We denote $K_2 = \left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right)$. Verify that the endomorphism $\sigma_0 = f_{K_2}$ is a reflection (orthogonal symmetry with respect to a line in the plane) and specify its eigenelements.

For all real $t$, specify the endomorphism $\sigma_t$ canonically associated with $R_t^{-1} K_2 R_t$ and in particular its eigenelements.

Let $A \in \mathcal{M}_2(\mathbb{R})$.
a) Determine the matrices whose eigenvalue circle has zero radius and characterize geometrically their canonically associated endomorphism.
b) When the eigenvalue circle is reduced to its center, specify the canonically associated endomorphism, on the one hand when this center belongs to the unit circle (with center the origin $O=(0,0)$ and radius 1) and on the other hand when this center belongs to the $x$-axis.
c) What can be said about the matrix $A$ and $f_A$ when the eigenvalue circle $\mathcal{CP}_A$ has zero radius and center belonging to the $y$-axis $\{0\} \times \mathbb{R}$?

Show that two matrices $A$ and $B$ of $\mathcal{M}_2(\mathbb{R})$ are directly orthogonally similar if and only if they have the same eigenvalue circle.

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ we consider the four points (possibly coinciding) $E = (d,-c)$, $F = (a,b)$, $G = (d,b)$ and $H = (a,-c)$.
In the case where $A = \left(\begin{array}{rr} 1 & 7 \\ -1 & 3 \end{array}\right)$, draw the circle and the quadrilateral $EHFG$.

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ we consider the four points (possibly coinciding) $E = (d,-c)$, $F = (a,b)$, $G = (d,b)$ and $H = (a,-c)$.
When the four points $E, F, G$ and $H$ are distinct show that they are the vertices of a rectangle, which we will call the eigenvalue rectangle of $A$.

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ we consider the four points (possibly coinciding) $E = (d,-c)$, $F = (a,b)$, $G = (d,b)$ and $H = (a,-c)$.
Specify the matrices for which some of these points coincide, that is, when the rectangle is flattened.

Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$. Show that there exists a unique triplet $(\alpha, \beta, \gamma)$ of $\mathbb{R}^2 \times \mathbb{R}_+$ that we will specify, such that $A$ is directly orthogonally similar to $\left(\begin{array}{cc} \alpha+\gamma & -\beta \\ \beta & \alpha-\gamma \end{array}\right)$.

Show that for every endomorphism $f$ of $\mathbb{R}^2$, there exist non-negative reals $k$ and $\ell$, a plane rotation $\rho_t$ and a reflection $\sigma_{t'}$ such that $f = k\rho_t + \ell\sigma_{t'}$.

Write a procedure or function in Maple or Mathematica that takes as input a quadruple $(a,b,c,d)$ of reals and returns a quadruple $(k, \ell, t, t')$ such that if $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ we have $f_A = k\rho_t + \ell\sigma_{t'}$.

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, intersecting the $x$-axis. Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$.
Characterize geometrically $f_A$ when $\Omega = O$, with $O = (0,0)$, and $r = 1$.

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, intersecting the $x$-axis. Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$.
Characterize geometrically $f_A$ when $\mathcal{CP}_A$ is the circle with diameter the segment $[O, I]$ with $I = (1,0)$.

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r \geqslant 0$ disjoint from the $x$-axis. We denote by $K$ the orthogonal projection of $\Omega$ onto the $x$-axis. Let $A$ be a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$.
Determine the points of $\mathcal{C}(\Omega, r)$ at which the tangent line to $\mathcal{C}(\Omega, r)$ contains $K$.

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ and $(x,y,z)$ in $\mathbb{R}^3$, we denote by $\psi_A(x,y,z)$ the real part of the determinant of the matrix $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$, where $\mathrm{i}$ is the complex affix of the point $J = (0,1)$.
Specify the nature of the quadric $\mathcal{H}_A$ with equation $\psi_A(x,y,z) = 0$.

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, let $\mathcal{H}_A$ be the quadric with equation $\psi_A(x,y,z) = 0$ where $\psi_A(x,y,z)$ is the real part of the determinant of $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$.
Specify the intersection of $\mathcal{H}_A$ with the plane with equation $z = 0$.

We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. We assume that $E$ is of dimension 4 and that $u$ has no real eigenvalue. Let $\widetilde{\mathcal{B}} = (z_1, z_2, y_1, -y_2)$ be the basis constructed in questions 16--18, where $\operatorname{Mat}_{\widetilde{\mathcal{B}}}(\omega) = J_4$.
Show that there exist $r > 0$ and $\theta \in \mathbb { R } \backslash \pi \mathbb { Z }$ such that $$\operatorname { Mat } _ { \widetilde { \mathcal { B } } } ( u ) = r \left( \begin{array} { c c } R _ { \theta } & 0 \\ 0 & R _ { - \theta } \end{array} \right)$$ where $R _ { \theta } = \left( \begin{array} { c c } \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right)$, and conclude that $$\operatorname { Mat } _ { \widetilde { \mathcal { B } } } ( \omega ) = J _ { 4 } \quad \text{and} \quad \operatorname { Mat } _ { \widetilde { \mathcal { B } } } \left( \omega _ { 1 } \right) = r \left( \begin{array} { c c } 0 & - R _ { - \theta } \\ R _ { \theta } & 0 \end{array} \right).$$