The semigroup generated by the similarity class of a singular matrix

TL;DR

This paper proves that the semigroup generated by the similarity class of a singular matrix consists exactly of all matrices with rank less than or equal to that of the original matrix, using canonical forms.

Contribution

It provides a new proof characterizing the semigroup generated by a singular matrix's similarity class via canonical forms.

Findings

The semigroup includes all matrices with rank ≤ that of A.

The proof applies to matrices over any field.

It clarifies the structure of the semigroup generated by similarity classes.

Abstract

Let A be a singular matrix of M_n(K), where K is an arbitrary field. Using canonical forms, we give a new proof that the sub-semigroup of (M_n(K),x) generated by the similarity class of A is the set of matrices of M_n(K) with a rank lesser than or equal to that of A.