Elements of \(M_n\) as Homomorphisms

The matrix algebra \(M_n(S)\) may also be viewed as the module \({\operatorname{Hom}}(S^{(n)}, S^{(n)})\). At present this will not happen automatically so that in order to treat elements of \(M_n(S)\) as homomorphisms, it is necessary to explicitly coerce the matrix into \({\operatorname{Hom}}(S^{(n)}, S^{(n)})\). However, two fundamental homomorphism-type operators are provided for elements of \(M_n(S)\).

Image(a): AlgMatElt -> ModTup

RowSpace(a): AlgMatElt -> ModTup

Given an element of \(M_n(S)\), return the image of the module \(S^{(n)}\) under the homomorphism represented by the matrix \(a\) (as an element of \(S^{(n)}\)).

Kernel(a): AlgMatElt -> ModTup

NullSpace(a): AlgMatElt -> ModTup

Al: MonStgElt                    Default: "Default"

Given an element of \(M_n(S)\), return the kernel of the homomorphism represented by the matrix \(a\) (as an element of \(S^{(n)}\)).

RowNullSpace(a): AlgMatElt -> ModTup

NullspaceOfTranspose(a): AlgMatVElt -> ModTupRng

Given an element of \(M_n(S)\), return the row nullspace of the homomorphism represented by the matrix \(a\) (as an element of \(S^{(n)}\)). This is equal to the kernel of the transpose of \(a\).

Restrict(a, V): AlgMatElt, ModTupRng -> AlgMatElt

Restrict(a, V): AlgMatElt, ModTupFld -> AlgMatElt

If \(V \simeq S^{(m)}\) is a free submodule of \(S^{(n)}\) stable by an element \(a\) of \(M_n(S)\), returns the restriction of the homomorphism \(a\) to the stable submodule \(V\), as an element of \(M_m(S)\).