Simple polynomial equations over $(2\times 2)$-matrices

Abstract

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 $(2\times 2)$-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 $n\ge 2$. We discuss its solution set $S$ supplied with the natural Euclidean topology. We completely describe $S$. We also show that $\dim S=2.$