Let $A \in M_p(\mathbb{C})$ and $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ as defined in V.A.4.
Deduce that $E(A)$ exists.

Show that for all $A$ in $\mathcal{M}_n(\mathbb{R})$ we have $A$ dos $A$, that for all $(A,B)$ in $\mathcal{M}_n(\mathbb{R})^2$ if $A$ dos $B$ then $B$ dos $A$, and that for all $(A,B,C)$ in $\mathcal{M}_n(\mathbb{R})^3$ if $A$ dos $B$ and $B$ dos $C$ then $A$ dos $C$.

What are the matrices directly orthogonally similar to $\alpha I_n$ for $\alpha$ real?

What are the matrices directly orthogonally similar to $A$ if $A$ belongs to $\mathrm{SO}(2)$?

What are the matrices directly orthogonally similar to $K_2 = \left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right)$?

Show that $\left(\begin{array}{ll} 0 & 0 \\ 0 & 2 \end{array}\right)$ and $\left(\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right)$ are directly orthogonally similar.

Show that $\left(\begin{array}{ll} 1 & 0 \\ 0 & 2 \end{array}\right)$ and $\left(\begin{array}{rr} 3 & 2 \\ -1 & 0 \end{array}\right)$ are similar but are not orthogonally similar.

Show that $\left(\begin{array}{rr} 3 & 2 \\ -1 & 0 \end{array}\right)$ and its transpose are orthogonally similar but are not directly orthogonally similar.

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 $r$, 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?

Calculate the product ${}^t\left(R_{p,q}(\theta)\right) R_{p,q}(\theta)$. What property of $R_{p,q}(\theta)$ is recognized?

For $i \in \mathbf{Z}$, we define $v_i$ in $\mathbf{C}^{\mathbf{Z}}$ by $$v_i(k) = \left\{\begin{array}{l} 1 \text{ if } k = i \\ 0 \text{ if } k \neq i \end{array}\right.$$
3a. Show that the family $\{v_i\}_{i \in \mathbf{Z}}$ is a basis of $V$.
3b. Calculate $E(v_i)$.

We are given $S \in \mathbf{S}_n$ and $R \in \mathbf{O}_n$. Verify that ${}^t R S R$ is symmetric and that it is similar to $S$.

Let $\lambda, \mu \in \mathbf{C}^{\mathbf{Z}}$. We define the linear maps $F, H \in \mathcal{L}(V)$ respectively by $$H(v_i) = \lambda(i) v_i \quad \text{and} \quad F(v_i) = \mu(i) v_{i+1}, \quad i \in \mathbf{Z}.$$ Show that $H \circ E = E \circ H + 2E$ if and only if for all $i \in \mathbf{Z}, \lambda(i) = \lambda(0) - 2i$.

Let $A \in \mathbf{M}_n$ and $U, V \in \mathbf{O}_n$. Show that $\|UAV\| = \|A\|$.

We assume that the conditions of question 4 are satisfied (i.e., $\lambda(i) = \lambda(0) - 2i$ for all $i \in \mathbf{Z}$). Let $\lambda, \mu \in \mathbf{C}^{\mathbf{Z}}$ and $F, H \in \mathcal{L}(V)$ defined by $H(v_i) = \lambda(i) v_i$ and $F(v_i) = \mu(i) v_{i+1}$. Show that $E \circ F = F \circ E + H$ if and only if for all $i \in \mathbf{Z}$, $$\mu(i) = \mu(0) + i(\lambda(0) - 1) - i^2.$$

We assume that the conditions of question 4 are satisfied (i.e., $\lambda(i) = \lambda(0) - 2i$ for all $i \in \mathbf{Z}$).
6a. Show that for $f \in V$, the vector space spanned by $H^n(f), n \in \mathbf{N}$, is finite-dimensional.
6b. Deduce that a non-zero subspace of $V$ stable under $H$ contains at least one of the $v_i$.

We are given a matrix $S \in \mathbf{S}_n$, an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ and integers $1 \leqslant p < q \leqslant n$ such that $s_{pq} \neq 0$. We define $S' = {}^t R_{p,q}(\theta) S R_{p,q}(\theta)$ and denote its coefficients by $s_{ij}'$.
Show that $s_{qq}' + s_{pp}' = s_{qq} + s_{pp}$.

We are given a matrix $S \in \mathbf{S}_n$, an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ and integers $1 \leqslant p < q \leqslant n$ such that $s_{pq} \neq 0$. We define $S' = {}^t R_{p,q}(\theta) S R_{p,q}(\theta)$ and denote its coefficients by $s_{ij}'$.
Express the coefficients $s_{ij}'$ of $S'$ in terms of those of $S$.

We assume that the conditions of questions 4 and 5 are satisfied and that $\lambda(0) = 0, \mu(0) = 1$. We denote by $\mathbf{C}[X]$ the algebra of polynomials with complex coefficients in one indeterminate $X$.
8a. Show that $\mathbf{C}[E]$ is isomorphic (as an algebra) to $\mathbf{C}[X]$.
8b. Show that $\mathbf{C}[F]$ is isomorphic (as an algebra) to $\mathbf{C}[X]$.
8c. Show that $\mathbf{C}[H]$ is isomorphic (as an algebra) to $\mathbf{C}[X]$.

We are given a matrix $S \in \mathbf{S}_n$, an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ and integers $1 \leqslant p < q \leqslant n$ such that $s_{pq} \neq 0$. We define $S' = {}^t R_{p,q}(\theta) S R_{p,q}(\theta)$ and denote its coefficients by $s_{ij}'$.
We seek an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ for which we have $s_{pq}' = 0$.
(a) Show that $s_{pq}' = 0$ if and only if $t = \tan\theta$ satisfies the equation $$t^2 + \frac{s_{pp} - s_{qq}}{s_{pq}} t - 1 = 0 \tag{1}$$
(b) Show that this equation admits one solution $t_0 \in ]-1, 1]$ and another $t_1 \notin ]-1, 1]$. What is the relationship between the angles $\theta_0$ and $\theta_1$ that correspond to these roots?
(c) In all that follows, we choose one of the two roots $t$ of equation (1). We thus have $s_{pq}' = 0$. A more precise choice will be made starting from question 12. Verify that $s_{pp}' - s_{pp} = t s_{pq}$; establish an analogous formula for $s_{qq}' - s_{qq}$.
(d) We decompose $S$ in the form $S = D + E$ with $D$ diagonal and $E$ with zero diagonal. We similarly decompose $S' = D' + E'$. Calculate $\|E'\|^2$ in terms of $\|E\|^2$ and $\left(s_{pq}\right)^2$.
(e) By justifying that $\|S'\| = \|S\|$, deduce an expression for $\|D'\|^2$ in terms of $\|D\|^2$ and $\left(s_{pq}\right)^2$.

We are given a matrix $S \in \mathbf{S}_n$, an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ and integers $1 \leqslant p < q \leqslant n$ such that $s_{pq} \neq 0$. We define $S' = {}^t R_{p,q}(\theta) S R_{p,q}(\theta)$ and denote its coefficients by $s_{ij}'$.
Show that the coefficients of $S'$ are expressed uniquely in terms of those of $S$ and the root ($t_0$ or $t_1$) that we have chosen.

Let $W_{\ell} = \bigoplus_{0 \leq i < \ell} \mathbf{C} v_i$ and $a \in \mathbf{C}^*$. We consider the element $G_a$ of $\mathcal{L}(W_{\ell})$ whose matrix in the basis $\{v_i\}_{0 \leq i < \ell}$ is: $$\left(\begin{array}{cccccc} 0 & 0 & 0 & \cdots & 0 & a \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ \vdots & 0 & \ddots & \ddots & \vdots & \vdots \\ 0 & \vdots & \ddots & \ddots & 0 & 0 \\ 0 & 0 & \ldots & 0 & 1 & 0 \end{array}\right)$$
10a. Calculate $G_a^{\ell}$. Show that $G_a$ is diagonalizable.
10b. Let $b$ be an $\ell$-th root of $a$. Calculate the eigenvectors of $G_a$ and the associated eigenvalues in terms of $b, q$ and the $v_i$.