Let $A, B$ be two diagonalizable matrices of $\mathcal{M}_n(\mathbb{C})$. Show that $A$ and $B$ are similar if and only if they have the same characteristic polynomial.

Justify that for all integer $k \geqslant 1$, $A^k$ is similar in $\mathcal{M}_n(\mathbb{C})$ to a triangular matrix, whose diagonal coefficients we will specify.

In the case $n=1$: Let $M$ be a matrix of size $2 \times 2$ that is symmetric and symplectic. Show that $M$ is diagonalizable and that its eigenvalues are inverses of each other. Show that there exists a matrix $P$ that is both orthogonal and symplectic such that $P^{-1} M P$ is diagonal.

Prove the following property: if $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, there exists $P \in \mathcal{O}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$ such that $P^{\top} M P$ is diagonal with diagonal coefficients $d_{1}, \ldots, d_{2n}$ satisfying for all $k \in \{1, \ldots, n\}$, $d_{k+n} = 1/d_{k}$.

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Show that there exists $P \in \mathcal{O}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$ such that $P^{\top} M^{2} P$ is diagonal with diagonal coefficients $d_{1}, \ldots, d_{2n}$ satisfying for all $k \in \{1, \ldots, n\}$, $d_{k+n} = 1/d_{k}$.

Show that if $\sigma \in \mathfrak{S}_n$, then $\chi_\sigma(X) = \prod_{\ell=1}^{n} \left(X^\ell - 1\right)^{c_\ell(\sigma)}$.
One may justify that $P_\sigma$ is similar to a block diagonal matrix whose blocks are matrices of the form $\Gamma_\ell$ ($\ell \geq 1$), where $\Gamma_\ell$ is defined above if $\ell \geq 2$ and where $\Gamma_\ell = (1)$ if $\ell = 1$.

Deduce property (S): The permutation matrices $P_\sigma$ and $P_\tau$ are similar if and only if the permutations $\sigma$ and $\tau$ are conjugate.
One may compute $T_\sigma D$ where $T_\sigma$ is the cycle type of $\sigma$ and $D$ is the divisor matrix defined in I.D.

In this subsection, $E$ is a $\mathbb{C}$-vector space of dimension $n \geq 1$. We say that an endomorphism $u$ of $E$ is a permutation endomorphism if there exists a basis $(e_1, \ldots, e_n)$ of $E$ and a permutation $\sigma \in \mathfrak{S}_n$ such that $u(e_j) = e_{\sigma(j)}$ for all $j \in \llbracket 1, n \rrbracket$.
Let $u$ be a permutation endomorphism of $E$. Show that $u$ is diagonalizable and that its trace belongs to $\llbracket 0, n \rrbracket$.

Let $A, B$ be two diagonalizable matrices of $\mathcal{M}_n(\mathbb{C})$. Show that $A$ and $B$ are similar if and only if they have the same characteristic polynomial.

We denote $D_{A} = \operatorname{diag}\left(\lambda_{1}(A), \ldots, \lambda_{n}(A)\right)$ and $D_{B} = \operatorname{diag}\left(\lambda_{1}(B), \ldots, \lambda_{n}(B)\right)$. Show that there exists an orthogonal matrix $P = \left(p_{i,j}\right)_{1 \leqslant i,j \leqslant n}$ such that $\|A - B\|_{F}^{2} = \left\|D_{A}P - PD_{B}\right\|_{F}^{2}$.

We denote $D_{A} = \operatorname{diag}(\lambda_{1}(A), \ldots, \lambda_{n}(A))$ and $D_{B} = \operatorname{diag}(\lambda_{1}(B), \ldots, \lambda_{n}(B))$. Show that there exists an orthogonal matrix $P = (p_{i,j})_{1 \leqslant i,j \leqslant n}$ such that $\|A - B\|_{F}^{2} = \|D_{A}P - PD_{B}\|_{F}^{2}$.

Prove that a matrix $A \in \mathcal{M}_{n}(\mathbb{R})$ is orthodiagonalizable if and only if it is symmetric.

Let $A$ be the matrix in $M_{2}(\mathbf{R})$ defined by: $$A = \left(\begin{array}{cc} 1 & 1 \\ -1 & 3 \end{array}\right)$$ Is the matrix $A$ semi-simple?

Let $N$ be a matrix in $M_{n}(\mathbf{R})$. Give the factored form of $\chi_{N}$ in $\mathbf{C}[X]$, specifying in the notation the real roots and the complex conjugate roots. Deduce that if $N$ is semi-simple then it is similar in $M_{n}(\mathbf{R})$ to an almost diagonal matrix.

In this part, $E$ denotes a $\mathbf{C}$-vector space of dimension $n$ and $u$ denotes an endomorphism of $E$.
Assume that every vector subspace of $E$ has a complement in $E$, stable under $u$. Prove that $u$ is diagonalizable. Deduce a characterization of diagonalizable matrices in $M_{n}(\mathbf{C})$.
Hint: one may reason by contradiction and introduce a vector subspace, whose existence one will justify, of dimension $n-1$ and containing the sum of the eigenspaces of $u$.

We say that the element $M$ of $\mathcal{M}_{n}(\mathbf{C})$ satisfies $\mathcal{P}$ if, for every $(i,j)$ in $\{1,\ldots,n\}^{2}$, there exists an element $P_{M,i,j}$ of $\mathbf{C}_{n-1}[X]$ such that $$\forall z \in \mathbf{C} \backslash \sigma(M), \quad \left(R_{z}(M)\right)_{i,j} = \frac{P_{M,i,j}(z)}{\chi_{M}(z)}$$
Show that the diagonalizable matrices of $\mathcal{M}_{n}(\mathbf{C})$ satisfy $\mathcal{P}$. Begin with the case of diagonal matrices.

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Show that $A^{-1}M$ is similar to a real symmetric matrix.
Hint: You may use question 3.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Show that $A ^ { - 1 } M$ is similar to a real symmetric matrix.
Hint: You may use question 3.

Let $J \in M_{n}(\mathbb{C})$ be the matrix defined by
$$J = \left( \begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & 0 & 1 \\ 1 & 0 & \ldots & 0 & 0 \end{array} \right).$$
Calculate $J^{n}$ and show that $J$ is diagonalisable.

Let $f$ be a symmetric endomorphism of $\mathbf{R}^n$. We denote by $\lambda_1 \leqslant \ldots \leqslant \lambda_n$ the eigenvalues ordered in increasing order of $f$.
Justify the existence of an orthonormal basis $(e_1, \ldots, e_n)$ of $\mathbf{R}^n$ formed of eigenvectors of $f$, the vector $e_i$ being associated with $\lambda_i$ for all $i \in \{1, \ldots, n\}$. We keep this basis henceforth.

Show that the matrices $M$ and $\left( m _ { \rho ( i ) , \rho ( j ) } \right) _ { 1 \leq i , j \leq n }$ are similar. Deduce that if $G = ( S , A )$ is a non-empty graph, and if $\sigma$ and $\sigma ^ { \prime }$ are two indexings of $S$, then $M _ { G , \sigma }$ and $M _ { G , \sigma ^ { \prime } }$ are similar.

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. We assume throughout this part that $A$ is diagonalizable in $\mathscr{M}_n(\mathbb{C})$ and we denote by $\lambda_1, \cdots, \lambda_\ell$ its eigenvalues with $\lambda_i \neq \lambda_j$ if $i \neq j$. Let $B \in \mathscr{M}_n(\mathbb{C})$ be an invertible matrix. Show that $$u(BAB^{-1}) = B\, u(A)\, B^{-1}.$$

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. We assume throughout this part that $A$ is diagonalizable in $\mathscr{M}_n(\mathbb{C})$ and we denote by $\lambda_1, \cdots, \lambda_\ell$ its eigenvalues with $\lambda_i \neq \lambda_j$ if $i \neq j$. Let $D \in \mathscr{M}_n(\mathbb{C})$ be a diagonal matrix and $S \in \mathscr{M}_n(\mathbb{C})$ be an invertible matrix such that $A = SDS^{-1}$.
(a) Show that $u(D)$ is diagonal and that $$\forall i \in \llbracket 1; n \rrbracket, [u(D)]_{i,i} = U([D]_{i,i}).$$ (b) Deduce an expression for $u(A)$.

Let $B \in \mathbf{M}_n$ be a diagonalizable matrix. Suppose that the characteristic polynomial of $B$ is reciprocal or antireciprocal. Prove that $B$ is invertible and similar to its inverse.

Show that the matrix $B = \left(\begin{array}{cccc} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & \frac{1}{2} & 1 \\ 0 & 0 & 0 & \frac{1}{2} \end{array}\right)$ is not similar to its inverse (although its characteristic polynomial $(X-2)^2\left(X-\frac{1}{2}\right)^2$ is reciprocal).
One may determine the eigenspaces of $B$ and $B^{-1}$ for the eigenvalue 2.