Simple polynomial equations over (mxm)-matrices

TL;DR

This paper studies solutions to polynomial matrix equations over 3x3 real matrices, describing the solution set and its dimension, providing insights into the structure of such equations for matrices of size at least 3.

Contribution

It characterizes the solution set of polynomial equations over (mxm)-matrices for m ≥ 3, especially detailing the case m=3 and computing the dimension of the solution set.

Findings

Solution set for m=3 described explicitly.

Dimension of the solution set calculated for m=3.

Provides a topological description of solutions in Euclidean space.

Abstract

Let $m$ be any integer $\geq 3$. We consider the polynomial equation $$X^n + a_{n-1}\cdot X^{n-1} + \dots + a_1 \cdot X + a_0 \cdot I = O,$$ over $(m \times m)$-matrices $X$ with the real entries, where $I$ is the identity matrix, $O$ is the null matrix, $a_i \in \mathbb R$ for each $i$ and $n \geq 1$. We discuss its solution set $S$ supplied with the natural Euclidean topology. In particular, we describe the solution set $S$ for $m=3$ and calculate its dimension.