We say that a matrix $S \in \mathbf{M}_n$ is a symmetry matrix if $S^2 = I_n$.
Prove that if $S_1$ and $S_2$ are two symmetry matrices, the product matrix $A = S_1 S_2$ is invertible and similar to its inverse.

We say that a matrix $S \in \mathbf{M}_n$ is a symmetry matrix if $S^2 = I_n$.
If a matrix $A$ is a product of two symmetry matrices, is the same true for every matrix similar to $A$?

Let $B$ and $C$ be two matrices of $\mathbf{GL}_n$. Let $A \in \mathbf{M}_{2n}$ be the block matrix defined as follows: $$A = \left(\begin{array}{cc} B & 0_n \\ 0_n & C \end{array}\right)$$
Deduce that if $C$ is similar to $B^{-1}$, then $A$ is a product of two symmetry matrices.

For every $\lambda \in \mathbb{C}$ nonzero, we set $J_n(\lambda) = \lambda I_n + N$, where $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise, and $N'$ is the matrix such that $J_n(\lambda)^{-1} = \frac{1}{\lambda} I_n + N'$.
Calculate $(N')^n$ and deduce that $J_n(\lambda)^{-1}$ is similar to $J_n\left(\frac{1}{\lambda}\right)$.

For every polynomial $P = P(X) \in \mathbf{C}_{n-1}[X]$ we set $$\left\{\begin{array}{l} s_1(P) = P(-X) \\ s_2(P) = P(1-X) \\ g(P) = P(X+1) - P(X) \end{array}\right.$$ For every $\lambda \in \mathbb{C}$ nonzero, $J_n(\lambda) = \lambda I_n + N$ where $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise.
Deduce from the previous questions that the matrix $J_n(1)$ is a product of two symmetry matrices.

Let $A$ be a matrix of $\mathbf{GL}_n$ similar to its inverse. We admit that $A$ is similar to a block diagonal matrix of the form $$A' = \left(\begin{array}{cccc} J_{n_1}(\lambda_1) & 0 & \cdots & 0 \\ 0 & J_{n_2}(\lambda_2) & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & J_{n_r}(\lambda_r) \end{array}\right)$$ where the $\lambda_i$ are the eigenvalues of $A$ (not necessarily distinct) and $r$ as well as the $n_i$, $1 \leq i \leq r$, are nonzero natural integers. Moreover the matrix $A'$ is unique up to the order of the blocks.
Prove that $A^{-1}$ is similar to $\left(\begin{array}{cccc} J_{n_1}\left(\frac{1}{\lambda_1}\right) & 0 & \cdots & 0 \\ 0 & J_{n_2}\left(\frac{1}{\lambda_2}\right) & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & J_{n_r}\left(\frac{1}{\lambda_r}\right) \end{array}\right)$.

Let $A$ be a matrix of $\mathbf{GL}_n$ similar to its inverse. We admit that $A$ is similar to a block diagonal matrix of the form $$A' = \left(\begin{array}{cccc} J_{n_1}(\lambda_1) & 0 & \cdots & 0 \\ 0 & J_{n_2}(\lambda_2) & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & J_{n_r}(\lambda_r) \end{array}\right)$$ where the $\lambda_i$ are the eigenvalues of $A$ (not necessarily distinct) and $r$ as well as the $n_i$, $1 \leq i \leq r$, are nonzero natural integers. Moreover the matrix $A'$ is unique up to the order of the blocks. One may also use that $J_n(-1)$ is a product of two symmetry matrices.
Using the results established in the previous parts, prove that $A$ is a product of two symmetry matrices.

Let $B \in \mathbf{M}_n$ be a diagonalizable matrix. We assume 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 $\left(X-2\right)^2\left(X-\frac{1}{2}\right)^2$ is reciprocal). One may determine the eigenspaces of $B$ and $B^{-1}$ for the eigenvalue 2.

We say that a matrix $S \in \mathbf{M}_n$ is a symmetry matrix if $S^2 = I_n$. Prove that if $S_1$ and $S_2$ are two symmetry matrices, the product matrix $A = S_1 S_2$ is invertible and similar to its inverse.

We say that a matrix $S \in \mathbf{M}_n$ is a symmetry matrix if $S^2 = I_n$. If a matrix $A$ is a product of two symmetry matrices, is the same true for every matrix similar to $A$?

Let $B$ and $C$ be two matrices of $\mathbf{GL}_n$. Let $A \in \mathbf{M}_{2n}$ be the matrix defined by blocks as follows: $$A = \left(\begin{array}{cc} B & 0_n \\ 0_n & C \end{array}\right)$$ Deduce that if $C$ is similar to $B^{-1}$, then $A$ is a product of two symmetry matrices.

For every $\lambda \in \mathbf{C}$ nonzero, we set $J_n(\lambda) = \lambda I_n + N$, and $N'$ is the matrix such that $J_n(\lambda)^{-1} = \frac{1}{\lambda} I_n + N'$. Calculate $(N')^n$ and deduce that $J_n(\lambda)^{-1}$ is similar to $J_n\left(\frac{1}{\lambda}\right)$.

Let $A$ be a matrix of $\mathbf{GL}_n$ similar to its inverse. We admit that $A$ is similar to a block diagonal matrix of the form $$A' = \left(\begin{array}{cccc} J_{n_1}(\lambda_1) & 0 & \ldots & 0 \\ 0 & J_{n_2}(\lambda_2) & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \ldots & 0 & J_{n_r}(\lambda_r) \end{array}\right)$$ where the $\lambda_i$ are the eigenvalues of $A$ (not necessarily distinct) and $r$ as well as the $n_i$, $1 \leq i \leq r$, are nonzero natural integers. Moreover the matrix $A'$ is unique up to the order of the blocks. Prove that $A^{-1}$ is similar to $\left(\begin{array}{cccc} J_{n_1}\left(\frac{1}{\lambda_1}\right) & 0 & \cdots & 0 \\ 0 & J_{n_2}\left(\frac{1}{\lambda_2}\right) & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & J_{n_r}\left(\frac{1}{\lambda_r}\right) \end{array}\right)$.

Problem 2, Part 1: Adapted norms
We denote by $\mathrm { M } _ { d } ( \mathbb { C } )$ the space of $d \times d$ square matrices with complex coefficients.
a. Let $u \in \mathcal { L } \left( \mathbb { C } ^ { d } \right)$ be an endomorphism of $\mathbb { C } ^ { d }$ and $M = \left( m _ { i , j } \right) _ { 1 \leqslant i , j \leqslant d }$ the matrix of $u$ in a basis $\mathcal { B } = \left( e _ { 1 } , \ldots , e _ { d } \right)$. Express the matrix $M ^ { \prime } = \left( m _ { i , j } ^ { \prime } \right) _ { 1 \leqslant i , j \leqslant d }$ of $u$ in the basis $\mathcal { B } ^ { \prime } = \left( \alpha _ { 1 } e _ { 1 } , \ldots , \alpha _ { d } e _ { d } \right)$, where the $\alpha _ { i }$ are complex numbers. b. Suppose that $M$ is upper triangular. Show that for all $\varepsilon > 0$ we can choose the $\alpha _ { i }$ such that for $j > i$ we have $\left| m _ { i , j } ^ { \prime } \right| < \varepsilon$.

Restriction of a diagonalizable endomorphism to a stable subspace Let $V$ be a finite-dimensional vector space, let $h$ be an endomorphism of $V$ and let $W$ be a subspace stable by $h$. We denote by $h_W$ the endomorphism of $W$ induced by $h$, that is $h_W : W \rightarrow W$, $v \mapsto h(v)$. Prove that if $h$ is diagonalizable, then $h_W$ is also diagonalizable.

``Graded'' version of the decomposition theorem In this part, we are given: a finite-dimensional vector space $V$; a nilpotent endomorphism $u$ of $V$; a nonzero natural integer $N$ and $\zeta = \exp\frac{2\mathrm{i}\pi}{N}$; an invertible endomorphism $h$ of $V$ such that $h^N = \operatorname{id}_V$ and $h \circ u \circ h^{-1} = \zeta u$.
a) Let $n$ be the index of $u$, that is, the integer such that $u^{n-1} \neq 0$ and $u^n = 0$. Prove that there exists a vector $v$ such that $v$ is an eigenvector of $h$ and $u^{n-1}(v) \neq 0$.
b) Prove that there exists a basis of $V$ in which the matrices of $u$ and $h$ are block diagonal and the diagonal blocks are respectively of the form $$J_r \quad \text{and} \quad D_{r,a} = \operatorname{diag}(\zeta^a, \zeta^{a+1}, \ldots, \zeta^{a+r-1})$$ for $r \in \mathbb{N}^*$ and $a \in \{0, \ldots, N-1\}$ suitable. We call $(r, a)$ the type of such a pair of matrices $(J_r, D_{r,a})$.

An example In this question, we further assume that $N = 4$ and $\ker(h - \operatorname{id}_V) = \{0\}$. For $j \in \{0, \ldots, 3\}$, we denote $V_j = \ker(h - \zeta^j \operatorname{id}_V)$. According to $11^\circ$b), the data of $u$ is equivalent to the data of the two linear maps $u_1 : V_1 \rightarrow V_2$ and $u_2 : V_2 \rightarrow V_3$ induced by $u$.
a) Verify that $u^3 = 0$.
b) Construct pairs $(u, h)$ that give rise to six different types of pairs of diagonal blocks $(J_r, D_{r,a})$ in the ``graded'' version of the decomposition theorem.
c) Prove that the number of blocks of each type is determined by the data of the three dimensions $d_j = \dim V_j$ ($1 \leqslant j \leqslant 3$) and the three ranks $r_1 = \operatorname{rg} u_1$, $r_2 = \operatorname{rg} u_2$ and $r_{21} = \operatorname{rg}(u_2 \circ u_1)$.

A reduction We fix two nonzero natural integers $m$ and $n$. For $(A, B)$ in $\mathcal{M}_{n,m}(\mathbb{C}) \times \mathcal{M}_{m,n}(\mathbb{C})$ we define the following $(m+n) \times (m+n)$ matrices: $$M_{A,B} = \begin{pmatrix} 0_m & B \\ A & 0_n \end{pmatrix} \quad \text{and} \quad H = \begin{pmatrix} \mathrm{I}_m & 0_{m,n} \\ 0_{n,m} & -\mathrm{I}_n \end{pmatrix}.$$ Let $(A, B)$ and $(A', B')$ be two pairs of matrices in $\mathcal{M}_{n,m}(\mathbb{C}) \times \mathcal{M}_{m,n}(\mathbb{C})$. Prove that the following conditions are equivalent:
(i) $(A, B)$ and $(A', B')$ are simultaneously equivalent;
(ii) there exist $P \in \mathrm{GL}_m(\mathbb{C})$ and $Q \in \mathrm{GL}_n(\mathbb{C})$ such that $A' = QAP^{-1}$ and $B' = PBQ^{-1}$;
(iii) there exists $R \in \mathrm{GL}_{m+n}(\mathbb{C})$ such that $M_{A',B'} = RM_{A,B}R^{-1}$ and $H = RHR^{-1}$.

Two linear maps: nilpotent case We fix two nonzero natural integers $m$ and $n$, matrices $(A,B) \in \mathcal{M}_{n,m}(\mathbb{C}) \times \mathcal{M}_{m,n}(\mathbb{C})$, and denote $M = M_{A,B} = \begin{pmatrix} 0_m & B \\ A & 0_n \end{pmatrix}$ and $H = \begin{pmatrix} \mathrm{I}_m & 0_{m,n} \\ 0_{n,m} & -\mathrm{I}_n \end{pmatrix}$. In this question, we assume that $M$ is nilpotent.
Prove that $(M, H)$ is simultaneously similar to a pair of block diagonal matrices whose diagonal blocks are respectively of the form $$\begin{pmatrix} 0_r & B_0 \\ A_0 & 0_s \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} \mathrm{I}_r & 0 \\ 0 & -\mathrm{I}_s \end{pmatrix},$$ where $r$ and $s$ are natural integers with $|r - s| \leqslant 1$ and $A_0$ and $B_0$ form one of the following pairs: $$A_0 = \begin{pmatrix} 1 & 0 & \cdots & 0 & 0 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{pmatrix}_{s \times (s+1)} \quad \text{and} \quad B_0 = \begin{pmatrix} 0 & \cdots & \cdots & 0 \\ 1 & \ddots & & \vdots \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{pmatrix}_{(s+1) \times s};$$ $$A_0 = \mathrm{I}_r \quad \text{and} \quad B_0 = J_r;$$ $$A_0 = J_r \quad \text{and} \quad B_0 = \mathrm{I}_r;$$ $$A_0 = \begin{pmatrix} 0 & \cdots & \cdots & 0 \\ 1 & \ddots & & \vdots \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{pmatrix}_{(r+1) \times r} \quad \text{and} \quad B_0 = \begin{pmatrix} 1 & 0 & \cdots & 0 & 0 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{pmatrix}_{r \times (r+1)}.$$

Two linear maps: invertible case We fix two nonzero natural integers $m$ and $n$, matrices $(A,B) \in \mathcal{M}_{n,m}(\mathbb{C}) \times \mathcal{M}_{m,n}(\mathbb{C})$, and denote $M = M_{A,B} = \begin{pmatrix} 0_m & B \\ A & 0_n \end{pmatrix}$ and $H = \begin{pmatrix} \mathrm{I}_m & 0_{m,n} \\ 0_{n,m} & -\mathrm{I}_n \end{pmatrix}$. In this question, we assume that $M$ is invertible.
a) Prove that $m = n$ and that $A$ and $B$ are invertible.
b) Prove that $(M, H)$ is simultaneously similar to a pair of block diagonal matrices whose diagonal blocks are of even size and are respectively of the form $$\begin{pmatrix} 0_r & B_1 \\ A_1 & 0_r \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} \mathrm{I}_r & 0_r \\ 0_r & -\mathrm{I}_r \end{pmatrix},$$ where $A_1 = \mathrm{I}_r$ and $B_1 = \lambda \mathrm{I}_r + J_r$ for $r$ nonzero integer and $\lambda$ nonzero complex suitable.

Let $K \in \mathcal{M}_n(\mathbb{R})$ be a matrix of rank 1, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$ be such that $K = \mathbf{u v}^T$.
(a) Show that $\operatorname{Tr}(K) = \langle \mathbf{v}, \mathbf{u} \rangle$.
(b) Show that $K^2 = \operatorname{Tr}(K) K$.
(c) Deduce that $K$ is diagonalizable if and only if $\operatorname{Tr}(K) \neq 0$.

Explain why the matrix $J_n$ is diagonalizable.

Let $(u,v,r,s) \in (\mathbb{N}^*)^4$. Let $A = (a_{ij})_{\substack{1 \leqslant i \leqslant u \\ 1 \leqslant j \leqslant v}} \in \mathcal{M}_{u,v}(\mathbb{R})$ and $B \in \mathcal{M}_{r,s}(\mathbb{R})$. We define the Kronecker product of $A$ by $B$, and we denote $A \otimes B$, the matrix of $\mathcal{M}_{ur,vs}(\mathbb{R})$ which is defined by $uv$ blocks of size $r \times s$ in such a way that, for all $(i,j) \in \llbracket 1,u \rrbracket \times \llbracket 1,v \rrbracket$, the block with index $(i,j)$ is $a_{i,j}B$.
Show that if $A \in \mathcal{M}_u(\mathbb{R})$ and $B \in \mathcal{M}_r(\mathbb{R})$ are diagonalizable, then $A \otimes B$ is diagonalizable and $$\operatorname{Sp}(A \otimes B) = \{\lambda\mu \mid \lambda \in \operatorname{Sp}(A), \mu \in \operatorname{Sp}(B)\}.$$ One may start by determining the inverse of the Kronecker product of two invertible matrices.

Let $K \in \mathcal{M}_n(\mathbb{R})$ be a matrix of rank 1, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$ be such that $K = \mathbf{u v}^T$.
(a) Show that $\operatorname{Tr}(K) = \langle \mathbf{v}, \mathbf{u} \rangle$.
(b) Show that $K^2 = \operatorname{Tr}(K) K$.
(c) Deduce that $K$ is diagonalizable if and only if $\operatorname{Tr}(K) \neq 0$.