Sums and products of square-zero matrices

TL;DR

This paper reviews and consolidates research on which matrices can be expressed as sums or products of square-zero matrices, providing a comprehensive theoretical overview of the topic.

Contribution

It offers a unified, holistic presentation of 25 years of research on sums and products of square-zero matrices, serving as an introductory resource.

Findings

Characterization of matrices as sums of square-zero matrices

Conditions for matrices to be expressed as products of square-zero matrices

Summary of existing results within the broader context of nilpotent matrices

Abstract

Which matrices can be written as sums or products of square-zero matrices? This question is the central premise of this dissertation. Over the past 25 years a significant body of research on products and linear combinations of square-zero matrices has developed, and it is the aim of this study to present this body of research in a consolidated, holistic format, that could serve as a theoretical introduction to the subject. The content of the research is presented in three parts: first results within the broader context of sums and products of nilpotent matrices are discussed, then products of square-zero matrices, and finally sums of square-zero matrices.