Sums Of K-potent Matrices

TL;DR

This paper investigates conditions under which complex matrices can be expressed as sums of $k$-potent matrices and extends previous results to matrices over fields, providing new decomposition insights.

Contribution

It generalizes existing results by characterizing sums of $k$-potent matrices and finite order elements, including a specific decomposition into 14 matrices over fields.

Findings

Conditions for expressing matrices as sums of $k$-potent matrices

Extension of results to matrices over fields

Any matrix in the space can be expressed as a sum of 14 $(k+1)$-potent matrices

Abstract

We study sums of $k$-potent matrices. We show the conditions by which a complex matrix $A$ can be expressed as a sums of $k$-potent matrices. Also we obtain conditions by which a complex matrix $A$ can be expressed as a sum of finite order elements. This generalize some results obtain by Wu. Also we study the sum of $k$-potent matrices in $\mathcal{M}_{\mathcal{C}f}(F)$ with $F$ be a field and proof that any matrix in this space can be expressed as a sum of $14$ $(k+1)$-potent matrices preserving the result obtain by Slowik.