In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, intersecting the $x$-axis. We denote by $L_1$ and $L_2$, with coordinates respectively $(\lambda_1, 0)$ and $(\lambda_2, 0)$, with $\lambda_1 < \lambda_2$, the two intersection points of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$. We keep the notations $E, F, G, H$ from III.D. Show that $A$ is diagonalizable.
In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, intersecting the $x$-axis. We denote by $L_1$ and $L_2$, with coordinates respectively $(\lambda_1, 0)$ and $(\lambda_2, 0)$, with $\lambda_1 < \lambda_2$, the two intersection points of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$. We keep the notations $E, F, G, H$ from III.D.
Show that $A$ is diagonalizable.