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^1 w^5 ]
  [ 0 w^3 w^4 ]
  [ w^1 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 ]