Let $A \in \mathcal{M}_{m,n}(\mathbb{R})$ be a matrix with $m$ rows and $n$ columns. We denote by $\langle u, v \rangle_{\mathbb{R}^n}$ the inner product between two vectors $u$ and $v$ of $\mathbb{R}^n$ and $\langle \mu, \nu \rangle_{\mathbb{R}^m}$ that between two vectors $\mu$ and $\nu$ of $\mathbb{R}^m$.
(a) Show that for all $(x, \nu) \in \mathbb{R}^n \times \mathbb{R}^m$, we have $$\langle Ax, \nu \rangle_{\mathbb{R}^m} = \left\langle x, A^\top \nu \right\rangle_{\mathbb{R}^n},$$ where $A^\top$ denotes the transpose matrix of $A$.
(b) Deduce that $\ker A \subset (\operatorname{Im} A^\top)^\perp$ where $E^\perp$ denotes the orthogonal complement of $E$ for the inner product on $\mathbb{R}^n$ for any vector subspace $E$ of $\mathbb{R}^n$.
(c) Show that $\ker A = (\operatorname{Im} A^\top)^\perp$.

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, and $P$ the matrix of order $n$ defined by $$P = I_n - \frac{1}{n} \mathbf{e} \cdot \mathbf{e}^T.$$
Show that $P$ is symmetric and that the endomorphism of $\mathbf{R}^n$ canonically associated is an orthogonal projection onto $\operatorname{Vect}(\mathbf{e})^\perp$.

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. We assume throughout this part that $A$ is diagonalizable in $\mathscr{M}_n(\mathbb{C})$ and we denote by $\lambda_1, \cdots, \lambda_\ell$ its eigenvalues with $\lambda_i \neq \lambda_j$ if $i \neq j$. For every $k \in \llbracket 1; \ell \rrbracket$ we define the polynomial: $$Q_k^A(X) = \prod_{j=1, j\neq k}^{\ell} \frac{X - \lambda_j}{\lambda_k - \lambda_j}$$
(a) Show that $$u(A) = \sum_{k=1}^{\ell} U(\lambda_k) Q_k^A(A).$$
(b) Show that for every $k \in \llbracket 1; \ell \rrbracket$, $Q_k^A(A)$ is a projection whose image and kernel we will specify.
(c) Deduce that $$\sum_{k=1}^{\ell} Q_k^A(A) = I_n.$$

Let $Z \in \mathscr{M}_{d}(\mathbb{R})$ be an invertible matrix with singular value decomposition $Z = VDU^{T}$ where $U = (u_{1}|\ldots|u_{d})$, $V = (v_{1}|\ldots|v_{d})$ are orthogonal matrices and $D = \operatorname{Diag}(\sqrt{\lambda_{1}}, \ldots, \sqrt{\lambda_{d}})$. We assume that $\operatorname{det}(Z) > 0$.

• [(a)] Show that if $R \in \mathrm{SO}_{d}(\mathbb{R})$ then $V^{T}RU \in \mathrm{SO}_{d}(\mathbb{R})$.
• [(b)] Show that $$\sup_{R \in \mathrm{SO}_{d}(\mathbb{R})} \langle Z, R \rangle = \sup_{R \in \mathrm{SO}_{d}(\mathbb{R})} \langle D, R \rangle$$

We consider $R \in \mathrm{O}_{d}(\mathbb{R})$ with $\operatorname{det}(R) = -1$.

• [(a)] Show that there exists an orthonormal basis $(u_{1}, \ldots, u_{d})$ of $\mathbb{R}^{d}$ such that $Ru_{d} = -u_{d}$ and $u_{d}^{T} Rx = 0$ for all $x \in E_{1}$ where $E_{1} = \operatorname{Vect}(u_{1}, \ldots, u_{d-1})$.
• [(b)] Deduce that $R(E_{1}) \subset E_{1}$ and then that $R(E_{1}) = E_{1}$.

We consider $R \in \mathrm{O}_{d}(\mathbb{R})$ with $\operatorname{det}(R) = -1$, and an orthonormal basis $(u_{1}, \ldots, u_{d})$ of $\mathbb{R}^{d}$ such that $Ru_{d} = -u_{d}$ and $R(E_{1}) = E_{1}$ where $E_{1} = \operatorname{Vect}(u_{1}, \ldots, u_{d-1})$. We consider a matrix $D = \operatorname{Diag}(\alpha_{1}, \ldots, \alpha_{d}) \in \mathscr{M}_{d}(\mathbb{R})$ diagonal with diagonal entries $\alpha_{i} \geqslant 0$ in decreasing order. We denote $U = (u_{1} | \ldots | u_{d})$.

• [(a)] Verify that $\langle D, R \rangle = \langle S, R^{\prime} \rangle$ where $R^{\prime} = U^{T}RU$ and $S = U^{T}DU$.
• [(b)] Show that if $R_{0} = (R_{ij}^{\prime})_{1 \leqslant i,j \leqslant d-1} \in \mathscr{M}_{d-1}(\mathbb{R})$ then $R_{0} \in \mathrm{O}_{d-1}(\mathbb{R})$.

We consider $R \in \mathrm{O}_{d}(\mathbb{R})$ with $\operatorname{det}(R) = -1$, and the notation from question 23. We set $S_{0} = (S_{ij})_{1 \leqslant i,j \leqslant d-1} \in \mathscr{M}_{d-1}(\mathbb{R})$.

• [(a)] Show that $\langle D, R \rangle = \operatorname{tr}(S_{0} R_{0}) - S_{dd}$.
• [(b)] Show that $\operatorname{tr}(S_{0} R_{0}) \leqslant \operatorname{tr}(S_{0})$.
• [(c)] Show that $\operatorname{tr}(S_{0}) + S_{dd} = \operatorname{tr}(D)$ and deduce that $\langle D, R \rangle \leqslant \operatorname{tr}(D) - 2S_{dd}$.

For every matrix $M \in S_n(\mathbf{R})$ construct a vector subspace $F_M$ of $\mathcal{M}_{n,1}(\mathbf{R})$ of dimension $\pi(M)$ satisfying condition $(\mathcal{C}_M)$: $$\forall X \in F \setminus \{0_{n,1}\} \quad X^\top M X > 0.$$ We thus have $d(M) \geq \pi(M)$.

We want to show that for every matrix $M \in S_n(\mathbf{R})$ we have $\pi(M) = d(M)$. By contradiction, assuming the existence of a vector subspace $G$ of $\mathcal{M}_{n,1}(\mathbf{R})$ of dimension $\dim G > \pi(M)$ satisfying condition $(\mathcal{C}_M)$, show $\dim(F_M^\perp \cap G) \geq 1$, deduce a contradiction and conclude.

Search for a stable complement In this part, we are given: a finite-dimensional vector space $V$; a nilpotent endomorphism $u$ of $V$; a nonzero natural integer $N$ and $\zeta = \exp\frac{2\mathrm{i}\pi}{N}$; an invertible endomorphism $h$ of $V$ such that $h^N = \operatorname{id}_V$ and $h \circ u \circ h^{-1} = \zeta u$. Let $W$ be a vector subspace of $V$ stable by $u$ and $h$. We assume that $W$ admits a complement $W'$ stable by $u$ and we seek a complement of $W$ stable by $u$ and $h$. Let $p$ be the projector onto $W$ parallel to $W'$.
a) Verify that $u$ and $p$ commute.
We denote $$\bar{p} = \frac{1}{N}\sum_{k=0}^{N-1} h^k \circ p \circ h^{-k}.$$
b) Prove that the image of $\bar{p}$ is contained in $W$ and that for $w$ in $W$, we have $\bar{p}(w) = w$.
c) Deduce that $\bar{p}$ is a projector and that its image is $W$.
d) Prove carefully that $\bar{p}$ commutes with $u$ and $h$.
e) Deduce that the kernel of $\bar{p}$ is a complement of $W$ and that it is stable under $u$ and $h$.

Let $P \in \mathcal{M}_n(\mathbb{R})$. Show that $P$ is an orthogonal projector of rank 1 if and only if there exists $\mathbf{y} \in \mathbb{R}^n$ with $\|\mathbf{y}\| = 1$ such that $P = \mathbf{y y}^T$.

Let $\left(\mathbf{v}_1, \ldots, \mathbf{v}_n\right)$ be any orthonormal basis of $\mathbb{R}^n$. Show that $$\mathbb{I}_n = \sum_{k=1}^n \mathbf{v}_k \mathbf{v}_k^T$$

Let $P \in \mathcal{M}_n(\mathbb{R})$. Show that $P$ is an orthogonal projector of rank 1 if and only if there exists $\mathbf{y} \in \mathbb{R}^n$ with $\|\mathbf{y}\| = 1$ such that $P = \mathbf{y y}^T$.

Let $W$ be a vector subspace of $V$ stable by $u$ and $h$. We assume that $W$ admits a complement $W'$ stable by $u$ and we seek a complement of $W$ stable by $u$ and $h$. Let $p$ be the projector onto $W$ parallel to $W'$. Verify that $u$ and $p$ commute.

We denote $$\bar{p} = \frac{1}{N} \sum_{k=0}^{N-1} h^k \circ p \circ h^{-k}.$$ Prove that the image of $\bar{p}$ is contained in $W$ and that for $w$ in $W$, we have $\bar{p}(w) = w$.

Deduce that $\bar{p}$ is a projector and that its image is $W$.

Prove carefully that $\bar{p}$ commutes with $u$ and $h$.

Deduce that the kernel of $\bar{p}$ is a complement of $W$ and that it is stable under $u$ and $h$.