In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, intersecting the $x$-axis. We denote by $L_1$ and $L_2$, with coordinates respectively $(\lambda_1, 0)$ and $(\lambda_2, 0)$, with $\lambda_1 < \lambda_2$, the two intersection points of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$. We keep the notations $E, F, G, H$ from III.D.
Show that if $c \neq 0$, then $\left(\overrightarrow{L_1 E}, \overrightarrow{L_2 E}\right)$ is a basis of $\mathbb{R}^2$ consisting of eigenvectors for $f_A$.

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, intersecting the $x$-axis. We denote by $L_1$ and $L_2$, with coordinates respectively $(\lambda_1, 0)$ and $(\lambda_2, 0)$, with $\lambda_1 < \lambda_2$, the two intersection points of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$. We keep the notations $E, F, G, H$ from III.D.
When $c = 0$, can we give a basis of eigenvectors for $f_A$ using the eigenvalue circle and the eigenvalue rectangle?

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, intersecting the $x$-axis. We denote by $L_1$ and $L_2$, with coordinates respectively $(\lambda_1, 0)$ and $(\lambda_2, 0)$, with $\lambda_1 < \lambda_2$, the two intersection points of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$. We keep the notations $E, F, G, H$ from III.D.
Show that the square of the cosine of the angle between two eigenvectors of $A$ associated with two distinct eigenvalues is determined by the circle $\mathcal{C}(\Omega, r)$, and does not depend on the choice of a matrix $A$ whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$ (one may, if deemed useful, introduce the orthogonal projection of $\Omega$ onto the $x$-axis). What about if $A$ is symmetric?

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 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 $\mathcal{C}(\Omega, r)$.
Is the matrix $A$ diagonalizable? Is it trigonalizable?

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 $\mathcal{C}(\Omega, r)$. We keep the notations $E, F, G, H$ from III.D.
Can we give an eigenvector using the points $L, E, F, G$ and $H$?

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, tangent to the $x$-axis. Let $A$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$.
What can be said about matrices whose eigenvalue circle is tangent to the $x$-axis and whose center is located on the $y$-axis?

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 $\mathcal{C}(\Omega, r)$, 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 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 $\mathcal{C}(\Omega, r)$.
Does there exist a matrix $P$ in $\mathrm{GL}_2(\mathbb{R})$ such that the matrix $P^{-1}AP$ is diagonal? Does there exist a matrix $P$ in $\mathrm{GL}_2(\mathbb{R})$ such that the matrix $P^{-1}AP$ is upper triangular?

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 $K$.

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 $\mathcal{C}(\Omega, r)$. Let $U$ be one of the points of $\mathcal{C}(\Omega, r)$ at which the tangent line contains $K$.
Express the eigenvalues of $A$, considered as an element of $\mathcal{M}_2(\mathbb{C})$, using the abscissa of $K$ and the distance $KU$ from $K$ to $U$.

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 $\Omega = (0, \alpha)$ 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.

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 $J = (0,1)$.
Calculate $\psi_A(x,y,z)$.

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 $z = 0$.

Let $F$ and $G$ be two supplementary subspaces of $E$ and $s$ the symmetry with respect to $F$ parallel to $G$. a) Show that $F = F_s$ and $G = G_s$. b) Show that $s \circ s = \operatorname{Id}_E$. Deduce that $s$ is an automorphism of $E$. c) Determine the eigenvalues and eigenspaces of $s$. We will discuss according to the subspaces $F$ and $G$.

Let $s$ be an endomorphism of $E$ such that $s \circ s = \operatorname{Id}_E$. We set $F = \operatorname{Ker}(s - \operatorname{Id}_E)$ and $G = \operatorname{Ker}(s + \operatorname{Id}_E)$. a) Show that $F$ and $G$ are two supplementary subspaces of $E$. b) Deduce that $s$ is a symmetry and specify its elements.

Let $s$ and $t$ be two symmetries of $E$ that anticommute, that is, such that $s \circ t + t \circ s = 0$. a) Prove the equalities $t(F_s) = G_s$ and $t(G_s) = F_s$. b) Deduce that $F_s$ and $G_s$ have the same dimension and that $n$ is even.

Let $H = l^2(\mathbb{N})$, the vector space of real sequences that are square-summable: $$l^2(\mathbb{N}) = \left\{(u_n)_{n \in \mathbb{N}} \in \mathbb{R}^{\mathbb{N}}, \quad \sum_{n=0}^{+\infty} |u_n|^2 < +\infty\right\}$$ equipped with the norm: $$\|u\|_2 = \sqrt{\sum_{n=0}^{+\infty} |u_n|^2}$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$ in $H = l^2(\mathbb{N})$.
Calculate the point spectrum of $S$ and $V$.

We now work in the space of bounded real sequences $F = l^{\infty}(\mathbb{N})$ equipped with the norm $$\|u\|_{\infty} = \sup_{n \in \mathbb{N}} |u_n|$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$.
Calculate the point spectrum of $S$ and $V$ in $F$.

We now work in the space of bounded real sequences $F = l^{\infty}(\mathbb{N})$ equipped with the norm $$\|u\|_{\infty} = \sup_{n \in \mathbb{N}} |u_n|$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$.
Calculate the spectrum of $S$ and $V$ in $F$.

In this problem, $\mathbb{K}$ denotes the field $\mathbb{R}$ or the field $\mathbb{C}$ and $E$ is a non-zero $\mathbb{K}$-vector space. If $f$ is an endomorphism of $E$, for every subspace $F$ of $E$ stable by $f$ we denote by $f_F$ the endomorphism of $F$ induced by $f$.
Show that a line $F$ generated by a vector $u$ is stable by $f$ if and only if $u$ is an eigenvector of $f$.