Subsections

• Indexing
• Extracting and Inserting Blocks
• Joining Matrices

Given an element a belonging to the module Hom_R(M, N), return the i-th row of a as an element of the module N.

a[i] := u : ModMatRngElt, RngIntElt, RngElt -> ModMatRngElt

Given an element a belonging to the module Hom_R(M, N), an integer i lying in the range [1, n], and an element u belonging to N, replace the i-th row of a by the vector u.

a[i, j] : ModMatRngElt, RngIntElt, RngIntElt -> RngElt

Given an element a belonging to a submodule to the module Hom_R(M, N), where Rank(M) = m and Rank(N) = n, and positive integers i and j such that 1 <= i, j <= n, return the (i, j)-th component of a (as an element of the ring R).

a[i, j] := t : ModMatRngElt, RngIntElt, RngIntElt, RngElt -> ModMatRngElt

Given an element a belonging to a submodule to the module Hom_R(M, N), where Rank(M) = m and Rank(N) = n, and positive integers i and j such that 1 <= i, j <= n, replace the (i, j)-th component of a by t.

Eltseq(a) : ModMatRngElt -> [ RngElt ]

Given an element a = (a_(ij)), 1 <= i, j <= n, belonging to a submodule to the module Hom_R(M, N), return a as the sequence of elements of S: [a_(11), ..., a_(1n), a_(21), ..., a_(2n), ..., a_(n1), ..., a_(nn)].

> Z  := Integers();
> Z5 := RSpace(Z, 5);
> Z6 := RSpace(Z, 6);
> M  := Hom(Z5, Z6);
> A := M ! [ 3, 1, 0, -4, 2, -12,   
>            2, -4, -5, 5, 23, 6, 
>            8, 0, 0, 1, 5, 12, 
>           -2, -6, 3, 8, 9, 17,
>           11, 12, -6, 4, 2, 27 ];
> A;
[  3   1   0  -4   2 -12]
[  2  -4  -5   5  23   6]
[  8   0   0   1   5  12]
[ -2  -6   3   8   9  17]
[ 11  12  -6   4   2  27]
> A[3];
( 8  0  0  1  5 12)
> A[3, 4];
1
> A[2] := Z6 ! [ 5, -3, 0, 1, 0, 2 ];
> A;
[  3   1   0  -4   2 -12]
[  5  -3   0   1   0   2]
[  8   0   0   1   5  12]
[ -2  -6   3   8   9  17]
[ 11  12  -6   4   2  27]
> A[2, 3] := -6;
> A;
[  3   1   0  -4   2 -12]
[  5  -3  -6   1   0   2]
[  8   0   0   1   5  12]
[ -2  -6   3   8   9  17]
[ 11  12  -6   4   2  27]

ExtractBlock(a, i, j, p, q) : ModMatRngElt, RngIntElt, RngIntElt, RngIntElt, RngIntElt -> ModMatRngElt

Given a homomorphism a belonging to the module Hom_R(M, N), where Rank(M) = m, Rank(N) = n, and integers i, j, p and q satisfying the conditions 1 <= i + p <= m, 1 <= j + q <= n, create the element of Hom(P, Q), where Rank(P) = p, Rank(Q) = q, consisting of the p x q submatrix of a whose first entry is the (i, j)-th entry of a.

Let a be a homomorphism belonging to a submodule Hom_R(M, N), with Rank(M) = m, and Rank(N) = n, and let b be a homomorphism belonging to a submodule of Hom_R(P, Q), with Rank(P) = p, and Rank(Q) = q. Let i and j be integers satisfying the conditions 1 <= i + p <= m, 1 <= j + q <= n. This procedure modifies a so that the p x q block beginning at the (i, j)-th entry is replaced by b.

Given matrices X with r rows and c columns, and Y with r rows and d columns, both over the same coefficient ring R, return the matrix over R with r rows and (c + d) columns obtained by joining X and Y horizontally (placing Y to the right of X).

HorizontalJoin(Q) : [ ModMatRngElt ] -> ModMatRngElt

Given a sequence Q of matrices, each having the same number of rows and being over the same coefficient ring R, return the matrix over R obtained by joining the elements of Q horizontally in order.

Given matrices X with r rows and c columns and Y with s rows and c columns, both over the same coefficient ring R, return the matrix with (r + s) rows and c columns over R obtained by joining X and Y vertically (placing Y underneath X).

VerticalJoin(Q) : [ ModMatRngElt ] -> ModMatRngElt

Given a sequence Q of matrices, each having the same number of columns and being over the same coefficient ring R, return the matrix over R obtained by joining the elements of Q vertically in order.

Given matrices X with a rows and b columns and Y with c rows and d columns, both over the same coefficient ring R, return the matrix with (a + c) rows and (b + d) columns over R obtained by joining X and Y diagonally (placing Y diagonally to the right of and underneath X, with zero blocks above and below the diagonal).

DiagonalJoin(Q) : [ ModMatRngElt ] -> ModMatRngElt

Given a sequence Q of matrices, each being over the same coefficient ring R, return the matrix over R obtained by joining the elements of Q diagonally in order.