Deduce that if $A$ is a matrix in $\mathcal { M } _ { n } ( \mathbb { R } )$ satisfying $A ^ { \top } A = 0$ then $A = 0$.

Suppose that $M$ and $N$ are two nilpotent matrices that commute. Show that $M N$ and $M + N$ are nilpotent.

Suppose that $M , N$ and $M + N$ are nilpotent. By computing $( M + N ) ^ { 2 } - M ^ { 2 } - N ^ { 2 }$, show that $\operatorname { tr } ( M N ) = 0$.

Show that two shift-invariant endomorphisms of $\mathbb{K}[X]$ commute.

For $X, Y \in \mathscr{M}_{N,1}(\mathbf{R})^2$, we define $$\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$$ where $\pi \in \mathscr{M}_{1,N}(\mathbf{R})$ is a probability with $\pi[j] \neq 0$ for all $j$. Show that $(X, Y) \mapsto \langle X, Y \rangle$ is an inner product on $\mathscr{M}_{N,1}(\mathbf{R})$.

Let $\mathscr{P}$ be the set of row vectors of size $d$ with non-negative coefficients whose coordinate sum equals 1: $$\mathscr{P} = \left\{ u \in \mathscr{M}_{1,d}\left(\mathbb{R}_{+}\right) : \sum_{j=1}^{d} u_j = 1 \right\}.$$ We consider a square matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ such that for all $i \in \{1,\ldots,d\}$, $$\sum_{j=1}^{d} P_{i,j} = 1$$ We further assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$P_{i,j} \geqslant c\nu_j.$$
Show that if $u \in \mathscr{P}$, then $uP \in \mathscr{P}$.

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, $\Delta_n$ the set of EDM of order $n$, and $\Omega_n$ the set of symmetric positive matrices of order $n$ such that $M \cdot \mathbf{e} = 0$. The application $T: \Delta_n \to \mathcal{M}_n(\mathbb{R})$ associates to $D$ the matrix $T(D) = -\frac{1}{2}PDP$, and the application $K: \Omega_n \to \mathcal{M}_n(\mathbb{R})$ associates to $A$ the matrix $K(A) = \mathbf{e}\cdot\mathbf{a}^T + \mathbf{a}\cdot\mathbf{e}^T - 2A$ where $\mathbf{a}$ is the column of diagonal coefficients of $A$.
Show that the applications $T: \Delta_n \rightarrow \Omega_n$ and $K: \Omega_n \rightarrow \Delta_n$ satisfy: $$T \circ K = \operatorname{Id}_{\Omega_n}.$$

We assume in this question that $0 < R_u \leqslant 1$. Let $A \in \mathbb{M}_n(u)$ and $B \in \mathbb{M}_n(u)$ be two symmetric matrices such that $AB = BA$. Show that $AB \in \mathbb{M}_n(u)$.

Let $\alpha \in \mathbb{C}$ such that $|\alpha| < R_u$. Show that $$u(\alpha I_n) = U(\alpha) I_n$$

Let $B \in \mathbb{M}_n(u)$.
(a) Show that there exists a polynomial $R \in \mathbb{C}[X]$ such that $$u(A) = R(A) \text{ and } u(B) = R(B).$$ (b) We assume that $AB \in \mathbb{M}_n(u)$ and $BA \in \mathbb{M}_n(u)$. Show that $$A\, u(BA) = u(AB)\, A$$

Let $v = (v_k)_{k \geqslant 0}$ be another sequence of $\mathbb{C}$ such that $A \in \mathbb{M}_n(v)$. We assume in this question only that the values $\lambda_1, \cdots, \lambda_\ell$ are real. Show that $$(u \star v)(A) = u(A)\, v(A)$$ (after having justified that $A \in \mathbb{M}_n(u \star v)$).

Let $B$ and $C$ be two matrices of $\mathbf{GL}_n$. Let $A \in \mathbf{M}_{2n}$ be the block matrix defined as follows: $$A = \left(\begin{array}{cc} B & 0_n \\ 0_n & C \end{array}\right)$$
Let $S_1$ be the block matrix $$S_1 = \left(\begin{array}{cc} 0_n & P \\ Q & 0_n \end{array}\right)$$ where $P, Q$ are two elements of $\mathbf{GL}_n$.
Determine the conditions relating $B, C, P, Q$ for the matrices $S_1$ and $S_2 = S_1 A$ to be symmetry matrices.

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: $$A = \left(\begin{array}{cc} B & 0_n \\ 0_n & C \end{array}\right)$$ Let $S_1$ be the block matrix $$S_1 = \left(\begin{array}{cc} 0_n & P \\ Q & 0_n \end{array}\right),$$ where $P, Q$ are two elements of $\mathbf{GL}_n$. Determine the conditions relating $B, C, P, Q$ for the matrices $S_1$ and $S_2 = S_1 A$ to be symmetry matrices.

For every polynomial $P = P(X) \in \mathbf{C}_{n-1}[X]$ we set $$\left\{\begin{array}{l} s_1(P) = P(-X) \\ s_2(P) = P(1-X) \\ g(P) = P(X+1) - P(X) \end{array}\right.$$ We thus define three endomorphisms of the vector space $\mathbf{C}_{n-1}[X]$. Calculate $s_1^2$, $s_2^2$ and express $s_1 \circ s_2$ in terms of $g$ and $Id_{\mathbf{C}_{n-1}[X]}$.

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 $\lambda \in \mathbf{C}$ 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 $$A' = \left(\begin{array}{cccc} J_{n_1}(\lambda_1) & 0 & \ldots & 0 \\ 0 & J_{n_2}(\lambda_2) & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \ldots & 0 & J_{n_r}(\lambda_r) \end{array}\right)$$ 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. Using the results established in the previous parts, prove that $A$ is a product of two symmetry matrices.

For every integer $j \in \llbracket 1, n \rrbracket$, we define the matrices $$B_j = S - \alpha_j I_n \quad \text{and} \quad C_j = I_n - \alpha_j S$$
Prove that $$J(p) = \sum_{j=1}^{n} f_j(S)^\top \left(C_j^\top C_j - B_j^\top B_j\right) f_j(S)$$

For every integer $j \in \llbracket 1, n \rrbracket$, we define the matrices $$B_j = S - \alpha_j I_n \quad \text{and} \quad C_j = I_n - \alpha_j S$$
Let $j \in \llbracket 1, n \rrbracket$. Show that $C_j^\top C_j - B_j^\top B_j = (1 - \alpha_j^2) U U^\top$.

Let $\mathbf{u}, \mathbf{v}, \mathbf{x}, \mathbf{y} \in \mathbb{R}^n \backslash \{0\}$. Show that $\mathbf{u v}^T = \mathbf{x y}^T$ if and only if there exists $\lambda \in \mathbb{R} \backslash \{0\}$ such that $$\mathbf{u} = \lambda \mathbf{x}, \quad \text{and} \quad \mathbf{v} = \frac{1}{\lambda} \mathbf{y}$$

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Calculate the block matrix product $$\left(\begin{array}{cc} \mathbb{I}_n & 0 \\ \mathbf{v}^T & 1 \end{array}\right) \left(\begin{array}{cc} \mathbb{I}_n + \mathbf{u}\mathbf{v}^T & \mathbf{u} \\ 0 & 1 \end{array}\right) \left(\begin{array}{cc} \mathbb{I}_n & 0 \\ -\mathbf{v}^T & 1 \end{array}\right)$$

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 $a_{i,j}B$.
Besides $u,v,r$ and $s$, we are also given two non-zero natural integers, $w$ and $t$.
Show that $\otimes$ is a bilinear map from $\mathcal{M}_{u,v}(\mathbb{R}) \times \mathcal{M}_{r,s}(\mathbb{R})$ to $\mathcal{M}_{ur,vs}(\mathbb{R})$.

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 $a_{i,j}B$.
Besides $u,v,r$ and $s$, we are also given two non-zero natural integers, $w$ and $t$.
Show that, for all matrices $A \in \mathcal{M}_{u,v}(\mathbb{R})$, $A' \in \mathcal{M}_{v,w}(\mathbb{R})$, $B \in \mathcal{M}_{r,s}(\mathbb{R})$ and $B' \in \mathcal{M}_{s,t}(\mathbb{R})$, $$(A \otimes B)(A' \otimes B') = (AA') \otimes (BB').$$

Let $\mathbf{u}, \mathbf{v}, \mathbf{x}, \mathbf{y} \in \mathbb{R}^n \backslash \{\mathbf{0}\}$. Show that $\mathbf{u v}^T = \mathbf{x y}^T$ if and only if there exists $\lambda \in \mathbb{R} \backslash \{0\}$ such that $$\mathbf{u} = \lambda \mathbf{x}, \quad \text{and} \quad \mathbf{v} = \frac{1}{\lambda} \mathbf{y}.$$

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Calculate the block matrix product $$\left(\begin{array}{cc} \mathbb{I}_n & 0 \\ \mathbf{v}^T & 1 \end{array}\right) \left(\begin{array}{cc} \mathbb{I}_n + \mathbf{u v}^T & \mathbf{u} \\ 0 & 1 \end{array}\right) \left(\begin{array}{cc} \mathbb{I}_n & 0 \\ -\mathbf{v}^T & 1 \end{array}\right).$$