In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, tangent to the $x$-axis. We call $L$, with coordinates $(\lambda, 0)$, the point of tangency of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$. Show that there exists a unique non-zero real $\alpha$ such that $A$ is directly orthogonally similar to the matrix $T_{\lambda,\alpha} = \left(\begin{array}{cc} \lambda & \alpha \\ 0 & \lambda \end{array}\right)$. Specify $\alpha$ using the elements of the matrix $A$. Where can we find this number on the eigenvalue circle?
In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, tangent to the $x$-axis. We call $L$, with coordinates $(\lambda, 0)$, the point of tangency of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$.
Show that there exists a unique non-zero real $\alpha$ such that $A$ is directly orthogonally similar to the matrix $T_{\lambda,\alpha} = \left(\begin{array}{cc} \lambda & \alpha \\ 0 & \lambda \end{array}\right)$. Specify $\alpha$ using the elements of the matrix $A$. Where can we find this number on the eigenvalue circle?