Let $V$ and $V^{\prime}$ be two subspaces of $E$ of dimension $p$ (we recall that $p \geqslant 1$).
(a) Show that there exist $u_1 \in V$ and $u_1^{\prime} \in V^{\prime}$ of norm 1 such that $$\left\langle u_1, u_1^{\prime}\right\rangle = \sup\left\{\left\langle a, a^{\prime}\right\rangle \mid\left(a, a^{\prime}\right) \in V \times V^{\prime},\|a\|=\left\|a^{\prime}\right\|=1\right\}.$$
(b) Extend this result by showing that there exist a family $u = (u_1, \ldots, u_p)$ of vectors of $V$ and a family $u^{\prime} = (u_1^{\prime}, \ldots, u_p^{\prime})$ of vectors of $V^{\prime}$ such that $u$ and $u^{\prime}$ are orthonormal and satisfy the following two conditions:
(i) For $k = 1$, we have $$\left\langle u_1, u_1^{\prime}\right\rangle = \sup\left\{\left\langle a, a^{\prime}\right\rangle \mid\left(a, a^{\prime}\right) \in V \times V^{\prime},\|a\|=\left\|a^{\prime}\right\|=1\right\}.$$
(ii) For $k \in \llbracket 2, p \rrbracket$, we have $$\left\langle u_k, u_k^{\prime}\right\rangle = \sup\left\{\left\langle a, a^{\prime}\right\rangle \mid\left(a, a^{\prime}\right) \in V \times V^{\prime},\|a\|=\left\|a^{\prime}\right\|=1,\right.$$ $$\left.\left\langle a, u_l\right\rangle=\left\langle a^{\prime}, u_l^{\prime}\right\rangle=0 \text{ for all } l \in \llbracket 1, k-1 \rrbracket\right\}.$$
(Hint: One may construct the vectors $u_k$ and $u_k^{\prime}$ by induction on the integer $k$.)

Let $V$ and $V^{\prime}$ be two subspaces of $E$ of dimension $p$ (we recall that $p \geqslant 1$).
(a) Show that there exist $u_1 \in V$ and $u_1^{\prime} \in V^{\prime}$ of norm 1 such that $$\left\langle u_1, u_1^{\prime}\right\rangle = \sup\left\{\left\langle a, a^{\prime}\right\rangle \mid\left(a, a^{\prime}\right) \in V \times V^{\prime},\|a\|=\left\|a^{\prime}\right\|=1\right\}.$$
(b) Extend this result by showing that there exist a family $u = (u_1, \ldots, u_p)$ of vectors of $V$ and a family $u^{\prime} = (u_1^{\prime}, \ldots, u_p^{\prime})$ of vectors of $V^{\prime}$ such that $u$ and $u^{\prime}$ are orthonormal and satisfy the following two conditions:
(i) For $k = 1$, we have $$\left\langle u_1, u_1^{\prime}\right\rangle = \sup\left\{\left\langle a, a^{\prime}\right\rangle \mid\left(a, a^{\prime}\right) \in V \times V^{\prime},\|a\|=\left\|a^{\prime}\right\|=1\right\}.$$
(ii) For $k \in \llbracket 2, p \rrbracket$, we have $$\left\langle u_k, u_k^{\prime}\right\rangle = \sup\left\{\left\langle a, a^{\prime}\right\rangle \mid\left(a, a^{\prime}\right) \in V \times V^{\prime},\|a\|=\left\|a^{\prime}\right\|=1,\right.$$ $$\left.\left\langle a, u_l\right\rangle=\left\langle a^{\prime}, u_l^{\prime}\right\rangle=0 \text{ for all } l \in \llbracket 1, k-1 \rrbracket\right\}.$$
(Hint: One may construct the vectors $u_k$ and $u_k^{\prime}$ by induction on the integer $k$.)

Let $A$ and $B$ be two matrices in $\mathcal { M } _ { n } ( \mathbb { R } )$ such that
$$\forall ( X , Y ) \in \left( \mathcal { M } _ { n , 1 } ( \mathbb { R } ) \right) ^ { 2 } , \quad X ^ { \top } A Y = X ^ { \top } B Y .$$
Show that $A = B$.

Let $A$ and $B$ be two matrices in $\mathcal { M } _ { n } ( \mathbb { R } )$ such that
$$\forall ( X , Y ) \in \left( \mathcal { M } _ { n , 1 } ( \mathbb { R } ) \right) ^ { 2 } , \quad X ^ { \top } A Y = X ^ { \top } B Y .$$
Show that $A = B$.

Prove that the application $$\begin{array} { | c l l } \mathcal { M } _ { n } ( \mathbb { R } ) & \rightarrow & \mathbb { R } \\ M & \mapsto & \operatorname { tr } ( M ) \end{array}$$ is a linear form and that $$\forall ( A , B ) \in \left( \mathcal { M } _ { n } ( \mathbb { R } ) \right) ^ { 2 } , \quad \operatorname { tr } ( A B ) = \operatorname { tr } ( B A ) .$$

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 $$\begin{gathered} \forall M \in \mathcal{M}_{n}(\mathbf{C}), \quad \|M\|_{\mathrm{op}} = \max\left\{\frac{\|MX\|}{\|X\|}; X \in \mathcal{M}_{n,1}(\mathbf{C}) \backslash \{0\}\right\}, \\ \forall (M, M^{\prime}) \in \mathcal{M}_{n}(\mathbf{C})^{2}, \quad \|M^{\prime}M\|_{\mathrm{op}} \leq \|M^{\prime}\|_{\mathrm{op}} \|M\|_{\mathrm{op}}. \end{gathered}$$

For a triangular matrix $T = \left( \begin{array} { l l } \lambda & a \\ 0 & \mu \end{array} \right) \in \mathbf { M } _ { 2 } ( \mathbb { C } )$, explicitly compute the successive powers $T ^ { n }$ for $n$ a strictly positive integer.

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 $$\left\| A ^ { n } x \right\| \leqslant \beta ( \rho ( A ) + \epsilon ) ^ { n } \| x \|$$

For a triangular matrix $T = \left( \begin{array} { l l } \lambda & a \\ 0 & \mu \end{array} \right) \in \mathbf { M } _ { 2 } ( \mathbb { C } )$, explicitly compute the successive powers $T ^ { n }$ for $n$ a strictly positive integer.

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 $$\left\| A ^ { n } x \right\| \leqslant \beta ( \rho ( A ) + \epsilon ) ^ { n } \| x \|$$

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

The bilinear form $B$ is defined by $$B:\left(\begin{pmatrix}x\\y\\z\end{pmatrix},\begin{pmatrix}x'\\y'\\z'\end{pmatrix}\right)\mapsto 3xx'+3yy'-zz'.$$ Given a vector $v\in V$, the pseudo-orthogonal of $v$ is $v^\perp = \{w\in V \mid B(v,w)=0\}$.
Let $v$ be a non-zero vector of $V$. Show that $v^\perp$ is a vector subspace of $V$ of codimension 1, and that $v^\perp$ is a complement of the line generated by $v$ if and only if $B(v,v)\neq 0$.

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 $v_1 = v_2$.

Deduce that if $v\in\mathcal{H}$, then the restriction of $B$ to $v^\perp$ is an inner product.

$\mathbf{3}$ ▷ 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 $V$ and $V^{\prime}$ be two subspaces of $E$ of dimension $p$. Let $u = (u_1, \ldots, u_p)$ and $u^{\prime} = (u_1^{\prime}, \ldots, u_p^{\prime})$ be the orthonormal families constructed in 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 $t \mapsto u_k(t) = \frac{u_k + t u_l}{\|u_k + t u_l\|}$ for all $t \in \mathbb{R}$ and $l \in \llbracket k+1, p \rrbracket$ 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.

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 $\left\| e^{A} \right\| \leq e^{\|A\|}$.

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.

We consider the three vectors $$w_1 = \begin{pmatrix}0\\1\\0\end{pmatrix}, \quad w_2 = \begin{pmatrix}1\\-1\\0\end{pmatrix}, \quad w_3 = \begin{pmatrix}-1\\0\\-1\end{pmatrix}.$$ Verify that $s_{w_1}$, $s_{w_2}$ and $s_{w_3}$ belong to $\Gamma$ and calculate the corresponding matrices.

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.