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 $\mathcal{C}(\Omega, r)$, and let $\alpha$ be the unique non-zero real such that $A$ is directly orthogonally similar to $T_{\lambda,\alpha} = \left(\begin{array}{cc} \lambda & \alpha \\ 0 & \lambda \end{array}\right)$. Show that there exists an orthonormal direct basis $(e_1, e_2)$ of the plane such that for all $u$ in $\mathbb{R}^2$, we have $f_A(u) = \lambda u + \alpha (e_2 \mid u) e_1$.
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 $\mathcal{C}(\Omega, r)$, and let $\alpha$ be the unique non-zero real such that $A$ is directly orthogonally similar to $T_{\lambda,\alpha} = \left(\begin{array}{cc} \lambda & \alpha \\ 0 & \lambda \end{array}\right)$.
Show that there exists an orthonormal direct basis $(e_1, e_2)$ of the plane such that for all $u$ in $\mathbb{R}^2$, we have $f_A(u) = \lambda u + \alpha (e_2 \mid u) e_1$.