We set $A_1 = \left(\begin{array}{ccc} 3 & -2 & 4 \\ -2 & 6 & 2 \\ 4 & 2 & 3 \end{array}\right)$.
Orthodiagonalize $A_1$.

Show that, if $A$ is nilpotent, that is, if there exists $p \in \mathbb{N}^\star$ such that $A^p = 0_n$, then the spectral radius of $A$ is zero.

In this part, $E$ denotes a $\mathbf{C}$-vector space of dimension $n$ and $u$ denotes an endomorphism of $E$.
Assume that every vector subspace of $E$ has a complement in $E$, stable under $u$. Prove that $u$ is diagonalizable. Deduce a characterization of diagonalizable matrices in $M_{n}(\mathbf{C})$.
Hint: one may reason by contradiction and introduce a vector subspace, whose existence one will justify, of dimension $n-1$ and containing the sum of the eigenspaces of $u$.

Let $J \in M_{n}(\mathbb{C})$ be the matrix defined by
$$J = \left( \begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & 0 & 1 \\ 1 & 0 & \ldots & 0 & 0 \end{array} \right).$$
Calculate $J^{n}$ and show that $J$ is diagonalisable.

Let $J \in M_{n}(\mathbb{C})$ be the matrix defined by
$$J = \left( \begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & 0 & 1 \\ 1 & 0 & \ldots & 0 & 0 \end{array} \right).$$
Calculate the eigenvalues of $J$.

Show that a matrix $S \in S _ { n } ( \mathbf { R } )$ belongs to $S _ { n } ^ { + } ( \mathbf { R } )$ if, and only if, $\operatorname { Sp } ( S ) \subset \mathbf { R } _ { + }$.

Show that, if $A \in S _ { n } ^ { + + } ( \mathbf { R } )$, there exists $S \in S _ { n } ^ { + + } ( \mathbf { R } )$ such that $A = S ^ { 2 }$.

In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$.
Show that there exist a non-zero natural number $r$, distinct complex numbers $\lambda _ { 1 } , \lambda _ { 2 } , \ldots$, $\lambda _ { r }$, and non-zero natural numbers $m _ { 1 } , m _ { 2 } , \ldots , m _ { r }$, such that: $$\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$$ where for $i \in \llbracket 1 ; r \rrbracket , E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$.

In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with the projections $p_i$ and inclusions $q_i$ as defined.
Let $( i , j ) \in \llbracket 1 ; r \rrbracket ^ { 2 }$. Express $p _ { i } q _ { j }$ and then $\sum _ { i = 1 } ^ { r } q _ { i } p _ { i }$ in terms of the endomorphisms $id _ { \mathbf { C } ^ { n } }$ and $id _ { E _ { j } }$.

In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with the projections $p_i$, inclusions $q_i$, and $a_i = p_i a q_i$ the endomorphism of $E_i$.
Show that: $a = \sum _ { i = 1 } ^ { r } q _ { i } a _ { i } p _ { i }$.

Let $E$ be a Euclidean space of dimension $N$. We denote by $(|)$ the inner product and $\|\cdot\|$ the associated Euclidean norm. Let $u$ be a self-adjoint endomorphism of $E$. We define $q_u : E \rightarrow \mathbf{R}$ by $q_u : x \mapsto (u(x) \mid x)$ and we assume that for all $x \in E$, $q_u(x) \geq 0$. State the spectral theorem for the endomorphism $u$. What can be said about the eigenvalues of $u$?

Let $E$ be a Euclidean space of dimension $N$. Let $u$ be a self-adjoint endomorphism of $E$ such that for all $x \in E$, $q_u(x) = (u(x) \mid x) \geq 0$. We assume that 0 is a simple eigenvalue of $u$ and we denote by $\lambda_2$ the smallest nonzero eigenvalue of $u$. We denote by $p : E \rightarrow E$ the orthogonal projection onto the vector line $\ker(u)$. Show that for all $x \in E$, $q_u(x - p(x)) \geq \lambda_2 \|x - p(x)\|^2$.

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$ and we set $\chi = \operatorname { det } \left( X I _ { n } - M \right) \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$.
Show that $M _ { \mid t = 0 }$ admits a real eigenvalue.

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$ and we set $\chi = \operatorname { det } \left( X I _ { n } - M \right) \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$. By Theorem 1, there exists $\rho _ { 1 } \in \mathbb { R } _ { + } ^ { * }$, $\rho _ { 1 } \leqslant \rho$ such that $\chi$ factors in the form $\chi = F G$ with $F \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathbb { R } _ { d } [ X ] \right)$ and $G \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathbb { R } _ { n - d } [ X ] \right)$ and $F _ { \mid t = 0 } = ( X - \lambda ) ^ { d }$.
Only in this question, we assume that $d = n$. Show that there exists a symmetric matrix $M _ { 0 } \in \mathscr { D } _ { \rho _ { 1 } } \left( S _ { n } ( \mathbb { R } ) \right)$ such that $M = \lambda I _ { n } + t M _ { 0 }$ for all $t \in U _ { \rho _ { 1 } }$.

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$ and $B = G ( M )$; we thus have $A , B \in \mathscr { D } _ { \rho _ { 1 } } \left( S _ { n } ( \mathbb { R } ) \right)$. For $a \in U _ { \rho _ { 1 } }$, we set $A _ { a } = A _ { \mid t = a }$ and $B _ { a } = B _ { \mid t = a }$.
Show that there exist two matrices $U \in \mathscr { M } _ { n , d } ( \mathbb { R } )$ and $V \in \mathscr { M } _ { n , n - d } ( \mathbb { R } )$ such that:

• $\operatorname { im } \left( B _ { 0 } U \right) = \operatorname { im } \left( B _ { 0 } \right)$,
• $\operatorname { im } \left( A _ { 0 } V \right) = \operatorname { im } \left( A _ { 0 } \right)$ and
• the block matrix $\left( B _ { 0 } U \mid A _ { 0 } V \right)$ is invertible.

We use the notations of the previous parts. We assume that there exists an eigenvalue $\lambda > 0$ and an associated eigenvector column $h \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$ with $Mh = \lambda h$, and that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that $M_{i,j} \geqslant c\nu_j$ for all $i,j$. Let $\pi \in \mathscr{P}$ be such that $\pi M = \lambda \pi$, and let $C > 0$, $\gamma \in [0,1[$ be as in question 17.
(a) Show that for all $n \geqslant 0$ and $u \in \mathscr{M}_{d,1}(\mathbb{R})$ such that $\langle u, \pi \rangle = 0$, $$\left\| M^n u \right\|_1 \leqslant C(\lambda\gamma)^n \|u\|_1.$$
(b) Deduce that there exists $C_1 \geqslant 0$ such that for all $n \geqslant 0$ and $u \in \mathscr{M}_{d,1}(\mathbb{R})$ column vector such that $\langle u, \pi \rangle = 0$, $$\mathbb{E}\left(\langle X_n, u \rangle^2\right) \leqslant C_1 \|u\|_1^2 \left(\lambda^{2n} \left(\sum_{k=0}^{n-1} \lambda^{-k} \gamma^{2n-2k}\right) + (\lambda\gamma)^{2n}\right).$$

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\varepsilon_0 > 0$ be such that for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$. Let $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$ and $\varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM)$. Show that $\varphi_\alpha$ is twice differentiable at 0 and that $$\varphi_\alpha''(0) = \operatorname{det}^{-\alpha}(A)\left(\alpha \operatorname{Tr}^2(A^{-1}M) + \operatorname{Tr}\left((A^{-1}M)^2\right)\right).$$

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$ and $B = G ( M )$, and $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ where $U \in \mathscr { M } _ { n , d } ( \mathbb { R } )$ and $V \in \mathscr { M } _ { n , n - d } ( \mathbb { R } )$ are as in question 20.
Show that there exists $\rho _ { 2 } \in \mathbb { R } _ { + } ^ { * }$, $\rho _ { 2 } \leqslant \rho _ { 1 }$ such that $Q \in \operatorname { GL } _ { n } \left( \mathscr { D } _ { \rho _ { 2 } } ( \mathbb { R } ) \right)$. (One may use the result of question 6.)

We suppose in the rest of this part that $\lambda > 1$ and we introduce the random row vector $$W_n = \lambda^{-n}\left(X_n - \|X_n\|_1 \pi\right).$$
(a) Show that the series $\displaystyle\sum_{n \geqslant 1} \left(\sum_{k=0}^{n-1} \lambda^{-k} \gamma^{2n-2k}\right)$ converges.
(b) Let $w \in (\mathbb{R}_{+})^d$ and let $e_0 = (1,\ldots,1)$. Show that $$\left\langle w - \|w\|_1 \pi, \pi \right\rangle = \left\langle w, \pi - \langle \pi, \pi \rangle e_0 \right\rangle$$ and that the vector $\pi - \langle \pi, \pi \rangle e_0$ is orthogonal to $\pi$.
(c) Show that the series $\displaystyle\sum_{n \geqslant 0} \mathbb{E}\left(\|W_n\|_2^2\right)$ is convergent. Deduce that the sequence $\left(\mathbb{E}\left(\|W_n\|_2^2\right)\right)_{n \geqslant 0}$ tends to $0$. (One may for example decompose $X_n$ in a well-chosen orthonormal basis of $\mathbb{R}^d$.)
(d) Show that for all $\varepsilon > 0$, $$\lim_{n \rightarrow \infty} \mathbb{P}\left(\|W_n\|_2 \geqslant \varepsilon\right) = 0.$$

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Show that $A^{-1}M$ is similar to a real symmetric matrix.
Hint: You may use question 3.

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$, $B = G ( M )$, $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$. For $a \in U _ { \rho _ { 2 } }$, we set $A _ { a } = A _ { \mid t = a }$ and $B _ { a } = B _ { \mid t = a }$. We consider a real number $a \in U _ { \rho _ { 2 } }$.
22a. Show that $\operatorname { im } \left( B _ { a } U \right) \oplus \operatorname { im } \left( A _ { a } V \right) = \mathbb { R } ^ { n }$.
22b. Show the equalities:

• $\operatorname { im } \left( B _ { a } U \right) = \operatorname { im } \left( B _ { a } \right) = \operatorname { ker } \left( A _ { a } \right)$ and
• $\operatorname { im } \left( A _ { a } V \right) = \operatorname { im } \left( A _ { a } \right) = \operatorname { ker } \left( B _ { a } \right)$.

(One may begin by showing the inclusions from left to right, then use a dimension argument.)

We suppose that $\lambda > 1$ and we use the random row vector $W_n = \lambda^{-n}\left(X_n - \|X_n\|_1 \pi\right)$.
Show that the event $\left\{\lim_{n \rightarrow +\infty} W_n = 0_{\mathbb{R}^d}\right\}$ is almost surely true. (One may begin by computing the probability of the event $$\left\{ \forall m \geqslant 0, \exists k \geqslant m \mid \|W_k\|_2 \geqslant \varepsilon \right\}$$ for all $\varepsilon > 0$.)

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$ and $\varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM)$. Using the fact that $A^{-1}M$ is similar to a real symmetric matrix, deduce that $\varphi_\alpha''(0) \geq 0$.

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$, $B = G ( M )$, $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$.
Show that $Q ^ { - 1 } \cdot M \cdot Q = \operatorname { Diag } \left( M _ { 1 } , M _ { 2 } \right)$ with $M _ { 1 } \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { d } ( \mathbb { R } ) \right) , M _ { 2 } \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n - d } ( \mathbb { R } ) \right)$.

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$, $B = G ( M )$, $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$.
Show that, for all $a \in U _ { \rho _ { 2 } }$, the direct sum $\operatorname { im } \left( B _ { a } U \right) \oplus \operatorname { im } \left( A _ { a } V \right) = \mathbb { R } ^ { n }$ of question 22a is orthogonal for the standard inner product on $\mathbb { R } ^ { n }$.