Matrix Spaces

Matrix Space Constructors

A matrix space in AbstractAlgebra.jl represents a collection of all matrices with given dimensions and base ring.

In order to construct matrices in AbstractAlgebra.jl, one can first construct the matrix space itself. This is accomplished with the following constructor. We discuss creation of matrix algebras separately in a dedicated section elsewhere in the documentation.

matrix_space(R::Ring, rows::Int, cols::Int)

Construct the space of matrices with the given number of rows and columns over the given base ring.

Here are some examples of creating matrix spaces and making use of the resulting parent objects to coerce various elements into the matrix space.

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()
[0   0   0]
[0   0   0]
[0   0   0]

julia> B = S(12)
[12    0    0]
[ 0   12    0]
[ 0    0   12]

julia> C = S(R(11))
[11    0    0]
[ 0   11    0]
[ 0    0   11]

Basic matrix space functionality

number_of_rows — Method

number_of_rows(a::MatSpace)

Return the number of rows of the given matrix space.

number_of_columns — Method

number_of_columns(a::MatSpace)

Return the number of columns of the given matrix space.

zero — Method

zero(a::MatSpace)

Return the zero matrix in the given matrix space.

one — Method

one(a::MatSpace)

Return the identity matrix of given matrix space. The matrix space must contain square matrices or else an error is thrown.

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 = S([R(2) R(3) R(1); t t + 1 t + 2; R(-1) t^2 t^3])
[ 2       3       1]
[ t   t + 1   t + 2]
[-1     t^2     t^3]

julia> r = number_of_rows(B)
3

julia> c = number_of_columns(B)
3

julia> length(B)
9

julia> isempty(B)
false

julia> M = A + B
[  t + 3         t + 3                   2]
[t^2 + t       2*t + 1             2*t + 2]
[     -3   t^2 + t + 2   t^3 + t^2 + t + 1]

julia> N = 2 + A
[t + 3       t             1]
[  t^2   t + 2             t]
[   -2   t + 2   t^2 + t + 3]

julia> M1 = deepcopy(A)
[t + 1       t             1]
[  t^2       t             t]
[   -2   t + 2   t^2 + t + 1]

julia> A != B
true

julia> isone(one(S))
true

julia> V = A[1:2, :]
[t + 1   t   1]
[  t^2   t   t]

julia> W = A^3
[    3*t^4 + 4*t^3 + t^2 - 3*t - 5            t^4 + 5*t^3 + 10*t^2 + 7*t + 4                 2*t^4 + 7*t^3 + 9*t^2 + 8*t + 1]
[t^5 + 4*t^4 + 3*t^3 - 7*t^2 - 4*t               4*t^4 + 8*t^3 + 7*t^2 + 2*t                 t^5 + 5*t^4 + 9*t^3 + 7*t^2 - t]
[  t^5 + 3*t^4 - 10*t^2 - 16*t - 2   t^5 + 6*t^4 + 12*t^3 + 11*t^2 + 5*t - 2   t^6 + 3*t^5 + 8*t^4 + 15*t^3 + 10*t^2 + t - 5]

julia> Z = divexact(2*A, 2)
[t + 1       t             1]
[  t^2       t             t]
[   -2   t + 2   t^2 + t + 1]

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

julia> M[1, 2] = BigInt(4)
4

julia> c = M[1, 1]
2

Inverses

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.

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

julia> K, = residue_field(R, x^3 + 3x + 1); a = K(x);

julia> S = matrix_space(K, 3, 3)
Matrix space of 3 rows and 3 columns
  over residue field of R modulo x^3 + 3*x + 1

julia> A = S([K(0) 2a + 3 a^2 + 1; a^2 - 2 a - 1 2a; a^2 + 3a + 1 2a K(1)])
[            0   2*x + 3   x^2 + 1]
[      x^2 - 2     x - 1       2*x]
[x^2 + 3*x + 1       2*x         1]

julia> X = inv(A)
[-343//7817*x^2 + 717//7817*x - 2072//7817   -4964//23451*x^2 + 2195//23451*x - 11162//23451    -232//23451*x^2 - 4187//23451*x - 1561//23451]
[ 128//7817*x^2 - 655//7817*x + 2209//7817      599//23451*x^2 - 2027//23451*x - 1327//23451   -1805//23451*x^2 + 2702//23451*x - 7394//23451]
[ 545//7817*x^2 + 570//7817*x + 2016//7817     -1297//23451*x^2 - 5516//23451*x - 337//23451   8254//23451*x^2 - 2053//23451*x + 16519//23451]

julia> is_invertible(A)
true

julia> is_invertible_with_inverse(A)
(true, [-343//7817*x^2+717//7817*x-2072//7817 -4964//23451*x^2+2195//23451*x-11162//23451 -232//23451*x^2-4187//23451*x-1561//23451; 128//7817*x^2-655//7817*x+2209//7817 599//23451*x^2-2027//23451*x-1327//23451 -1805//23451*x^2+2702//23451*x-7394//23451; 545//7817*x^2+570//7817*x+2016//7817 -1297//23451*x^2-5516//23451*x-337//23451 8254//23451*x^2-2053//23451*x+16519//23451])

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

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

julia> A = S([R(0) 2x + 3 x^2 + 1; x^2 - 2 x - 1 2x; x^2 + 3x + 1 2x R(1)])
[            0   2*x + 3   x^2 + 1]
[      x^2 - 2     x - 1       2*x]
[x^2 + 3*x + 1       2*x         1]

julia> X, d = pseudo_inv(A)
([4*x^2-x+1 -2*x^3+3 x^3-5*x^2-5*x-1; -2*x^3-5*x^2-2*x-2 x^4+3*x^3+2*x^2+3*x+1 -x^4+x^2+2; -x^3+2*x^2+2*x-1 -2*x^3-9*x^2-11*x-3 2*x^3+3*x^2-4*x-6], -x^5 - 2*x^4 - 15*x^3 - 18*x^2 - 8*x - 7)

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

julia> K, = residue_field(R, x^3 + 3x + 1); a = K(x);

julia> S = matrix_space(K, 3, 3)
Matrix space of 3 rows and 3 columns
  over residue field of R modulo x^3 + 3*x + 1

julia> A = S([K(0) 2a + 3 a^2 + 1; a^2 - 2 a - 1 2a; a^2 - 2 a - 1 2a])
[      0   2*x + 3   x^2 + 1]
[x^2 - 2     x - 1       2*x]
[x^2 - 2     x - 1       2*x]

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

julia> r, d, P, L, U = fflu(A)
(2, 3*x^2 - 10*x - 8, (1,2), [x^2-2 0 0; 0 3*x^2-10*x-8 0; x^2-2 0 1], [x^2-2 x-1 2*x; 0 3*x^2-10*x-8 -4*x^2-x-2; 0 0 0])

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

julia> K, = residue_field(R, x^3 + 3x + 1); a = K(x);

julia> S = matrix_space(K, 3, 3)
Matrix space of 3 rows and 3 columns
  over residue field of R modulo x^3 + 3*x + 1

julia> M = S([K(0) 2a + 3 a^2 + 1; a^2 - 2 a - 1 2a; a^2 + 3a + 1 2a K(1)])
[            0   2*x + 3   x^2 + 1]
[      x^2 - 2     x - 1       2*x]
[x^2 + 3*x + 1       2*x         1]

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

julia> is_rref(A)
true

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

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

julia> M = S([R(0) 2x + 3 x^2 + 1; x^2 - 2 x - 1 2x; x^2 + 3x + 1 2x R(1)])
[            0   2*x + 3   x^2 + 1]
[      x^2 - 2     x - 1       2*x]
[x^2 + 3*x + 1       2*x         1]

julia> r, A, d = rref_rational(M)
(3, [-x^5-2*x^4-15*x^3-18*x^2-8*x-7 0 0; 0 -x^5-2*x^4-15*x^3-18*x^2-8*x-7 0; 0 0 -x^5-2*x^4-15*x^3-18*x^2-8*x-7], -x^5 - 2*x^4 - 15*x^3 - 18*x^2 - 8*x - 7)

julia> is_rref(A)
true

Matrix element constructors

There are a few ways to construct matrices other than by coercing elements as shown above. The first method is from an array of elements.

This can be done with either two or one dimensional arrays.

(S::MatSpace{T})(A::Matrix{S}) where {S <: RingElement, T <: RingElement}
(S::MatRing{T})(A::Matrix{S}) where {S <: RingElement, T <: RingElement}

Create the matrix in the given space/algebra whose $(i, j)$ entry is given by A[i, j], where S is the type of elements that can be coerced into the base ring of the matrix.

(S::MyMatSpace{T})(A::Vector{S}) where {S <: RingElem, T <: RingElem}
(S::MyMatAlgebra{T})(A::Vector{S}) where {S <: RingElem, T <: RingElem}

Create the matrix in the given space/algebra of matrices (with dimensions $m\times n$ say), whose $(i, j)$ entry is given by A[i*(n - 1) + j] and where S is the type of elements that can be coerced into the base ring of the matrix.

We also provide the following syntax for constructing literal matrices (similar to how Julia arrays can be be constructed).

R[a b c...;...]

Create the matrix over the base ring $R$ consisting of the given rows (separated by semicolons). Each entry is coerced into $R$ automatically. Note that parentheses may be placed around individual entries if the lists would otherwise be ambiguous, e.g. R[1 2; 2 (- 3)].

Also see the Matrix interface for a list of other ways to create matrices.

Examples

julia> S = matrix_space(QQ, 2, 3)
Matrix space of 2 rows and 3 columns
  over rationals

julia> T = matrix_ring(QQ, 2)
Matrix ring of degree 2
  over rationals

julia> M1 = S(Rational{BigInt}[2 3 1; 1 0 4])
[2//1   3//1   1//1]
[1//1   0//1   4//1]

julia> M2 = S(BigInt[2 3 1; 1 0 4])
[2//1   3//1   1//1]
[1//1   0//1   4//1]

julia> M3 = S(BigInt[2, 3, 1, 1, 0, 4])
[2//1   3//1   1//1]
[1//1   0//1   4//1]

julia> N1 = T(Rational{BigInt}[2 3; 1 0])
[2//1   3//1]
[1//1   0//1]

julia> N2 = T(BigInt[2 3; 1 0])
[2//1   3//1]
[1//1   0//1]

julia> N3 = T(BigInt[2, 3, 1, 1])
[2//1   3//1]
[1//1   1//1]

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> M = R[t + 1 1; t^2 0]
[t + 1   1]
[  t^2   0]

julia> N = R[t + 1 2 t] # create a row vector
[t + 1   2   t]

julia> P = R[1; 2; t] # create a column vector
[1]
[2]
[t]

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

julia> is_hessenberg(A)
true