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

TL;DR

This paper completely characterizes the solution set of polynomial equations over 2x2 real matrices, showing that the set is a 2-dimensional space and providing a detailed description of its structure.

Contribution

It offers a complete description of solutions to polynomial matrix equations over 2x2 matrices, including their topological and dimensional properties.

Findings

Solution set is fully described.

The solution set has dimension 2.

The structure of solutions is characterized in detail.

Abstract

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