Using the results of Q30 and Q31, deduce that there exists $j \in \llbracket 1, q \rrbracket$ such that $\lambda = 2\cos\frac{j\pi}{q+1}$.

Let $N = \left(\begin{array}{ccccc} 0 & 0 & \cdots & \cdots & 0 \\ 1 & 0 & & & \vdots \\ 0 & \ddots & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)$.
Show that the set of matrices that commute with $N$ is the set of lower triangular Toeplitz matrices.

Determine the spectrum of $B$ (the square matrix of order $q$ with coefficient $(i,j)$ equal to 1 if $|i-j|=1$ and 0 otherwise) and a basis of eigenvectors of $B$.

Show that if $i$ and $j$ are in $\llbracket -n+1, n-1 \rrbracket$, if $A \in \Delta_i$ and $B \in \Delta_j$, then $AB \in \Delta_{i+j}$.

With $A = (1-2r)I_{q} + rB$, $r = \frac{\tau}{\delta^2}$, $\delta = \frac{1}{q+1}$, give a necessary and sufficient condition on $r$ for the sequence $\left(F_{n}\right)_{n \in \mathbb{N}}$ to be bounded regardless of the choices of $q$ and $F_{0}$.

Deduce that if $A \in H_i$ and $B \in H_j$, then $AB \in H_{i+j}$.

Let $C$ be a nilpotent matrix. Show that $I_n + C$ is invertible and that $$\left(I_n + C\right)^{-1} = I_n - C + C^2 + \cdots + (-1)^{n-1} C^{n-1}$$

We assume that $k \geqslant 0$ and that $C$ is a matrix in $\Delta_{k+1}$. We set $P = I_n + C$. Show that $P$ is invertible and that $P^{-1} \in \bigoplus_{p=0}^{n-1} \Delta_{p(k+1)}$.

We assume that $k \geqslant 0$ and that $C$ is a matrix in $\Delta_{k+1}$. We set $P = I_n + C$. We consider the endomorphism $\varphi$ of $\mathcal{M}_n(\mathbb{R})$ defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), \varphi: M \mapsto P^{-1}MP$.
Let $i \in \llbracket 0, k \rrbracket$ and $M \in \Delta_i$. Show that there exists $M'$ in $H_{k+1}$ such that $\varphi(M) = M + M'$.

We assume that $k \geqslant 0$ and that $C$ is a matrix in $\Delta_{k+1}$. We set $P = I_n + C$. We consider the endomorphism $\varphi$ of $\mathcal{M}_n(\mathbb{R})$ defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), \varphi: M \mapsto P^{-1}MP$. The matrix $N$ being the matrix defined in III.A.4, show that there exists $N'$ in $H_{k+1}$ such that $$\varphi(N) = N + NC - CN + N'$$

We assume that $k \geqslant 0$ and that $C$ is a matrix in $\Delta_{k+1}$. We set $P = I_n + C$. We consider the endomorphism $\varphi$ of $\mathcal{M}_n(\mathbb{R})$ defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), \varphi: M \mapsto P^{-1}MP$. Let $T$ be an upper triangular matrix. We set $A = N + T$, $B = \varphi(A)$. Show that $B \in H_{-1}$ and that $$\begin{cases} \forall i \in \llbracket -1, k-1 \rrbracket, \quad B^{(i)} = A^{(i)} \\ B^{(k)} = A^{(k)} + NC - CN \end{cases}$$

We define the operators $$\mathcal{S}: \begin{cases} \mathcal{M}_n(\mathbb{R}) \rightarrow \mathcal{M}_n(\mathbb{R}) \\ X \mapsto NX - XN \end{cases} \quad \text{and} \quad \mathcal{S}^*: \begin{cases} \mathcal{M}_n(\mathbb{R}) \rightarrow \mathcal{M}_n(\mathbb{R}) \\ X \mapsto {}^t N X - X {}^t N \end{cases}$$ Show that the kernel of $\mathcal{S}$ is the set of real Toeplitz matrices that are lower triangular. We admit that the kernel of $\mathcal{S}^*$ is the set of real Toeplitz matrices that are upper triangular.

Let $M \in \mathcal{M}_n(\mathbb{K})$. Show that $M$ and $M^{\top}$ have the same spectrum.

When $x \in \mathbb{C}^n$, verify that $\|x\|_2^2 = \bar{x}^T x$.

Are the subsets $\mathrm{T}_{n}(\mathbb{K})$ and $\mathrm{T}_{n}^{+}(\mathbb{K})$ subalgebras of $\mathcal{M}_{n}(\mathbb{K})$?

Let $M \in \mathcal{M}_n(\mathbb{K})$. Show that $M^{\top}$ is diagonalisable if and only if $M$ is diagonalisable.

Let $U \in \mathcal{M}_n(\mathbb{C})$ be a unitary matrix. Show that $\|Ux\|_2 = \|x\|_2$ for all $x \in \mathbb{C}^n$.

Are the subsets $S_{2}(\mathbb{K})$ and $A_{2}(\mathbb{K})$ subalgebras of $\mathcal{M}_{2}(\mathbb{K})$?

If $D \in \mathcal{M}_n(\mathbb{C})$ is a diagonal matrix whose diagonal coefficients are $d_0, \ldots, d_{n-1}$, show that $\|D\| = \max_{0 \leq i \leq n-1} |d_i|$.

Suppose $n \geqslant 3$. Are the subsets $\mathrm{S}_{n}(\mathbb{K})$ and $\mathrm{A}_{n}(\mathbb{K})$ subalgebras of $\mathcal{M}_{n}(\mathbb{K})$?

Let $A, B \in \mathcal{M}_n(\mathbb{C})$. Suppose that there exists a unitary matrix $U \in \mathcal{M}_n(\mathbb{C})$ such that $B = UAU^{-1}$. Show that $\|A\| = \|B\|$.

Let $F$ be a vector subspace of $E$ of dimension $p$ and $\mathcal{A}_{F}$ the set of endomorphisms of $E$ that stabilise $F$, that is $\mathcal{A}_{F} = \{ u \in \mathcal{L}(E) \mid u(F) \subset F \}$.
Show that $\mathcal{A}_{F}$ is a subalgebra of $\mathcal{L}(E)$.

Show that $f$ is cyclic if and only if there exists a basis $\mathcal{B}$ of $E$ in which the matrix of $f$ is of the form $C_Q$, where $Q$ is a monic polynomial of degree $n$.

We fix a polynomial $f \in \mathbb{C}[X]$ of degree $n \geq 1$. We consider a complex number $z \in \overline{\mathbb{D}}$ and we define the matrices $M \in \mathcal{M}_{n+1}(\mathbb{C})$ and $P \in \mathcal{M}_{n+1,1}(\mathbb{C})$ by $$M = \left(\begin{array}{cccccc} z & 0 & 0 & \ldots & 0 & \sqrt{1-|z|^2} \\ \sqrt{1-|z|^2} & 0 & 0 & \ldots & 0 & -\bar{z} \\ 0 & & & & & 0 \\ 0 & & & & & 0 \\ \vdots & & I_{n-1} & & & \vdots \\ 0 & & & & & 0 \end{array}\right)$$ and $$P = \left(\begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \end{array}\right)$$ Show that $M$ is a unitary matrix.

Let $F$ be a vector subspace of $E$ of dimension $p$ and $\mathcal{A}_{F}$ the set of endomorphisms of $E$ that stabilise $F$, that is $\mathcal{A}_{F} = \{ u \in \mathcal{L}(E) \mid u(F) \subset F \}$.
Show that $\operatorname{dim} \mathcal{A}_{F} = n^{2} - pn + p^{2}$.
One may consider a basis of $E$ in which the matrix of every element of $\mathcal{A}_{F}$ is block triangular.