Simple polynomial equations over (mxm)-matrices

Abstract

Let $m$ be any integer $\ge 3$. We consider the polynomial equation $$X^n+a_{n-1}\cdot X^{n-1}+\cdots +a_1\cdot X+a_0\cdot I=O,$$ 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 $n\ge 1$. 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.