Let $M \in \mathrm { GL } _ { n } ( \mathbb { R } )$. Show that the eigenvalues of $M ^ { \top } M$ are all strictly positive.
Deduce that there exists a symmetric matrix $S$ with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$.

Show that the application $$\begin{array} { | c c c } \left( \mathcal { M } _ { n } ( \mathbb { R } ) \right) ^ { 2 } & \rightarrow & \mathbb { R } \\ ( A , B ) & \mapsto & \operatorname { tr } \left( A ^ { \top } B \right) \end{array}$$ is an inner product on $\mathcal { M } _ { n } ( \mathbb { R } )$.

If $U \in \mathcal{M}_{n,1}(\mathbf{C})$, show that $$\max\left\{\left|V^{T}U\right|; V \in \Sigma_{n}\right\} = \|U\|.$$ Deduce that, if $M$ is in $\mathcal{M}_{n}(\mathbf{C})$, then $$\max\left\{\left|X^{T}MY\right|; (X,Y) \in \Sigma_{n} \times \Sigma_{n}\right\} = \|M\|_{\mathrm{op}}.$$

$\mathbf{3}$ ▷ Conversely, suppose the relation $\forall t \in \mathbf{R} \quad e^{t(A+B)} = e^{tA} e^{tB}$ is satisfied. By differentiating this relation twice with respect to the real variable $t$, show that the matrices $A$ and $B$ commute.

Let $M$ be a matrix in $M_{2}(\mathbf{R})$. We assume that $M$ has two complex eigenvalues $\mu = a + ib$ and $\bar{\mu} = a - ib$ with $a \in \mathbf{R}$ and $b \in \mathbf{R}^{*}$. Prove that $M$ is semi-simple and similar in $M_{2}(\mathbf{R})$ to the matrix: $$\left(\begin{array}{cc} a & b \\ -b & a \end{array}\right)$$

We fix two orthonormal families $u = (u_1, \ldots, u_p)$ of vectors of $V$ and $u^{\prime} = (u_1^{\prime}, \ldots, u_p^{\prime})$ of vectors of $V^{\prime}$ satisfying the conditions of question (1).
(a) Show that $u$ is an orthonormal basis of $V$.
(b) Show that for $k \in \llbracket 1, p-1 \rrbracket$, we have $u_k^{\prime} \in \operatorname{Vect}(u_{k+1}, \ldots, u_p)^{\perp}$. (Hint: one may consider the map $t \mapsto u_k(t) = \frac{u_k + u_{\ell}}{\|u_k + t u_{\ell}\|}$ for all $t \in \mathbb{R}$ and $\ell \in \llbracket k+1, p \rrbracket$ as well as its derivative.)
(c) Show that $u_{k+1} \in \left(\operatorname{Vect}(u_1, \ldots, u_k) + \operatorname{Vect}(u_1^{\prime}, \ldots, u_k^{\prime})\right)^{\perp}$ for all $k$ element of $\llbracket 1, p-1 \rrbracket$.
(d) Deduce that the subspaces $W_k = \operatorname{Vect}(u_k, u_k^{\prime})$ for $k \in \llbracket 1, p \rrbracket$ are pairwise orthogonal.

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

Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded. For $M \in \mathcal{B}_{n}$, we set $$b(M) = \sup\left\{\left\|M^{k}\right\|_{\mathrm{op}}; k \in \mathbf{N}\right\}.$$
Let $M \in \mathcal{B}_{n}$, $X \in \mathcal{M}_{n,1}(\mathbf{C})$. Show that the sequence $\left(\left\|M^{k}X\right\|\right)_{k \in \mathbf{N}}$ is bounded. If $\lambda \in \sigma(M)$, if $X$ is an eigenvector of $M$ associated with $\lambda$, express for $k \in \mathbf{N}$, the vector $M^{k}X$ in terms of $\lambda$, $k$ and $X$. Deduce that $\sigma(M) \subset \mathbb{D}$.

$\mathbf{4}$ ▷ For any matrix $A \in \mathcal{M}_n(\mathbf{K})$, prove the relation $\left\| e^{A} \right\| \leq e^{\|A\|}$.

Let $M$ be a matrix in $M_{2}(\mathbf{R})$. Prove that $M$ is semi-simple if and only if one of the following conditions is satisfied:

1. [i)] $M$ is diagonalizable in $M_{2}(\mathbf{R})$;
2. [ii)] $\chi_{M}$ has two complex conjugate roots with non-zero imaginary part.

We fix two orthonormal families $u = (u_1, \ldots, u_p)$ of vectors of $V$ and $u^{\prime} = (u_1^{\prime}, \ldots, u_p^{\prime})$ of vectors of $V^{\prime}$ satisfying the conditions of question (1).
(a) Show that there exist $0 \leqslant \theta_1 \leqslant \cdots \leqslant \theta_p \leqslant \pi/2$ such that $\cos(\theta_k) = \langle u_k, u_k^{\prime}\rangle$ for all $k \in \llbracket 1, p \rrbracket$.
(b) Calculate the value of $\det(\operatorname{Gram}(u, u^{\prime}))$ as a function of the $\cos(\theta_k)$.
(c) Deduce that $\det(\operatorname{Gram}(u, u^{\prime})) \leqslant 1$. What can be said about $V$ and $V^{\prime}$ in the case of equality?

Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded.
Assume that $n \geq 2$. Indicate, with justification, a matrix $M$ in $\mathcal{M}_{n}(\mathbf{C})$, upper triangular, such that $\sigma(M) \subset \mathbb{D}$, but not belonging to $\mathcal{B}_{n}$.

For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \det(\operatorname{Gram}(e, u))$$
(a) Verify that the map $(x_1, \ldots, x_p) \mapsto [x_1, \ldots, x_p]$ belongs to $\mathcal{A}_p(\mathbb{R}^p, \mathbb{R})$.
(b) Verify that if $F$ is a vector space over $\mathbb{R}$ and if $f : F \rightarrow \mathbb{R}^p$ is linear, then $g : F^p \rightarrow \mathbb{R}$ defined for $u = (u_1, \ldots, u_p) \in F^p$ by $g(u) = [f(u_1), \ldots, f(u_p)]$ is an element of $\mathcal{A}_p(F, \mathbb{R})$.

Determine the trace and the determinant of a nilpotent matrix in $\mathcal { M } _ { n } ( \mathbb { R } )$.

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.

Write the matrix $H$ of the inner product $\phi(P,Q) = \int_0^1 P(t)Q(t)\,\mathrm{d}t$ in the canonical basis of $\mathbb{R}_{n-1}[X]$, that is, the matrix with general term $h_{i,j} = \phi\left(X^i, X^j\right)$ where the indices $i$ and $j$ vary between 0 and $n-1$.

In this part, we denote by $A$ and $B$ two arbitrary matrices in $\mathcal{M}_n(\mathbf{K})$. For every nonzero natural integer $k$, we set $$X_k = \exp\left(\frac{A}{k}\right) \exp\left(\frac{B}{k}\right) \text{ and } Y_k = \exp\left(\frac{A+B}{k}\right).$$
$\mathbf{6}$ ▷ Prove the inequalities $$\forall k \in \mathbf{N}^* \quad \left\| X_k \right\| \leq \exp\left(\frac{\|A\| + \|B\|}{k}\right) \text{ and } \left\| Y_k \right\| \leq \exp\left(\frac{\|A\| + \|B\|}{k}\right).$$

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.

For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \det(\operatorname{Gram}(e, u))$$
(a) Show that for all $e \in E^p$, we have $\Omega_p(e) \in \mathcal{A}_p(E, \mathbb{R})$.
(b) Verify that for all $(e, u) \in E^p \times E^p$, we have $\Omega_p(e)(u) = \Omega_p(u)(e)$.
(c) Show that $\Omega_p \in \mathcal{A}_p(E, \mathcal{A}_p(E, \mathbb{R}))$.

Show that, if $M \in \mathcal { M } _ { n } ( \mathbb { R } )$ is nilpotent, then $M ^ { 2 }$ is nilpotent.

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 $$\forall z \in \mathbf{C} \backslash \sigma(M), \quad \left(R_{z}(M)\right)_{i,j} = \frac{P_{M,i,j}(z)}{\chi_{M}(z)}$$
We admit that every matrix in $\mathcal{M}_{n}(\mathbf{C})$ satisfies $\mathcal{P}$. Deduce that, if $M \in \mathcal{M}_{n}(\mathbf{C})$ and $(X,Y) \in \mathcal{M}_{n,1}(\mathbf{C})^{2}$, there exists an element $P_{M,X,Y}$ of $\mathbf{C}_{n-1}[X]$ such that $$\forall z \in \mathbf{C} \backslash \sigma(M), \quad X^{T}R_{z}(M)Y = \frac{P_{M,X,Y}(z)}{\chi_{M}(z)}.$$

Let $H$ be the matrix of the inner product $\phi(P,Q) = \int_0^1 P(t)Q(t)\,\mathrm{d}t$ in the canonical basis of $\mathbb{R}_{n-1}[X]$, with general term $h_{i,j} = \phi(X^i, X^j)$. Let $U \in \mathcal{M}_{n,1}(\mathbb{R})$. Express the product $U^\top H U$ using $\phi$ and the coefficients of $U$.

In this part, we denote by $A$ and $B$ two arbitrary matrices in $\mathcal{M}_n(\mathbf{K})$. For every nonzero natural integer $k$, we set $$X_k = \exp\left(\frac{A}{k}\right) \exp\left(\frac{B}{k}\right) \text{ and } Y_k = \exp\left(\frac{A+B}{k}\right).$$
We introduce the function $$\begin{aligned} h : \mathbf{R} & \longrightarrow \mathcal{M}_n(\mathbf{K}) \\ t & \longmapsto h(t) = e^{tA} e^{tB} - e^{t(A+B)} \end{aligned}$$
$\mathbf{7}$ ▷ Show that $$X_k - Y_k = O\left(\frac{1}{k^2}\right) \text{ as } k \rightarrow +\infty.$$

In this part, $E$ denotes a $\mathbf{C}$-vector space of dimension $n$ and $u$ denotes an endomorphism of $E$. We assume that $u$ is diagonalizable. We denote by $\mathcal{B} = (v_{1}, v_{2}, \ldots, v_{n})$ a basis of $E$ formed of eigenvectors of $u$. Let $F$ be a vector subspace of $E$, different from $\{0_{E}\}$ and from $E$.
Prove that there exists $k \in \llbracket 1; n \rrbracket$ such that $v_{k} \notin F$ and that then $F$ and the vector line spanned by $v_{k}$ are in direct sum.

For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \det(\operatorname{Gram}(e, u))$$
(a) Let $M \in \mathcal{M}_p(\mathbb{R})$, $e = (e_1, \ldots, e_p)$ and $e^{\prime} = (e_1^{\prime}, \ldots, e_p^{\prime})$ in $E^p$ satisfying $e_i^{\prime} = \sum_{j=1}^p M_{ij} e_j$ for all $i \in \llbracket 1, p \rrbracket$. Show that $\Omega_p(e^{\prime}) = \det(M) \Omega_p(e)$.
(b) Let $e \in E^p$. Show that $\Omega_p(e) \neq 0$ if and only if $e$ is a free family.
(c) Verify that $\Omega_p(e)(e) \geqslant 0$ for all family $e \in E^p$. In the sequel for all $e \in E^p$, we call the $p$-volume of $e$ the quantity $$\operatorname{vol}_p(e) = \sqrt{\Omega_p(e)(e)} = (\det(\operatorname{Gram}(e, e)))^{1/2}$$