Sums of two nilpotent quaternionic matrices

TL;DR

This paper characterizes when square matrices over a quaternion division algebra can be expressed as the sum of two nilpotent matrices, extending previous results that showed such matrices can be expressed as the sum of three.

Contribution

It provides a simple criterion for representing quaternionic matrices as sums of two nilpotent matrices, including special cases involving scalar and unispectral matrices.

Findings

Matrices with pure quaternionic trace are sums of three nilpotent matrices.

Characterization of matrices as sums of two nilpotent matrices for dimensions n ≥ 3.

Special cases include scalar matrices, rank-1 perturbations, and 3x3 unispectral diagonalisable matrices.

Abstract

Let $\mathcal{Q}$ be a quaternion division algebra over a field, and $n \geq 2$ be an integer. In a recent article, de La Cruz et al have proved that every $n$-by-$n$ matrix with entries in $\mathcal{Q}$ and pure quaternionic trace is the sum of three nilpotent matrices, and they have shown that some are not the sum of two nilpotent matrices. Here, we give a simple characterization of the square matrices with entries in $\mathcal{Q}$ that are the sum of two nilpotent ones. When $n \geq 3$, the special cases involve the scalar matrices and their perturbations by rank $1$ matrices, as well as the very special case of $3$-by-$3$ unispectral diagonalisable matrices.