transforming a matrix over Z to a diagonal matrix by applying elements of GLT Z.

Smith Normal Form of a matrix.

Property Let m be in Matrix Z and 'SmithNormalForm o ds a b = smithNormalForm m, then holds: (a *> m) <* b == diagonal (rows m) (cols m) ds where ds = (takeN o $ repeat 1) ++ ds'.

Smith Normal Form of a diagonal matrix over Z with strict positive entries.

Property Let d = DiagonalFormStrictPositive d' (DiagonalForm ds a b) be in DiagonalFormStrictPositive Z, m = dgfMatrix d' and s = smithNormalForm' d, then holds: m == dgfMatrix (snfDiagonalForm s).

transforming a matrix over Z to a diagonal matrix with strict positive entries by applying elements of GLT Z.

Property Let m be in Matrix Z and DiagonalFormStrictPositive d = zmxDiagonalFormStrictPositive m, then holds: (a *> m) <* b == diagonal (rows m) (cols m) ds where DiagonalForm ds a b = d.

Note The entries of the diagonal may not be successively divisible, as such it is a pre-form of the Smith Normal Form.

the smith normal form.

Properties Let s = SmithNormalForm o ds a b be in SmithNormalForm Z, then holds:

1. snfDiagonalForm s is valid.
2. For all k in ks holds: 0 < k.
3. For all ..k:k'.. in ks holds: mod k' k == 0.

the underlying diagonal form.

validating diagonal and Smith Normal form for Z-matrices.