0]
[0   0]

number_of_rows — Method

number_of_rows(a::MatrixElem{T}) where T <: NCRingElement

Return the number of rows of the given matrix.

number_of_columns — Method

number_of_columns(a::MatrixElem{T}) where T <: NCRingElement

Return the number of columns of the given matrix.

length — Method

length(a::MatrixElem{T}) where T <: NCRingElement

Return the number of entries in the given matrix.

isempty — Method

isempty(a::MatrixElem{T}) where T <: NCRingElement

Return true if a does not contain any entry (i.e. length(a) == 0), and false otherwise.

identity_matrix — Method

identity_matrix(R::NCRing, n::Int)

Return the $n \times n$ identity matrix over $R$ .

identity_matrix — Method

identity_matrix(M::MatElem{T}) where T <: NCRingElement

Construct the identity matrix in the same matrix space as M, i.e. with ones down the diagonal and zeroes elsewhere. M must be square. This is an alias for one(M).

ones_matrix — Method

ones_matrix(R::Ring, r::Int, c::Int)

Return the $r \times c$ ones matrix over $R$ .

scalar_matrix — Method

scalar_matrix(R::NCRing, n::Int, a::NCRingElement)
scalar_matrix(n::Int, a::NCRingElement)

Return the $n \times n$ matrix over R with a along the main diagonal and zeroes elsewhere. If R is not specified, it defaults to parent(a).

diagonal_matrix — Method

diagonal_matrix(x::NCRingElement, m::Int, [n::Int])

Return the $m \times n$ matrix over $R$ with x along the main diagonal and zeroes elsewhere. If n is not specified, it defaults to m.

Examples

julia> diagonal_matrix(ZZ(2), 2, 3)
[2   0   0]
[0   2   0]

julia> diagonal_matrix(QQ(-1), 3)
[-1//1    0//1    0//1]
[ 0//1   -1//1    0//1]
[ 0//1    0//1   -1//1]

zero — Method

zero(x::MatElem{T}, R::NCRing, r::Int, c::Int) where T <: NCRingElement
zero(x::MatElem{T}, r::Int, c::Int) where T <: NCRingElement
zero(x::MatElem{T}, R::NCRing) where T <: NCRingElement
zero(x::MatElem{T}) where T <: NCRingElement

Create an zero matrix over the given ring and dimensions, with defaults based upon the given source matrix x.

one — Method

one(a::MatrixElem{T}) where T <: NCRingElement

Return the identity matrix in the same matrix space as $a$ . If the space does not contain square matrices, an error is thrown.

transpose — Method

transpose(x::MatrixElem{T}) where T <: NCRingElement

Return the transpose of the given matrix.

Examples

julia> R, t = polynomial_ring(QQ, :t)
(Univariate polynomial ring in t over rationals, t)

julia> S = matrix_space(R, 3, 3)
Matrix space of 3 rows and 3 columns
  over univariate polynomial ring in t over rationals

julia> A = S([t + 1 t R(1); t^2 t t; R(-2) t + 2 t^2 + t + 1])
[t + 1       t             1]
[  t^2       t             t]
[   -2   t + 2   t^2 + t + 1]

julia> B = transpose(A)
[t + 1   t^2            -2]
[    t     t         t + 2]
[    1     t   t^2 + t + 1]

transpose(A::SMat) -> SMat

Returns the transpose of $A$ .

tr — Method

tr(x::MatrixElem{T}) where T <: NCRingElement

Return the trace of the matrix $a$ , i.e. the sum of the diagonal elements. We require the matrix to be square.

Examples

julia> R, t = polynomial_ring(QQ, :t)
(Univariate polynomial ring in t over rationals, t)

julia> S = matrix_space(R, 3, 3)
Matrix space of 3 rows and 3 columns
  over univariate polynomial ring in t over rationals

julia> A = S([t + 1 t R(1); t^2 t t; R(-2) t + 2 t^2 + t + 1])
[t + 1       t             1]
[  t^2       t             t]
[   -2   t + 2   t^2 + t + 1]

julia> b = tr(A)
t^2 + 3*t + 2

tr(x::AbstractAssociativeAlgebraElem{T}) where T -> T

Returns the trace of $x$ .

det — Method

det(M::MatrixElem{T}) where {T <: RingElement}

Return the determinant of the matrix $M$ . We assume $M$ is square.

Examples

julia> R, x = polynomial_ring(QQ, :x)
(Univariate polynomial ring in x over rationals, x)

julia> A = R[x 1; 1 x^2];

julia> d = det(A)
x^3 - 1

rank — Method

rank(M::MatrixElem{T}) where {T <: RingElement}

Return the rank of the matrix $M$ .

Examples

julia> A = QQ[1 2; 3 4];

julia> d = rank(A)
2

lower_triangular_matrix — Method

lower_triangular_matrix(L::AbstractVector{T}) where {T <: NCRingElement}

Return the $n$ by $n$ matrix whose entries on and below the main diagonal are the elements of L, and which has zeroes elsewhere. The value of $n$ is determined by the condition that L has length $n(n+1)/2$ .

An exception is thrown if there is no integer $n$ with this property.

Examples

julia> lower_triangular_matrix([1, 2, 3])
[1   0]
[2   3]

upper_triangular_matrix — Method

upper_triangular_matrix(L::AbstractVector{T}) where {T <: NCRingElement}

Return the $n$ by $n$ matrix whose entries on and above the main diagonal are the elements of L, and which has zeroes elsewhere. The value of $n$ is determined by the condition that L has length $n(n+1)/2$ .

An exception is thrown if there is no integer $n$ with this property.

Examples

julia> upper_triangular_matrix([1, 2, 3])
[1   2]
[0   3]

strictly_lower_triangular_matrix — Method

strictly_lower_triangular_matrix(L::AbstractVector{T}) where {T <: NCRingElement}

Return the $n$ by $n$ matrix whose entries below the main diagonal are the elements of L, and which has zeroes elsewhere. The value of $n$ is determined by the condition that L has length $(n-1)n/2$ .

An exception is thrown if there is no integer $n$ with this property.

Examples

julia> strictly_lower_triangular_matrix([1, 2, 3])
[0   0   0]
[1   0   0]
[2   3   0]

strictly_upper_triangular_matrix — Method

strictly_upper_triangular_matrix(L::AbstractVector{T}) where {T <: NCRingElement}

Return the $n$ by $n$ matrix whose entries above the main diagonal are the elements of L, and which has zeroes elsewhere. The value of $n$ is determined by the condition that L has length $(n-1)n/2$ .

An exception is thrown if there is no integer $n$ with this property.

Examples

julia> strictly_upper_triangular_matrix([1, 2, 3])
[0   1   2]
[0   0   3]
[0   0   0]

is_lower_triangular — Method

is_lower_triangular(A::MatrixElem)

Return true if $A$ is an lower triangular matrix, that is, all entries above the main diagonal are zero. Note that this definition also applies to non-square matrices.

Alias for LinearAlgebra.istril.

Examples

julia> is_lower_triangular(QQ[1 2 ; 0 4])
false

julia> is_lower_triangular(QQ[1 0 ; 3 4])
true

julia> is_lower_triangular(QQ[1 2 ;])
false

julia> is_lower_triangular(QQ[1 ; 2])
true

is_upper_triangular — Method

is_upper_triangular(A::MatrixElem)

Return true if $A$ is an upper triangular matrix, that is, all entries below the main diagonal are zero. Note that this definition also applies to non-square matrices.

Alias for LinearAlgebra.istriu.

Examples

julia> is_upper_triangular(QQ[1 2 ; 0 4])
true

julia> is_upper_triangular(QQ[1 0 ; 3 4])
false

julia> is_upper_triangular(QQ[1 2 ;])
true

julia> is_upper_triangular(QQ[1 ; 2])
false

is_diagonal — Method

is_diagonal(A::MatrixElem)

Return true if $A$ is a diagonal matrix, that is, all entries off the main diagonal are zero. Note that this definition also applies to non-square matrices.

Alias for LinearAlgebra.isdiag.

Examples

julia> is_diagonal(QQ[1 0 ; 0 4])
true

julia> is_diagonal(QQ[1 2 ; 3 4])
false

julia> is_diagonal(QQ[1 0 ;])
true

change_base_ring — Method

change_base_ring(R::NCRing, M::MatrixElem{T}) where T <: NCRingElement

Return the matrix obtained by coercing each entry into R.

map — Method

map(f, a::MatrixElem{T}) where T <: NCRingElement

Transform matrix a by applying f on each element. This is equivalent to map_entries(f, a).

map! — Method

map!(f, dst::MatrixElem{T}, src::MatrixElem{U}) where {T <: NCRingElement, U <: NCRingElement}

Like map, but stores the result in dst rather than a new matrix. This is equivalent to map_entries!(f, dst, src).

Inverse

inv — Method

inv(M::MatrixElem{T}) where {T <: RingElement}

Given a non-singular $n\times n$ matrix over a ring, return an $n\times n$ matrix $X$ such that $MX = I_n$ , where $I_n$ is the $n\times n$ identity matrix. If $M$ is not invertible over the base ring an exception is raised.

is_invertible — Method

is_invertible(A::MatrixElem{T}) where {T <: RingElement}

Return true if a given square matrix is invertible, false otherwise. If the inverse should also be computed, use is_invertible_with_inverse.

is_invertible_with_inverse — Method

is_invertible_with_inverse(A::MatrixElem{T}; side::Symbol = :left) where {T <: RingElement}

Given an $n \times m$ matrix $A$ over a ring, return a tuple (flag, B). If side is :right and flag is true, $B$ is a right inverse of $A$ i.e. $A B$ is the $n \times n$ unit matrix. If side is :left and flag is true, $B$ is a left inverse of $A$ i.e. $B A$ is the $m \times m$ unit matrix. If flag is false, no right or left inverse exists.

To get the space of all inverses, note that if $B$ and $C$ are both right inverses, then $A (B - C) = 0$ , and similar for left inverses. Hence from one inverse one can find all by making suitable use of kernel.

pseudo_inv — Method

pseudo_inv(M::MatrixElem{T}) where {T <: RingElement}

Given a non-singular $n\times n$ matrix $M$ over a ring return a tuple $X, d$ consisting of an $n\times n$ matrix $X$ and a denominator $d$ such that $MX = dI_n$ , where $I_n$ is the $n\times n$ identity matrix. The denominator will be the determinant of $M$ up to sign. If $M$ is singular an exception is raised.

Examples

julia> M = matrix(QQ, 3, 3, [1 2 3;4 5 6;0 0 1])
[1//1   2//1   3//1]
[4//1   5//1   6//1]
[0//1   0//1   1//1]

julia> X = inv(M)
[-5//3    2//3    1//1]
[ 4//3   -1//3   -2//1]
[ 0//1    0//1    1//1]

julia> is_invertible(M)
true

julia> is_invertible_with_inverse(M)
(true, [-5//3 2//3 1; 4//3 -1//3 -2; 0 0 1])

julia> pseudo_inv(M)
([5 -2 -3; -4 1 6; 0 0 -3], -3//1)

Submatrices

Submatrices are only available for matrix spaces, not for matrix algebras and generally only available for generic matrices built on Julia arrays.

Submatrices return a new matrix with the same entries as the submatrix with the given range of rows and columns. They are best illustrated with examples.

Examples

julia> M = matrix(ZZ, BigInt[1 2 3; 2 3 4; 3 4 5])
[1   2   3]
[2   3   4]
[3   4   5]

julia> N1 = M[1:2, :]
[1   2   3]
[2   3   4]

julia> N2 = M[:, :]
[1   2   3]
[2   3   4]
[3   4   5]

julia> N3 = M[2:3, 2:3]
[3   4]
[4   5]

As per Julia, AbstractAlgebra supports the construction of matrix views. These allow one to work with a submatrix of a given matrix. Modifying the submatrix also modifies the original matrix.

The syntax for views is as for Julia's own views.

Examples

julia> M = matrix(ZZ, 3, 3, BigInt[1, 2, 3, 2, 3, 4, 3, 4, 5])
[1   2   3]
[2   3   4]
[3   4   5]

julia> N1 = @view M[1:2, :]
[1   2   3]
[2   3   4]

julia> N2 = @view M[:, 1:2]
[1   2]
[2   3]
[3   4]

julia> R = N1*N2
[14   20]
[20   29]

Elementary row and column operations

add_column — Method

add_column(a::MatrixElem{T}, s::RingElement, i::Int, j::Int, rows = 1:nrows(a)) where T <: RingElement

Create a copy of $a$ and add $s$ times the $i$ -th row to the $j$ -th row of $a$ .

By default, the transformation is applied to all rows of $a$ . This can be changed using the optional rows argument.

add_column! — Method

add_column!(a::MatrixElem{T}, s::RingElement, i::Int, j::Int, rows = 1:nrows(a)) where T <: RingElement

Add $s$ times the $i$ -th row to the $j$ -th row of $a$ .

By default, the transformation is applied to all rows of $a$ . This can be changed using the optional rows argument.

add_row — Method

add_row(a::MatrixElem{T}, s::RingElement, i::Int, j::Int, cols = 1:ncols(a)) where T <: RingElement

Create a copy of $a$ and add $s$ times the $i$ -th row to the $j$ -th row of $a$ .

By default, the transformation is applied to all columns of $a$ . This can be changed using the optional cols argument.

add_row! — Method

add_row!(a::MatrixElem{T}, s::RingElement, i::Int, j::Int, cols = 1:ncols(a)) where T <: RingElement

Add $s$ times the $i$ -th row to the $j$ -th row of $a$ .

By default, the transformation is applied to all columns of $a$ . This can be changed using the optional cols argument.

multiply_column — Method

multiply_column(a::MatrixElem{T}, s::RingElement, i::Int, rows = 1:nrows(a)) where T <: RingElement

Create a copy of $a$ and multiply the $i$ th column of $a$ with $s$ .

By default, the transformation is applied to all rows of $a$ . This can be changed using the optional rows argument.

multiply_column! — Method

multiply_column!(a::MatrixElem{T}, s::RingElement, i::Int, rows = 1:nrows(a)) where T <: RingElement

Multiply the $i$ th column of $a$ with $s$ .

By default, the transformation is applied to all rows of $a$ . This can be changed using the optional rows argument.

multiply_row — Method

multiply_row(a::MatrixElem{T}, s::RingElement, i::Int, cols = 1:ncols(a)) where T <: RingElement

Create a copy of $a$ and multiply the $i$ th row of $a$ with $s$ .

By default, the transformation is applied to all columns of $a$ . This can be changed using the optional cols argument.

multiply_row! — Method

multiply_row!(a::MatrixElem{T}, s::RingElement, i::Int, cols = 1:ncols(a)) where T <: RingElement

Multiply the $i$ th row of $a$ with $s$ .

By default, the transformation is applied to all columns of $a$ . This can be changed using the optional cols argument.

Examples

julia> M = ZZ[1 2 3; 2 3 4; 4 5 5]
[1   2   3]
[2   3   4]
[4   5   5]

julia> add_column(M, 2, 3, 1)
[ 7   2   3]
[10   3   4]
[14   5   5]

julia> add_row(M, 1, 2, 3)
[1   2   3]
[2   3   4]
[6   8   9]

julia> multiply_column(M, 2, 3)
[1   2    6]
[2   3    8]
[4   5   10]

julia> multiply_row(M, 2, 3)
[1    2    3]
[2    3    4]
[8   10   10]

Swapping rows and columns

swap_rows — Method

swap_rows(a::MatrixElem{T}, i::Int, j::Int) where T <: NCRingElement

Return a matrix $b$ with the entries of $a$ , where the $i$ th and $j$ th row are swapped.

Examples

julia> M = identity_matrix(ZZ, 3)
[1   0   0]
[0   1   0]
[0   0   1]

julia> swap_rows(M, 1, 2)
[0   1   0]
[1   0   0]
[0   0   1]

julia> M  # was not modified
[1   0   0]
[0   1   0]
[0   0   1]

swap_rows! — Method

swap_rows!(a::MatrixElem{T}, i::Int, j::Int) where T <: NCRingElement

Swap the $i$ th and $j$ th row of $a$ in place. The function returns the mutated matrix (since matrices are assumed to be mutable in AbstractAlgebra.jl).

Examples

julia> M = identity_matrix(ZZ, 3)
[1   0   0]
[0   1   0]
[0   0   1]

julia> swap_rows!(M, 1, 2)
[0   1   0]
[1   0   0]
[0   0   1]

julia> M  # was modified
[0   1   0]
[1   0   0]
[0   0   1]

swap_cols — Method

swap_cols(a::MatrixElem{T}, i::Int, j::Int) where T <: NCRingElement

Return a matrix $b$ with the entries of $a$ , where the $i$ th and $j$ th row are swapped.

swap_cols! — Method

swap_cols!(a::MatrixElem{T}, i::Int, j::Int) where T <: NCRingElement

Swap the $i$ th and $j$ th column of $a$ in place. The function returns the mutated matrix (since matrices are assumed to be mutable in AbstractAlgebra.jl).

Swap the rows of M in place. The function returns the mutated matrix (since matrices are assumed to be mutable in AbstractAlgebra.jl).

Concatenation

The following are only available for matrix spaces, not for matrix algebras.

hcat(M::T, N::T) where T <: MatElem

Return the horizontal concatenation of $M$ and $N$ . It is assumed that the number of rows of $M$ and $N$ are the same.

vcat(M::T, N::T) where T <: MatElem

Return the vertical concatenation of $M$ and $N$ . It is assumed that the number of columns of $M$ and $N$ are the same.

Examples

julia> M = matrix(ZZ, BigInt[1 2 3; 2 3 4; 3 4 5])
[1   2   3]
[2   3   4]
[3   4   5]

julia> N = matrix(ZZ, BigInt[1 0 1; 0 1 0; 1 0 1])
[1   0   1]
[0   1   0]
[1   0   1]

julia> P = hcat(M, N)
[1   2   3   1   0   1]
[2   3   4   0   1   0]
[3   4   5   1   0   1]

julia> Q = vcat(M, N)
[1   2   3]
[2   3   4]
[3   4   5]
[1   0   1]
[0   1   0]
[1   0   1]

Linear solving

See Linear Solving & Kernel

Block diagonal matrix constructors

It is also possible to create block diagonal matrices from a vector of existing matrices. It is also possible to construct them from Julia matrices if one supplies the base ring.

Note that if the input matrices are not square, the output matrix may not be square.

block_diagonal_matrix — Method

block_diagonal_matrix(V::Vector{<:MatElem{T}}) where T <: NCRingElement

Create the block diagonal matrix whose blocks are given by the matrices in V. There must be at least one matrix in V.

block_diagonal_matrix(xs::Vector{SMat})

Return the block diagonal matrix with the matrices in xs on the diagonal. Requires all blocks to have the same base ring.

block_diagonal_matrix — Method

block_diagonal_matrix(R::NCRing, V::Vector{<:Matrix{T}}) where T <: NCRingElement

Create the block diagonal matrix over the ring R whose blocks are given by the matrices in V. Entries are coerced into R upon creation.

Examples

julia> block_diagonal_matrix(ZZ, [[1 2; 3 4], [4 5 6; 7 8 9]])
[1   2   0   0   0]
[3   4   0   0   0]
[0   0   4   5   6]
[0   0   7   8   9]

julia> M = matrix(ZZ, [1 2; 3 4])
[1   2]
[3   4]

julia> N = matrix(ZZ, [4 5 6; 7 8 9])
[4   5   6]
[7   8   9]

julia> block_diagonal_matrix([M, N])
[1   2   0   0   0]
[3   4   0   0   0]
[0   0   4   5   6]
[0   0   7   8   9]

Similar and zero

Both similar and zero construct new matrices, but the entries are either undefined with similar or zero-initialized with zero.

similar(x::MatElem, R::Ring=base_ring(x))
zero(x::MatElem, R::Ring=base_ring(x))

Construct the matrix with the same dimensions as the given matrix, and the same base ring unless explicitly specified.

similar(x::MatElem, R::Ring, r::Int, c::Int)
similar(x::MatElem, r::Int, c::Int)
zero(x::MatElem, R::Ring, r::Int, c::Int)
zero(x::MatElem, r::Int, c::Int)

Construct the $r\times c$ matrix with R as base ring (which defaults to the base ring of the the given matrix). If $x$ belongs to a matrix algebra and $r \neq c$ , an exception is raised, and it's also possible to specify only one Int as the order (e.g. similar(x, n)).

Base.isassigned(M::MatElem, i, j)

Test whether the given matrix has a value associated with indices i and j.

Examples

julia> M = matrix(ZZ, BigInt[3 1 2; 2 0 1])
[3   1   2]
[2   0   1]

julia> isassigned(M, 1, 2)
true

julia> isassigned(M, 4, 4)
false

julia> A = similar(M)
[#undef   #undef   #undef]
[#undef   #undef   #undef]

julia> isassigned(A, 1, 2)
false

julia> B = zero(M)
[0   0   0]
[0   0   0]

julia> C = similar(M, 4, 5)
[#undef   #undef   #undef   #undef   #undef]
[#undef   #undef   #undef   #undef   #undef]
[#undef   #undef   #undef   #undef   #undef]
[#undef   #undef   #undef   #undef   #undef]

julia> base_ring(B)
Integers

julia> D = zero(M, QQ, 2, 2)
[0//1   0//1]
[0//1   0//1]

julia> base_ring(D)
Rationals

LU factorisation

lu — Method

lu(A::MatrixElem{T}, P = SymmetricGroup(nrows(A))) where {T <: FieldElement}

Return a tuple $r, p, L, U$ consisting of the rank of $A$ , a permutation $p$ of $A$ belonging to $P$ , a lower triangular matrix $L$ and an upper triangular matrix $U$ such that $p(A) = LU$ , where $p(A)$ stands for the matrix whose rows are the given permutation $p$ of the rows of $A$ .

fflu — Method

fflu(A::MatrixElem{T}, P = SymmetricGroup(nrows(A))) where {T <: RingElement}

Return a tuple $r, d, p, L, U$ consisting of the rank of $A$ , a denominator $d$ , a permutation $p$ of $A$ belonging to $P$ , a lower triangular matrix $L$ and an upper triangular matrix $U$ such that $p(A) = LDU$ , where $p(A)$ stands for the matrix whose rows are the given permutation $p$ of the rows of $A$ and such that $D$ is the diagonal matrix diag $(p_1, p_1p_2, \ldots, p_{n-2}p_{n-1}, p_{n-1}p_n)$ where the $p_i$ are the inverses of the diagonal entries of $L$ . The denominator $d$ is set to $\pm \mathrm{det}(S)$ where $S$ is an appropriate submatrix of $A$ ( $S = A$ if $A$ is square and nonsingular) and the sign is decided by the parity of the permutation.

Examples

julia> M = matrix(QQ, 3, 3, [1 2 3;4 5 6;0 0 1])
[1//1   2//1   3//1]
[4//1   5//1   6//1]
[0//1   0//1   1//1]

julia> r, P, L, U = lu(M)
(3, (), [1 0 0; 4 1 0; 0 0 1], [1 2 3; 0 -3 -6; 0 0 1])

julia> r, d, P, L, U = fflu(M)
(3, -3//1, (), [1 0 0; 4 -3 0; 0 0 -3], [1 2 3; 0 -3 -6; 0 0 -3])

Reduced row-echelon form

rref_rational — Method

rref_rational(M::MatrixElem{T}) where {T <: RingElement}

Return a tuple $(r, A, d)$ consisting of the rank $r$ of $M$ and a denominator $d$ in the base ring of $M$ and a matrix $A$ such that $A/d$ is the reduced row echelon form of $M$ . Note that the denominator is not usually minimal.

rref — Method

rref(M::MatrixElem{T}) where {T <: FieldElement}

Return a tuple $(r, A)$ consisting of the rank $r$ of $M$ and a reduced row echelon form $A$ of $M$ .

is_rref — Method

is_rref(M::MatrixElem{T}) where {T <: RingElement}

Return true if $M$ is in reduced row echelon form, otherwise return false.

is_rref — Method

is_rref(M::MatrixElem{T}) where {T <: RingElement}

Return true if $M$ is in reduced row echelon form, otherwise return false.

is_rref(M::MatrixElem{T}) where {T <: FieldElement}

Return true if $M$ is in reduced row echelon form, otherwise return false.

Examples

julia> M = matrix(QQ, 3, 3, [1 2 3;4 5 6;0 0 1])
[1//1   2//1   3//1]
[4//1   5//1   6//1]
[0//1   0//1   1//1]

julia> r1, A = rref(M)
(3, [1 0 0; 0 1 0; 0 0 1])

julia> N = matrix(ZZ, 3, 3, [1 2 3;4 5 6;0 0 1])
[1   2   3]
[4   5   6]
[0   0   1]

julia> r2, B = rref_rational(N)
(3, [-3 0 0; 0 -3 0; 0 0 -3], -3)

julia> is_rref(A)
true

julia> is_rref(B)
true

Other functionality

Symmetry testing

is_symmetric — Method

is_symmetric(M::MatrixElem)

Return true if the given matrix is symmetric with respect to its main diagonal, i.e., transpose(M) == M, otherwise return false.

Alias for LinearAlgebra.issymmetric.

Examples

julia> M = matrix(ZZ, [1 2 3; 2 4 5; 3 5 6])
[1   2   3]
[2   4   5]
[3   5   6]

julia> is_symmetric(M)
true

julia> N = matrix(ZZ, [1 2 3; 4 5 6; 7 8 9])
[1   2   3]
[4   5   6]
[7   8   9]

julia> is_symmetric(N)
false

is_skew_symmetric — Method

is_skew_symmetric(M::MatrixElem)

Return true if the given matrix is skew symmetric with respect to its main diagonal, i.e., transpose(M) == -M, otherwise return false.

Examples

julia> M = matrix(ZZ, [0 -1 -2; 1 0 -3; 2 3 0])
[0   -1   -2]
[1    0   -3]
[2    3    0]

julia> is_skew_symmetric(M)
true

Powering

powers — Method

powers(a::Union{NCRingElement, MatElem}, d::Int)

Return an array $M$ of "powers" of a where $M[i + 1] = a^i$ for $i = 0..d$ .

Examples

julia> M = ZZ[1 2 3; 2 3 4; 4 5 5]
[1   2   3]
[2   3   4]
[4   5   5]

julia> A = powers(M, 4)
5-element Vector{AbstractAlgebra.Generic.MatSpaceElem{BigInt}}:
 [1 0 0; 0 1 0; 0 0 1]
 [1 2 3; 2 3 4; 4 5 5]
 [17 23 26; 24 33 38; 34 48 57]
 [167 233 273; 242 337 394; 358 497 579]
 [1725 2398 2798; 2492 3465 4044; 3668 5102 5957]

Gram matrix

gram — Method

gram(x::MatElem)

Return the Gram matrix of $x$ , i.e. if $x$ is an $r\times c$ matrix return the $r\times r$ matrix whose entries $i, j$ are the dot products of the $i$ -th and $j$ -th rows, respectively.

Examples

julia> R, t = polynomial_ring(QQ, :t)
(Univariate polynomial ring in t over rationals, t)

julia> S = matrix_space(R, 3, 3)
Matrix space of 3 rows and 3 columns
  over univariate polynomial ring in t over rationals

julia> A = S([t + 1 t R(1); t^2 t t; R(-2) t + 2 t^2 + t + 1])
[t + 1       t             1]
[  t^2       t             t]
[   -2   t + 2   t^2 + t + 1]

julia> B = gram(A)
[2*t^2 + 2*t + 2   t^3 + 2*t^2 + t                   2*t^2 + t - 1]
[t^3 + 2*t^2 + t       t^4 + 2*t^2                       t^3 + 3*t]
[  2*t^2 + t - 1         t^3 + 3*t   t^4 + 2*t^3 + 4*t^2 + 6*t + 9]

Content

content — Method

content(x::MatrixElem{T}) where T <: RingElement

Return the content of the matrix $a$ , i.e. the greatest common divisor of all its entries, assuming it exists.

Examples

julia> R, t = polynomial_ring(QQ, :t)
(Univariate polynomial ring in t over rationals, t)

julia> S = matrix_space(R, 3, 3)
Matrix space of 3 rows and 3 columns
  over univariate polynomial ring in t over rationals

julia> A = S([t + 1 t R(1); t^2 t t; R(-2) t + 2 t^2 + t + 1])
[t + 1       t             1]
[  t^2       t             t]
[   -2   t + 2   t^2 + t + 1]

julia> b = content(A)
1

Permutation

* — Method

*(P::Perm, x::MatrixElem{T}) where T <: NCRingElement

Apply the pemutation $P$ to the rows of the matrix $x$ and return the result.

Examples

julia> R, t = polynomial_ring(QQ, :t)
(Univariate polynomial ring in t over rationals, t)

julia> S = matrix_space(R, 3, 3)
Matrix space of 3 rows and 3 columns
  over univariate polynomial ring in t over rationals

julia> G = SymmetricGroup(3)
Full symmetric group over 3 elements

julia> A = S([t + 1 t R(1); t^2 t t; R(-2) t + 2 t^2 + t + 1])
[t + 1       t             1]
[  t^2       t             t]
[   -2   t + 2   t^2 + t + 1]

julia> P = G([1, 3, 2])
(2,3)

julia> B = P*A
[t + 1       t             1]
[   -2   t + 2   t^2 + t + 1]
[  t^2       t             t]

Nilpotency

is_nilpotent — Method

is_nilpotent(A::MatrixElem{T}) where {T <: RingElement}

Return if A is nilpotent, i.e. if there exists a natural number $k$ such that $A^k = 0$ . If A is not square an exception is raised.

Minors

minors — Method

minors(A::MatElem, k::Int)

Return an array consisting of the k-minors of A.

Examples

julia> A = ZZ[1 2 3; 4 5 6]
[1   2   3]
[4   5   6]

julia> minors(A, 2)
3-element Vector{BigInt}:
 -3
 -6
 -3

Exterior power

exterior_power — Method

exterior_power(A::MatElem, k::Int) -> MatElem

Return the k-th exterior power of A.

Examples

julia> A = matrix(ZZ, 3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9]);

julia> exterior_power(A, 2)
[-3    -6   -3]
[-6   -12   -6]
[-3    -6   -3]

Pfaffian

pfaffian — Method

pfaffian(M::MatElem)

Return the Pfaffian of a skew-symmetric matrix M.

pfaffians — Method

pfaffians(M::MatElem, k::Int)

Return a vector consisting of the k-Pfaffians of a skew-symmetric matrix M.

Examples

julia> R, x = polynomial_ring(QQ, ["x$i" for i in 1:6])
(Multivariate polynomial ring in 6 variables over rationals, AbstractAlgebra.Generic.MPoly{Rational{BigInt}}[x1, x2, x3, x4, x5, x6])

julia> M = R[0 x[1] x[2] x[3]; -x[1] 0 x[4] x[5]; -x[2] -x[4] 0 x[6]; -x[3] -x[5] -x[6] 0]
[  0    x1    x2   x3]
[-x1     0    x4   x5]
[-x2   -x4     0   x6]
[-x3   -x5   -x6    0]

julia> pfaffian(M)
x1*x6 - x2*x5 + x3*x4

julia> pfaffians(M, 2)
6-element Vector{AbstractAlgebra.Generic.MPoly{Rational{BigInt}}}:
 x1
 x2
 x4
 x3
 x5
 x6

Nullspace

nullspace — Method

nullspace(M::MatElem{T}) where {T <: RingElement}

Return a tuple $(\nu, N)$ consisting of the nullity $\nu$ of $M$ and a basis $N$ (consisting of column vectors) for the right nullspace of $M$ , i.e. such that $MN$ is the zero matrix. If $M$ is an $m\times n$ matrix $N$ will be an $n\times \nu$ matrix. Note that the nullspace is taken to be the vector space kernel over the fraction field of the base ring if the latter is not a field. In AbstractAlgebra we use the name "kernel" for a function to compute an integral kernel.

Examples

julia> R, x = polynomial_ring(ZZ, :x)
(Univariate polynomial ring in x over integers, x)

julia> S = matrix_space(R, 4, 4)
Matrix space of 4 rows and 4 columns
  over univariate polynomial ring in x over integers

julia> M = S([-6*x^2+6*x+12 -12*x^2-21*x-15 -15*x^2+21*x+33 -21*x^2-9*x-9;
              -8*x^2+8*x+16 -16*x^2+38*x-20 90*x^2-82*x-44 60*x^2+54*x-34;
              -4*x^2+4*x+8 -8*x^2+13*x-10 35*x^2-31*x-14 22*x^2+21*x-15;
              -10*x^2+10*x+20 -20*x^2+70*x-25 150*x^2-140*x-85 105*x^2+90*x-50])
[  -6*x^2 + 6*x + 12   -12*x^2 - 21*x - 15    -15*x^2 + 21*x + 33     -21*x^2 - 9*x - 9]
[  -8*x^2 + 8*x + 16   -16*x^2 + 38*x - 20     90*x^2 - 82*x - 44    60*x^2 + 54*x - 34]
[   -4*x^2 + 4*x + 8    -8*x^2 + 13*x - 10     35*x^2 - 31*x - 14    22*x^2 + 21*x - 15]
[-10*x^2 + 10*x + 20   -20*x^2 + 70*x - 25   150*x^2 - 140*x - 85   105*x^2 + 90*x - 50]

julia> n, N = nullspace(M)
(2, [1320*x^4-330*x^2-1320*x-1320 1056*x^4+1254*x^3+1848*x^2-66*x-330; -660*x^4+1320*x^3+1188*x^2-1848*x-1056 -528*x^4+132*x^3+1584*x^2+660*x-264; 396*x^3-396*x^2-792*x 0; 0 396*x^3-396*x^2-792*x])

nullspace(M::MatElem{T}) where {T <: FieldElement}

Return a tuple $(\nu, N)$ consisting of the nullity $\nu$ of $M$ and a basis $N$ (consisting of column vectors) for the right nullspace of $M$ , i.e. such that $MN$ is the zero matrix. If $M$ is an $m\times n$ matrix $N$ will be an $n\times \nu$ matrix.

Hessenberg form

hessenberg — Method

hessenberg(A::MatrixElem{T}) where {T <: RingElement}

Return the Hessenberg form of $M$ , i.e. an upper Hessenberg matrix which is similar to $M$ . The upper Hessenberg form has nonzero entries above and on the diagonal and in the diagonal line immediately below the diagonal.

is_hessenberg — Method

is_hessenberg(A::MatrixElem{T}) where {T <: RingElement}

Return true if $M$ is in Hessenberg form, otherwise returns false.

Examples

julia> R, = residue_ring(ZZ, 7);

julia> M = matrix(R, 4, 4, [1 2 4 3; 2 5 1 0;6 1 3 2; 1 1 3 5])
[1   2   4   3]
[2   5   1   0]
[6   1   3   2]
[1   1   3   5]

julia> A = hessenberg(M)
[1   5   5   3]
[2   1   1   0]
[0   1   3   2]
[0   0   2   2]

julia> is_hessenberg(A)
true

Characteristic polynomial

charpoly — Method

charpoly(Y::MatrixElem{T}) where {T <: RingElement}
charpoly(S::PolyRing{T}, Y::MatrixElem{T}) where {T <: RingElement}

Return the characteristic polynomial $p$ of the square matrix $Y$ . If a polynomial ring $S$ over the same base ring as $Y$ is supplied, the resulting polynomial is an element of it.

Examples

julia> R, = residue_ring(ZZ, 7);

julia> S = matrix_space(R, 4, 4)
Matrix space of 4 rows and 4 columns
  over residue ring of integers modulo 7

julia> T, y = polynomial_ring(R, :y)
(Univariate polynomial ring in y over R, y)

julia> M = S([R(1) R(2) R(4) R(3); R(2) R(5) R(1) R(0);
              R(6) R(1) R(3) R(2); R(1) R(1) R(3) R(5)])
[1   2   4   3]
[2   5   1   0]
[6   1   3   2]
[1   1   3   5]

julia> A = charpoly(T, M)
y^4 + 2*y^2 + 6*y + 2

julia> A = charpoly(M)
x^4 + 2*x^2 + 6*x + 2

Minimal polynomial

minpoly — Method

minpoly(M::MatElem{T}) where {T <: RingElement}
minpoly(S::PolyRing{T}, M::MatElem{T}) where {T <: RingElement}

Return the minimal polynomial $p$ of the square matrix $M$ . If a polynomial ring $S$ over the same base ring as $Y$ is supplied, the resulting polynomial is an element of it.

Examples

julia> R = GF(13)
Finite field F_13

julia> S, y = polynomial_ring(R, :y)
(Univariate polynomial ring in y over R, y)

julia> M = R[7 6 1;
             7 7 5;
             8 12 5]
[7    6   1]
[7    7   5]
[8   12   5]

julia> A = minpoly(S, M)
y^2 + 10*y

julia> A = minpoly(M)
x^2 + 10*x

Transforms

similarity! — Method

similarity!(A::MatrixElem{T}, r::Int, d::T) where {T <: RingElement}

Applies a similarity transform to the $n\times n$ matrix $M$ in-place. Let $P$ be the $n\times n$ identity matrix that has had all zero entries of row $r$ replaced with $d$ , then the transform applied is equivalent to $M = P^{-1}MP$ . We require $M$ to be a square matrix. A similarity transform preserves the minimal and characteristic polynomials of a matrix.

Examples

julia> R, = residue_ring(ZZ, 7);

julia> S = matrix_space(R, 4, 4)
Matrix space of 4 rows and 4 columns
  over residue ring of integers modulo 7

julia> M = S([R(1) R(2) R(4) R(3); R(2) R(5) R(1) R(0);
              R(6) R(1) R(3) R(2); R(1) R(1) R(3) R(5)])
[1   2   4   3]
[2   5   1   0]
[6   1   3   2]
[1   1   3   5]

julia> similarity!(M, 1, R(3))

Hermite normal form

hnf — Method

hnf(A::MatrixElem{T}) where {T <: RingElement}

Return the upper right row Hermite normal form of $A$ .

hnf_with_transform — Method

hnf_with_transform(A)

Return the tuple $H, U$ consisting of the upper right row Hermite normal form $H$ of $A$ together with invertible matrix $U$ such that $UA = H$ .

is_hnf — Method

is_hnf(M::MatrixElem{T}) where T <: RingElement

Return true if the matrix is in Hermite normal form.

Examples

julia> A = matrix(ZZ, [2 3 -1; 3 5 7; 11 1 12])
[ 2   3   -1]
[ 3   5    7]
[11   1   12]

julia> H = hnf(A)
[1   0   255]
[0   1    17]
[0   0   281]

julia> is_hnf(H)
true

julia> H, U = hnf_with_transform(A)
([1 0 255; 0 1 17; 0 0 281], [-47 28 1; -3 2 0; -52 31 1])

julia> U*A
[1   0   255]
[0   1    17]
[0   0   281]

Smith normal form

is_snf — Method

is_snf(A::MatrixElem{T}) where T <: RingElement

Return true if $A$ is in Smith Normal Form.

snf — Method

snf(A::MatrixElem{T}) where {T <: RingElement}

Return the Smith normal form of $A$ .

snf_with_transform — Method

snf_with_transform(A)

Return the tuple $S, T, U$ consisting of the Smith normal form $S$ of $A$ together with invertible matrices $T$ and $U$ such that $TAU = S$ .

Examples

julia> A = matrix(ZZ, [2 3 -1; 3 5 7; 11 1 12])
[ 2   3   -1]
[ 3   5    7]
[11   1   12]

julia> S = snf(A)
[1   0     0]
[0   1     0]
[0   0   281]

julia> S, T, U = snf_with_transform(A)
([1 0 0; 0 1 0; 0 0 281], [1 0 0; 7 1 0; 229 31 1], [0 -3 26; 0 2 -17; -1 0 1])

julia> T*A*U
[1   0     0]
[0   1     0]
[0   0   281]

(Weak) Popov form

AbstractAlgebra.jl provides algorithms for computing the (weak) Popov of a matrix with entries in a univariate polynomial ring over a field.

is_weak_popov — Method

is_weak_popov(P::MatrixElem{T}, rank::Int) where T <: PolyRingElem

Return true if $P$ is a matrix in weak Popov form of the given rank.

weak_popov — Method

weak_popov(A::MatElem{T}) where {T <: PolyRingElem}

Return the weak Popov form of $A$ .

weak_popov_with_transform — Method

weak_popov_with_transform(A::MatElem{T}) where {T <: PolyRingElem}

Compute a tuple $(P, U)$ where $P$ is the weak Popov form of $A$ and $U$ is a transformation matrix so that $P = UA$ .

popov — Method

popov(A::MatElem{T}) where {T <: PolyRingElem}

Return the Popov form of $A$ .

popov_with_transform — Method

popov_with_transform(A::MatElem{T}) where {T <: PolyRingElem}

Compute a tuple $(P, U)$ where $P$ is the Popov form of $A$ and $U$ is a transformation matrix so that $P = UA$ .

Examples

julia> R, x = polynomial_ring(QQ, :x);

julia> A = matrix(R, map(R, Any[1 2 3 x; x 2*x 3*x x^2; x x^2+1 x^3+x^2 x^4+x^2+1]))
[1         2           3               x]
[x       2*x         3*x             x^2]
[x   x^2 + 1   x^3 + x^2   x^4 + x^2 + 1]

julia> P = weak_popov(A)
[   1                        2                    3   x]
[   0                        0                    0   0]
[-x^3   -2*x^3 + x^2 - 2*x + 1   -2*x^3 + x^2 - 3*x   1]

julia> P, U = weak_popov_with_transform(A)
([1 2 3 x; 0 0 0 0; -x^3 -2*x^3+x^2-2*x+1 -2*x^3+x^2-3*x 1], [1 0 0; -x 1 0; -x^3-x 0 1])

julia> U*A
[   1                        2                    3   x]
[   0                        0                    0   0]
[-x^3   -2*x^3 + x^2 - 2*x + 1   -2*x^3 + x^2 - 3*x   1]