Prove that $A$ is a compact subset of $M_3(\Bbb R) \cong \Bbb R^9$.

Question

Consider the set $A ⊂ M_3(\Bbb R)$ of $3 × 3$ real matrices with characteristic polynomial $x^3 − 3x^2 + 2x − 1$. Then $A$ is a compact subset of $M_3(\Bbb R) \cong \Bbb R^9$.

Now this polynomial is irreducible over rationals and it is equivalent to the companion matrix $$ \begin{pmatrix} 0 & 0 & 1 \\ 1& 0 & -2 \\ 0 & 1 & 3 \\ \end{pmatrix}. $$

Now am I missing any important theorems or details?

Answer 1

The statement is false. If $m\in\mathbb N$, then the characteristic polynomial of the matrix$$\begin{bmatrix}0&0&m\\\frac1{m^2}&0&-\frac2m\\0&m&3\end{bmatrix}$$is also $x^3-3x^2+2x-1$. Therefore, your set is unbounded. In particular, it is not compact.

Answer 2

Let $B$ be your matrix and $V_t=\operatorname{diag}\{ t,1,1\}$. Then $V_t B V_t^{-1}= \begin{bmatrix}0 & 0 & t \\ {1 \over t} & 0 & -2 \\ 0 & 1 & 3\end{bmatrix} \in A$ for all $t \neq 0$.