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# Understand the problem

True or false: Consider the set $A \subset M_3(\Bbb R)$ of $3 \times 3$ real matrices with characteristic polynomial
$x^3-3x^2+2x-1$. Then $A$ is a compact subset of $M_3(\Bbb R) \cong \Bbb R^9$.

##### Source of the problem
TIFR GS 2019, Part B Problem 8
Lie algebra
Hard
##### Suggested Book
Lie algebra; Brian Hall

# Start with hints

Do you really need a hint? Try it first!

Start with a companion matrix for that characteristic polynomial. Choose an $latex S$ such that $latex S$ and $S^{-1}$ have entries that are polynomial in $t$.

$S=\begin{pmatrix} 1&0&0\\ 0&1&0\\ t&0&1\\ \end{pmatrix}$ and $A=\begin{pmatrix} 0&0&1 \\ 1&0&-2 \\ 0&1&3\\ \end{pmatrix}$ where $A$ is the companion matrix. Now calculate $S^{-1}AS$.
The resulting matrix is $\begin{pmatrix} t & 0& 1 \\ 1-2t & 0 & -2 \\ -t^2+3t & 1 & 3-t \\ \end{pmatrix}$. It has that characteristic polynomial.
Now the first entry $a_{11}=t$ which is unbounded with $t \in \Bbb R$.

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