A note on Products of Nilpotent Matrices

TL;DR

This paper clarifies and rigorously proves the characterization of matrices over a field that can be expressed as a product of two nilpotent matrices, resolving issues in previous proofs and extending understanding of nilpotent matrix products.

Contribution

It provides a detailed, rigorous proof of the known characterization, fixing problematic details in earlier proofs by Sourour and Laffey.

Findings

A matrix over a field is a product of two nilpotent matrices if and only if it is singular, except for nonzero 2x2 nilpotent matrices.

The paper corrects and clarifies the proof strategies used in prior work.

It resolves specific issues in the original proofs to ensure the validity of the characterization.

Abstract

Let $\mathscr{F}$ be a field. A matrix $A$ of order $n \times n$ over $\mathscr{F}$ is a product of two nilpotent matrices if and only if it is singular, except if $A$ is a nonzero nilpotent matrix of order $2 \times 2$. This result was proved independently by Sourour and Laffey. While these results remain true and the general strategies and principles of the proofs correct, there are certain problematic details in the original proofs which are resolved in this article. A detailed and rigorous proof of the result based on Laffey's original proof is provided, and a problematic detail in the Proposition which is the main device underpinning the proof by Sourour is resolved.