In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r \geqslant 0$ disjoint from the $x$-axis. We denote by $K$ the orthogonal projection of $\Omega$ onto the $x$-axis. Let $A$ be a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$. Does there exist a matrix $P$ in $\mathrm{GL}_2(\mathbb{R})$ such that the matrix $P^{-1}AP$ is diagonal? Does there exist a matrix $P$ in $\mathrm{GL}_2(\mathbb{R})$ such that the matrix $P^{-1}AP$ is upper triangular?
In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r \geqslant 0$ disjoint from the $x$-axis. We denote by $K$ the orthogonal projection of $\Omega$ onto the $x$-axis. Let $A$ be a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$.
Does there exist a matrix $P$ in $\mathrm{GL}_2(\mathbb{R})$ such that the matrix $P^{-1}AP$ is diagonal? Does there exist a matrix $P$ in $\mathrm{GL}_2(\mathbb{R})$ such that the matrix $P^{-1}AP$ is upper triangular?