# Whether the set $A$ is a compact subset of $M_3(\Bbb R)$.

Consider $$X=\Big\{A \in M_3(\Bbb R): \rho_A(x)=x^3-3x^2+2x-1\Big\}$$ where $$\rho_A(x)$$ is the characteristic polynomial of $$A$$ and $$M_3(\Bbb R)$$ is the space of all $$3 \times 3$$ matrices over $$\Bbb R$$.

Is $$X$$ compact in $$M_3(\Bbb R)$$ ?

My try: I confused with only the $$2x$$ term in $$\rho_A(x)$$. Because , if the $$2x$$ term not appear , then $$X$$ becomes $$X=\{A \in M_3(\Bbb R): \text{trace}(A)=3,\det A=1\}$$ which is unbounded, since $$(\forall n \in \Bbb N):\begin{pmatrix} 1&0 & n\\0&1&0\\0&0&1 \end{pmatrix} \in X$$

But here the problem is, the appearance of $$2$$. I Know $$2=A_{11}+A_{22}+A_{33}$$

so I think in this case the set becomes bounded and closed

Any help?

• I hope my earlier incorrect comment did not mislead you. The characteristic polynomial has three distinct eigenvalues, so any $A \in X$ must have the same eigenvalues. – copper.hat Dec 10 '18 at 3:00
• If you were dealing with complex elements then the set $X$ would be unbounded (take an upper triangular matrix). – copper.hat Dec 10 '18 at 3:13

For example, $$\pmatrix{t & 0 & 1\cr 1-2t & 0 & -2\cr -t^2+3t & 1 & 3-t\cr}$$ has that characteristic polynomial. This is $$S^{-1} A S$$ where $$A = \pmatrix{0 & 0 & 1\cr 1 & 0 & -2\cr 0 & 1 & 3\cr},\ S = \pmatrix{1 & 0 & 0\cr 0 & 1 & 0\cr t & 0 & 1\cr}$$
• Start with a companion matrix for that characteristic polynomial. Choose an $S$ such that $S$ and $S^{-1}$ have entries that are polynomial in $t$... – Robert Israel Dec 10 '18 at 3:22
• Of course there's the trivial case where the minimal polynomial has degree $1$, so the matrix is a constant multiple of $I$. – Robert Israel Dec 10 '18 at 3:25