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 $\Omega = (0, \alpha)$ with $\alpha > 0$ and $r = \alpha/2$. 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 $B$. Make a drawing in the case where $\alpha = 6$ illustrating questions IV.C.2 and IV.C.3.
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 $\Omega = (0, \alpha)$ with $\alpha > 0$ and $r = \alpha/2$.
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 $B$. Make a drawing in the case where $\alpha = 6$ illustrating questions IV.C.2 and IV.C.3.