The functions defined here apply to matrices defined over Euclidean Domains.

EchelonForm(a) : ModMatRngElt -> ModMatRngElt, ModMatRngElt

The (row) echelon form of matrix a belonging to a submodule of the module Hom_R(M, N). This function returns two values:

• The (row) echelon form e of a; and
• A matrix b such that b * a = e, i.e. b is a product of elementary matrices that transforms a into echelon form.

The elementary divisors of the matrix a belonging to a submodule of the module Hom_R(M, N). The divisors are returned as a sequence [e_1, ..., e_d], e_i | e_(i + 1) (i=1 , ..., d - 1) of d elements of R, where d is the rank of a.

The row Hermite normal form of an matrix a belonging to a submodule of the module Hom_R(M, N). This function returns two values:

• The Hermite normal form h of a; and
• A matrix b such that b * a = h, i.e. b is the product of elementary matrices that transforms a into Hermite normal form.

Given a matrix a belonging to the matrix ring R over the Euclidean domain S, return the column rank of a.

The Smith normal form for the matrix a belonging to a submodule of the module Hom_R(M, N). This function returns two values:

• The Smith normal form s of a; and
• Matrices b and c such that b * a * c = s, i.e. b is a matrix that transforms a into Smith normal form.

> K<w> := GaloisField(8);
> V3 := VectorSpace(K, 3);
> V4 := VectorSpace(K, 4);
> M := Hom(V4, V3);
> A := M ! [1, w, w^5, 0,  w^3, w^4, w, 1,  w^6, w^3, 1, w^4 ];
> print A;
[  1   w w^5]
[  0 w^3 w^4]
[  w   1 w^6]
[w^3   1 w^4]
> print EchelonForm(A);
[  1   0   0]
[  0   1   0]
[  0   0   1]
[  0   0   0]

> Z  := Integers();
> Z4 := RSpace(Z, 4);
> Z5 := RSpace(Z, 5);
> M  := Hom(Z4, Z5);
> x  := M ! [ 2, -4, 12, 7, 0,   
>             3, -3,  5, -1,  4,
>             2, -1, -4, -5,-12,    
>             0,  3,  6, -2,  0 ];
> print x;
[  2  -4  12   7   0]
[  3  -3   5  -1   4]
[  2  -1  -4  -5 -12]
[  0   3   6  -2   0]
> print Rank(x);
4
> print HermiteForm(x);
[   1    1    1    6 -164]
[   0    3    0   16 -348]
[   0    0    2   13 -200]
[   0    0    0   19 -316]
> print SmithForm(x);
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 2 0]
> print ElementaryDivisors(x);
[ 2 ]