In this section, we consider in $\mathbb{R}^2$ a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r$ and $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$. In this question, $\Omega = (\alpha, \beta) \in \mathbb{R} \times \mathbb{R}^*$, $r = |\beta|$ and $E = (\alpha + |\beta|, \beta)$. Specify the eigenvalues of $A$ and give a matrix $B$ whose off-diagonal entries are opposite and which is directly orthogonally similar to $A$, as well as an orthogonal decomposition of the endomorphism canonically associated with $B$.
In this section, we consider in $\mathbb{R}^2$ a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r$ and $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$.
In this question, $\Omega = (\alpha, \beta) \in \mathbb{R} \times \mathbb{R}^*$, $r = |\beta|$ and $E = (\alpha + |\beta|, \beta)$.
Specify the eigenvalues of $A$ and give a matrix $B$ whose off-diagonal entries are opposite and which is directly orthogonally similar to $A$, as well as an orthogonal decomposition of the endomorphism canonically associated with $B$.