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 $C$.
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 $$\left\| A ^ { n } x \right\| \leqslant \beta ( \rho ( A ) + \epsilon ) ^ { n } \| x \|$$
Let $A \in \mathrm { M } _ { 2 } ( \mathbb { C } )$ be a matrix and let $\eta$ be a strictly positive real number. (a) For $x \in \mathbb { C } ^ { 2 }$, show that the series $$\sum _ { n } ( \rho ( A ) + \eta ) ^ { - n } \left\| A ^ { n } x \right\|$$ is convergent. We denote $$N ( x ) = \sum _ { n = 0 } ^ { \infty } ( \rho ( A ) + \eta ) ^ { - n } \left\| A ^ { n } x \right\|$$ the sum of this series. (b) Show that $x \mapsto N ( x )$ is a norm on $\mathbb { C } ^ { 2 }$, which satisfies the following inequality $$\forall x \in \mathbb { C } ^ { 2 } , \quad N ( A x ) \leqslant ( \rho ( A ) + \eta ) N ( x ) .$$ (c) Show that there exists a real $C > 0$ such that for all $x \in \mathbb { C } ^ { 2 }$ we have $$\| x \| \leqslant N ( x ) \leqslant C \| x \|$$
(a) If $B \in \mathrm { M } _ { \ell } ( \mathbb { C } )$ is diagonalizable, show that there exists a norm $\| \cdot \| _ { B }$ on $\mathbb { C } ^ { \ell }$ such that $\| B x \| _ { B } \leqslant \rho ( B ) \| x \| _ { B }$ for all $x \in \mathbb { C } ^ { \ell }$. Hint: one may verify that if $P \in \mathrm { GL } _ { \ell } ( \mathbb { C } )$, then $x \mapsto \| P x \|$ is a norm on $\mathbb { C } ^ { \ell }$. (b) Show that there exists a matrix $C \in \mathrm { M } _ { 2 } ( \mathbb { C } )$ such that, for every norm $N$ on $\mathbb { C } ^ { 2 }$ there exists $y \in \mathbb { C } ^ { 2 }$ such that $N ( C y ) > \rho ( C ) N ( y )$.
Let $\phi : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ be a map and let $x ^ { * }$ be a fixed point of $\phi$. Let $A \in \mathrm { M } _ { 2 } ( \mathbb { R } )$ be a matrix satisfying $\rho ( A ) < 1$, and let $M > 0$ be a real number. We assume that $\phi$ satisfies $$\forall x \in \mathbb { R } ^ { 2 } , \quad \left\| \phi ( x ) - \phi \left( x ^ { * } \right) - A \left( x - x ^ { * } \right) \right\| \leqslant M \left\| x - x ^ { * } \right\| ^ { 2 }$$ Show that there exists $\varepsilon > 0$ such that for all $x _ { 0 } \in \mathbb { R } ^ { 2 }$ satisfying $\left\| x _ { 0 } - x ^ { * } \right\| < \varepsilon$, the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ defined by $x _ { n + 1 } = \phi \left( x _ { n } \right)$ (for $n \geqslant 0$) converges to $x ^ { * }$ when $n \rightarrow + \infty$.
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. 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 $\mathcal{B}_{n}$.
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 $\mathscr{P}$ 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 $i,j \in \{1,\ldots,d\}$, $$P_{i,j} \geqslant c\nu_j.$$ Let $(x_n)_n \in \mathscr{P}^{\mathbb{N}}$ be defined by recursion by $x_0 \in \mathscr{P}$ and $$x_{n+1} = x_n P.$$ Show that the series $\sum_{n \geqslant 0} \|x_{n+1} - x_n\|_1$ is convergent.
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 $i,j \in \{1,\ldots,d\}$, $$P_{i,j} \geqslant c\nu_j.$$ Let $(x_n)_n \in \mathscr{P}^{\mathbb{N}}$ be defined by recursion by $x_0 \in \mathscr{P}$ and $x_{n+1} = x_n P$. Deduce that $(x_n)_n$ converges to an element of $\mathscr{P}$.
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 $\pi M = \lambda \pi$.
We use the notations of the previous parts. For $u \in \mathscr{M}_{d,1}(\mathbb{R})$, we denote $T(u) = (T_i(u))_{1 \leqslant i \leqslant d} \in \mathbb{R}^d$ the vector defined by $$T_i(u) = \operatorname{Var}\left(\langle L_i, u \rangle\right) \quad \text{for } i \in \{1,\ldots,d\}.$$ (a) Show that there exists $c_0 \geqslant 0$ such that for all $u \in \mathscr{M}_{d,1}(\mathbb{R})$, we have $\|T(u)\|_1 \leqslant c_0 \|u\|_2^2$. (b) Deduce the existence of $c_1 \geqslant 0$ such that for all $u \in \mathscr{M}_{d,1}(\mathbb{R})$, we have $\|T(u)\|_1 \leqslant c_1 \|u\|_1^2$.