For real 3x3 matrices X, the set of solutions to X^n + a_{n-1}X^{n-1} + ... + a_0 I = O is a union of similarity orbits whose covering dimension is always 0, 4, or 6, or else the set is empty.
Simple polynomial equations over $(2 \times 2)$-matrices
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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 $\dim S =2.$
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Simple polynomial equations over (mxm)-matrices
For real 3x3 matrices X, the set of solutions to X^n + a_{n-1}X^{n-1} + ... + a_0 I = O is a union of similarity orbits whose covering dimension is always 0, 4, or 6, or else the set is empty.