Deduce that $$\forall (A,B) \in \mathcal{S}_{n}(\mathbb{R})^{2}, \quad \sum_{i=1}^{n}\left(\lambda_{i}(A) - \lambda_{i}(B)\right)^{2} \leqslant \|A - B\|_{F}^{2}.$$

Show that, for every $M$ in $\mathcal{M}_{n}(\mathbb{R})$ and for all $P$ and $Q$ in $\mathcal{O}_{n}(\mathbb{R})$, we have $\|PMQ\|_{F} = \|M\|_{F}$.

We denote $\mathcal{B}_{n}(\mathbb{R})$ the set of bistochastic matrices in $\mathcal{M}_{n}(\mathbb{R})$, that is the set of matrices $M = (m_{i,j})_{1 \leqslant i,j \leqslant n}$ whose coefficients are all non-negative and such that $\sum_{j=1}^{n} m_{i,j} = \sum_{j=1}^{n} m_{j,i} = 1$ for every $i \in \llbracket 1, n \rrbracket$.
We denote $f : \left|\, \begin{array}{ccc} \mathcal{M}_{n}(\mathbb{R}) & \rightarrow & \mathbb{R} \\ M & \mapsto & \sum_{1 \leqslant i,j \leqslant n} m_{i,j}\left(\lambda_{i}(A) - \lambda_{j}(B)\right)^{2} \end{array}\right.$
Justify that $f$ admits a minimum on $\mathcal{B}_{n}(\mathbb{R})$.

We denote $\mathcal{B}_{n}(\mathbb{R})$ the set of bistochastic matrices in $\mathcal{M}_{n}(\mathbb{R})$, that is the set of matrices $M = (m_{i,j})_{1 \leqslant i,j \leqslant n}$ whose coefficients are all non-negative and such that $\sum_{j=1}^{n} m_{i,j} = \sum_{j=1}^{n} m_{j,i} = 1$ for every $i \in \llbracket 1, n \rrbracket$.
We denote $f : \left|\, \begin{array}{ccc} \mathcal{M}_{n}(\mathbb{R}) & \rightarrow & \mathbb{R} \\ M & \mapsto & \sum_{1 \leqslant i,j \leqslant n} m_{i,j}\left(\lambda_{i}(A) - \lambda_{j}(B)\right)^{2} \end{array}\right.$
Let $(i,j,k) \in \llbracket 1,n \rrbracket^{3}$ such that $j \geqslant i$ and $k \geqslant i$. Show that, for $M \in \mathcal{M}_{n}(\mathbb{R})$ and for $x \in \mathbb{R}^{+}$, $$f\left(M + xE_{ii} + xE_{jk} - xE_{ik} - xE_{ji}\right) - f(M) = 2x\left(\lambda_{i}(A) - \lambda_{j}(A)\right)\left(\lambda_{k}(B) - \lambda_{i}(B)\right) \leqslant 0$$

We denote $\mathcal{B}_{n}(\mathbb{R})$ the set of bistochastic matrices in $\mathcal{M}_{n}(\mathbb{R})$, that is the set of matrices $M = (m_{i,j})_{1 \leqslant i,j \leqslant n}$ whose coefficients are all non-negative and such that $\sum_{j=1}^{n} m_{i,j} = \sum_{j=1}^{n} m_{j,i} = 1$ for every $i \in \llbracket 1, n \rrbracket$.
We denote $f : \left|\, \begin{array}{ccc} \mathcal{M}_{n}(\mathbb{R}) & \rightarrow & \mathbb{R} \\ M & \mapsto & \sum_{1 \leqslant i,j \leqslant n} m_{i,j}\left(\lambda_{i}(A) - \lambda_{j}(B)\right)^{2} \end{array}\right.$
Let $n \geqslant 2$ and $M = (m_{i,j})_{1 \leqslant i,j \leqslant n} \in \mathcal{B}_{n}(\mathbb{R})$ a matrix different from the identity. We denote $i$ the smallest integer belonging to $\llbracket 1,n \rrbracket$ such that $m_{i,i} \neq 1$. Show that there exists a matrix $M^{\prime} = (m_{i,j}^{\prime})_{1 \leqslant i,j \leqslant n} \in \mathcal{B}_{n}(\mathbb{R})$ such that $f(M^{\prime}) \leqslant f(M)$ and $m_{j,j}^{\prime} = 1$ for every $j \in \llbracket 1, i \rrbracket$.

We denote $\mathcal{B}_{n}(\mathbb{R})$ the set of bistochastic matrices in $\mathcal{M}_{n}(\mathbb{R})$, that is the set of matrices $M = (m_{i,j})_{1 \leqslant i,j \leqslant n}$ whose coefficients are all non-negative and such that $\sum_{j=1}^{n} m_{i,j} = \sum_{j=1}^{n} m_{j,i} = 1$ for every $i \in \llbracket 1, n \rrbracket$.
We denote $f : \left|\, \begin{array}{ccc} \mathcal{M}_{n}(\mathbb{R}) & \rightarrow & \mathbb{R} \\ M & \mapsto & \sum_{1 \leqslant i,j \leqslant n} m_{i,j}\left(\lambda_{i}(A) - \lambda_{j}(B)\right)^{2} \end{array}\right.$
Deduce that $$\min\left\{f(M) \mid M \in \mathcal{B}_{n}(\mathbb{R})\right\} = f(I_{n})$$

Deduce that $$\forall (A,B) \in \mathcal{S}_{n}(\mathbb{R})^{2}, \quad \sum_{i=1}^{n} \left(\lambda_{i}(A) - \lambda_{i}(B)\right)^{2} \leqslant \|A - B\|_{F}^{2}.$$

We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive. We set $\varepsilon = \min _ { 1 \leqslant i , j \leqslant n } m _ { i , j }$. We are interested in the sequence $\left( M ^ { k } \right) _ { k \in \mathbb { N } }$ of powers of $M$. We denote by $m _ { i , j } ^ { ( k ) }$ the coefficient of the matrix $M ^ { k }$ located in row $i$ and column $j$.
For all $j \in \llbracket 1 , n \rrbracket$, we set $$\left\{ \begin{array} { l } \alpha _ { j } ^ { ( k ) } = \min _ { 1 \leqslant i \leqslant n } m _ { i , j } ^ { ( k ) } , \\ \beta _ { j } ^ { ( k ) } = \max _ { 1 \leqslant i \leqslant n } m _ { i , j } ^ { ( k ) } . \end{array} \right.$$
Deduce that $\beta _ { j } ^ { ( k + 1 ) } - \alpha _ { j } ^ { ( k + 1 ) } \leqslant ( 1 - 2 \varepsilon ) \left( \beta _ { j } ^ { ( k ) } - \alpha _ { j } ^ { ( k ) } \right)$.

We denote $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. Prove that $C$ is a closed subset of $\mathcal{M}_{n,1}(\mathbb{R})$.

We denote $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. Deduce that the application $U \mapsto \left| U^\top A U \right|$ admits a maximum on $C$.

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

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 $$X_k = \exp\left(\frac{A}{k}\right) \exp\left(\frac{B}{k}\right) \text{ and } Y_k = \exp\left(\frac{A+B}{k}\right).$$
The objective is to prove the relation $$\lim_{k \rightarrow +\infty} \left(e^{\frac{A}{k}} e^{\frac{B}{k}}\right)^k = e^{A+B}.$$
$\mathbf{8}$ ▷ Verify the relation $$X_k^k - Y_k^k = \sum_{i=0}^{k-1} X_k^i \left(X_k - Y_k\right) Y_k^{k-i-1}$$ Deduce from this the relation $\lim_{k \rightarrow +\infty} \left(e^{\frac{A}{k}} e^{\frac{B}{k}}\right)^k = e^{A+B}$.

Let $A \in \mathbf { M } _ { 2 } ( \mathbf { C } )$ be a matrix and let $\epsilon > 0$ be a real number.
(a) Show the existence of a real number $\alpha > 0$ such that for every non-negative integer $n$ the absolute values of the coefficients of $A ^ { n }$ are bounded by $\alpha ( \rho ( A ) + \epsilon ) ^ { n }$.
(b) Deduce the existence of a real number $\beta > 0$ such that for every non-negative integer $n$ and every $x \in \mathbb { C } ^ { 2 }$ we have $$\left\| A ^ { n } x \right\| \leqslant \beta ( \rho ( A ) + \epsilon ) ^ { n } \| x \|$$

Let $A \in \mathrm { M } _ { 2 } ( \mathbb { C } )$ be a matrix and let $\epsilon > 0$ be a real number.
(a) Show the existence of a real number $\alpha > 0$ such that for every positive integer $n$ the absolute values of the coefficients of $A ^ { n }$ are bounded by $\alpha ( \rho ( A ) + \epsilon ) ^ { n }$.
(b) Deduce the existence of a real number $\beta > 0$ such that for every positive integer $n$ and all $x \in \mathbb { C } ^ { 2 }$ we have $$\left\| A ^ { n } x \right\| \leqslant \beta ( \rho ( A ) + \epsilon ) ^ { n } \| x \|$$

Justify that if $M \in \mathcal{M}_{n}(\mathbf{C})$, the map $$X \in \Sigma_{n} \longmapsto \|MX\|$$ attains its maximum, which we denote by $\|M\|_{\text{op}}$. Establish the two properties $$\begin{gathered} \forall M \in \mathcal{M}_{n}(\mathbf{C}), \quad \|M\|_{\mathrm{op}} = \max\left\{\frac{\|MX\|}{\|X\|}; X \in \mathcal{M}_{n,1}(\mathbf{C}) \backslash \{0\}\right\}, \\ \forall (M, M^{\prime}) \in \mathcal{M}_{n}(\mathbf{C})^{2}, \quad \|M^{\prime}M\|_{\mathrm{op}} \leq \|M^{\prime}\|_{\mathrm{op}} \|M\|_{\mathrm{op}}. \end{gathered}$$

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 $$\max\left\{\left|X^{T}MY\right|; (X,Y) \in \Sigma_{n} \times \Sigma_{n}\right\} = \|M\|_{\mathrm{op}}.$$

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 $\sigma(M) \subset \mathbb{D}$.

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 $b(M) = \sup\left\{\left\|M^{k}\right\|_{\mathrm{op}}; k \in \mathbf{N}\right\}$. For $M \in \mathcal{B}_{n}$, we define the function $$\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}.$$
Deduce from the previous question the inequality $$\forall M \in \mathcal{B}_{n}, \quad \forall z \in \mathbf{C} \backslash \mathbb{D}, \quad \varphi_{M}(z) \leq b(M)$$

Let $v_1$ and $v_2$ be two vectors of $\mathcal{H} = \{v\in V \mid B(v,v)=-1 \text{ and } z_v > 0\}$. Show that $$B(v_1,v_2) \leq -1,$$ with equality if and only if $v_1 = v_2$.

We denote $\| u \| = \sup _ { \substack { x \in E \\ x \neq 0 } } \frac { \| u ( x ) \| } { \| x \| }$. Show that $\|.\|$ is a norm on $\mathcal { L } ( E )$.

Show that it is a sub-multiplicative norm, that is: $$\forall ( u , v ) \in \mathcal { L } ( E ) ^ { 2 } , \quad \| u v \| \leqslant \| u \| \cdot \| v \| ,$$ and deduce a bound on $\left\| u ^ { k } \right\|$, for any natural number $k$, in terms of $\| u \|$ and the integer $k$.

Let $M \in S_n^+(\mathbf{R})$ be a non-zero matrix. You may use without proving it the inequality: $$\forall (x_1, \ldots, x_n) \in (\mathbf{R}_+)^n, \quad 2\max\{x_1, \ldots, x_n\}\left(\frac{1}{n}\sum_{k=1}^n x_k - \prod_{k=1}^n x_k^{1/n}\right) \geq \frac{1}{n}\sum_{k=1}^n \left(x_k - \prod_{j=1}^n x_j^{1/n}\right)^2.$$ Deduce that $$\frac{\operatorname{Tr}(M)}{n} - \operatorname{det}^{1/n}(M) \geq \frac{\left\|M - \operatorname{det}^{1/n}(M) I_n\right\|_2^2}{2n\|M\|_2}.$$

Let $M \in S _ { n } ^ { + } ( \mathbf { R } )$ be a non-zero matrix. In the rest of this part, you may use without proof the inequality below $\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \left( \mathbf { R } _ { + } \right) ^ { n }$,
$$2 \max \left\{ x _ { 1 } , \ldots , x _ { n } \right\} \left( \frac { 1 } { n } \sum _ { k = 1 } ^ { n } x _ { k } - \prod _ { k = 1 } ^ { n } x _ { k } ^ { 1 / n } \right) \geq \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \left( x _ { k } - \prod _ { j = 1 } ^ { n } x _ { j } ^ { 1 / n } \right) ^ { 2 }$$
Deduce that
$$\frac { \operatorname { Tr } ( M ) } { n } - \operatorname { det } ^ { 1 / n } ( M ) \geq \frac { \left\| M - \operatorname { det } ^ { 1 / n } ( M ) I _ { n } \right\| _ { 2 } ^ { 2 } } { 2 n \| M \| _ { 2 } }$$

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 for all $u, v \in \mathscr{P}$, $$\|uP - vP\|_1 \leqslant (1-c)\|u - v\|_1.$$ (One may introduce $R = P - cN$ where $N = (n_{i,j})_{1 \leqslant i,j \leqslant d}$ with $n_{i,j} = \nu_j$ for all $1 \leqslant i,j \leqslant d$.)

Let $M \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$. We assume that the matrix $M$ has an eigenvalue $\lambda > 0$ and that there exists $h \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$ a column vector such that: $$Mh = \lambda h.$$ We also assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$M_{i,j} \geqslant c\nu_j.$$ We introduce the matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ defined for $1 \leqslant i,j \leqslant d$ by $$P_{i,j} = \frac{M_{i,j} h_j}{\lambda h_i}.$$
(a) Show that there exist $\mu \in \mathscr{P}$, $C > 0$ and $\gamma \in [0,1[$, such that $\mu P = \mu$ and for all $n \geqslant 0$, $$\sum_{i=1}^{d} \sum_{j=1}^{d} \left| \lambda^{-n} \left(M^n\right)_{i,j} - h_i \frac{\mu_j}{h_j} \right| \leqslant C\gamma^n.$$
(b) Prove that there exists a unique $\pi \in \mathscr{P}$ such that $\pi M = \lambda \pi$.