Let $A \in \mathcal{M}_n(\mathbb{R})$ be an irreducible matrix. For all $i$ in $\llbracket 1, n \rrbracket$, we denote $L_i = \{m \in \mathbb{N}^*, a_{i,i}^{(m)} > 0\}$ the (nonempty) set of lengths of circuits of $A$ passing through $i$, and we denote $d_i$ the gcd of the elements of $L_i$.
Establish that the coefficient of imprimitivity $p$ of $A$ is equal to $d_i$ for all $i$ in $\llbracket 1, n \rrbracket$ (this gcd does not depend on the index $i$).

Determine the spectral radius of the following matrices $$\left( \begin{array} { l l } 0 & 0 \\ 0 & 1 \end{array} \right) , \quad \left( \begin{array} { l l } 0 & 0 \\ 1 & 0 \end{array} \right) , \quad \left( \begin{array} { l l } 1 & 0 \\ 0 & 0 \end{array} \right) , \quad \left( \begin{array} { c c } 0 & - 1 \\ 2 & 0 \end{array} \right) , \quad \left( \begin{array} { l l } 3 & 2 \\ 1 & 2 \end{array} \right)$$

Let $x , y \in \mathbb { C } ^ { n } , \lambda , \mu \in \mathbb { C }$. Show that if $\lambda \neq \mu$, then the following implication holds $$\left( A x = \lambda x \quad \text { and } \quad { } ^ { t } A y = \mu y \right) \Longrightarrow { } ^ { t } x y = 0 .$$

Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$. Let $\lambda$ be an eigenvalue of $A$ with modulus $\rho ( A )$ and let $x \in \mathbb { C } ^ { n } \backslash \{ 0 \}$ be an eigenvector of $A$ associated with $\lambda$. We define the positive non-zero vector $v _ { 0 }$ by $\left( v _ { 0 } \right) _ { i } = \left| x _ { i } \right|$ for $1 \leqslant i \leqslant n$. a) Show that $A v _ { 0 } \geqslant \rho ( A ) v _ { 0 }$, then that $$A v _ { 0 } = \rho ( A ) v _ { 0 }$$ b) Deduce that $\rho ( A ) > 0$ and $$\forall i \in \llbracket 1 , n \rrbracket , \left( v _ { 0 } \right) _ { i } > 0 .$$ c) Show that $x$ is collinear with $v _ { 0 }$. Deduce that $\lambda = \rho ( A )$.

Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$. By applying the previous results to the matrix ${ } ^ { t } A$, we obtain the existence of $w _ { 0 } \in \mathbb { R } ^ { n }$, whose all components are strictly positive, such that ${ } ^ { t } A w _ { 0 } = \rho ( A ) w _ { 0 }$. We set $$F = \left\{ x \in \mathbb { C } ^ { n } \mid { } ^ { t } x w _ { 0 } = 0 \right\}$$ a) Show that $F$ is a vector subspace of $\mathbb { C } ^ { n }$ stable by $\varphi _ { A }$, and that $$\mathbb { C } ^ { n } = F \oplus \mathbb { C } v _ { 0 }$$ b) Show that if $v$ is an eigenvector of $A$ associated with an eigenvalue $\mu \neq \rho ( A )$, then $v \in F$. Deduce property (iii): if $v$ is an eigenvector of $A$ whose all components are positive, then $v \in \operatorname { ker } ( A - \rho ( A ) I _ { n } )$.

We consider $A \in \mathcal{M}_{n}(\mathbb{R})$. For every real eigenvalue $\lambda$ of $A$, show that $\min \operatorname{sp}_{\mathbb{R}}(A_{s}) \leqslant \lambda \leqslant \max \operatorname{sp}_{\mathbb{R}}(A_{s})$.
Deduce that if $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$ then $A$ is invertible.

Let $A \in \mathrm{O}_{n}(\mathbb{R})$. Show that the eigenvalues of $A_{s}$ are in $[-1,1]$.

Let $F$ be a vector subspace of $E_{n}$ of dimension $n-p$, where $1 \leqslant p \leqslant n-1$. We now assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$. Deduce that the real eigenvalues of $N^{\prime\top} A N^{\prime}$ are strictly positive.

Let $A \in \mathcal{M}_{2}(\mathbb{R})$. Show that $A$ is positively stable if and only if $\operatorname{tr}(A) > 0$ and $\operatorname{det}(A) > 0$.

a) Is the sum of two positively stable matrices of $\mathcal{M}_{2}(\mathbb{R})$ necessarily positively stable?
b) Let $A, B$ in $\mathcal{M}_{n}(\mathbb{R})$ be two positively stable matrices that commute. Show that $A + B$ is positively stable.

Let $A \in \mathcal{M}_{n}(\mathbb{R})$ such that $A_{s}$ is positive definite.
a) Let $X = Y + \mathrm{i} Z$ be a column matrix of $\mathcal{M}_{n,1}(\mathbb{C})$, where $Y$ and $Z$ belong to $\mathcal{M}_{n,1}(\mathbb{R})$. We set $\bar{X} = Y - \mathrm{i} Z$ and we identify the matrix $\bar{X}^{\top} A X \in \mathcal{M}_{1}(\mathbb{C})$ with the complex number equal to its unique entry.
Show that, if $X \neq 0$, then $\operatorname{Re}\left(\bar{X}^{\top} A X\right) > 0$, where $\operatorname{Re}(z)$ denotes the real part of $z \in \mathbb{C}$.
b) Show that $A$ is positively stable.

Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix. We consider the endomorphism $\Phi$ of $\mathcal{M}_{n}(\mathbb{R})$ such that $$\forall M \in \mathcal{M}_{n}(\mathbb{R}), \quad \Phi(M) = A^{\top} M + MA$$ Show that $\Phi$ is positively stable, that is, its matrix in any basis of $\mathcal{M}_{n}(\mathbb{R})$ is positively stable.

We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. We assume that $E$ is of dimension 4. Let $\mathcal { B }$ be a basis of $E$ such that $\operatorname { Mat } _ { \mathcal { B } } ( \omega ) = J _ { 4 }$. Let $U \in \mathcal { M } _ { 4 } ( \mathbb { R } )$ be the matrix of $u$ in the basis $\mathcal { B }$.
(a) What relation is there between the matrices $J _ { 4 }$ and $U$ ?
(b) Show that there exist $N \in \mathcal { M } _ { 2 } ( \mathbb { R } )$ and $\alpha , \beta \in \mathbb { R }$ such that $$U = \left( \begin{array} { c c } N & \alpha J _ { 2 } \\ \beta J _ { 2 } & { } ^ { t } N \end{array} \right)$$
(c) Determine, as a function of $N , \alpha$ and $\beta$, the coefficients of the polynomial $T$ defined by $T ( X ) = \operatorname { det } \left( N - X I _ { 2 } \right) + \alpha \beta$. Show that $T$ is an annihilating polynomial of $U$.

We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$. Let $\mathcal { S }$ be the set defined in question 13: $$\mathcal { S } = \left\{ u \in \mathrm { GL } ( E ) : \forall ( x , y ) \in E ^ { 2 } , \omega ( x , u ( y ) ) = \omega ( u ( x ) , y ) \right\}$$ Show that the set of elements of $\mathcal { S }$ whose characteristic polynomial $P$ has roots of multiplicity at most 2 in $\mathbb { C }$ is dense in $\mathcal { S }$.
Hint: You may use $r \left( P ^ { \prime } \right)$ where the map $r$ is defined in question 11.

Let $A = \left(\begin{array}{cc} a & b \\ c & a \end{array}\right)$ be a Toeplitz matrix of size $2 \times 2$, where $(a, b, c)$ are complex numbers. Give the characteristic polynomial of $A$.

Let $(a_0, \ldots, a_{n-1}) \in \mathbb{C}^n$. If $\lambda$ is a root of the polynomial identified in Q24, determine the eigenspace of $C(a_0, \ldots, a_{n-1})$ associated with the eigenvalue $\lambda$ and specify its dimension.

Let $M \in \mathcal{M}_n(\mathbb{K})$. Show that $M$ and $M^{\top}$ have the same spectrum.

Let $\left(a_0, a_1, \ldots, a_{n-1}\right) \in \mathbb{K}^n$ and $Q(X) = X^n + a_{n-1}X^{n-1} + \cdots + a_0$. We consider the matrix
$$C_Q = \left(\begin{array}{cccccc} 0 & \cdots & \cdots & \cdots & 0 & -a_0 \\ 1 & 0 & \cdots & \cdots & 0 & -a_1 \\ 0 & 1 & \ddots & & \vdots & -a_2 \\ \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ \vdots & & \ddots & 1 & 0 & -a_{n-2} \\ 0 & \cdots & \cdots & 0 & 1 & -a_{n-1} \end{array}\right).$$
Determine as a function of $Q$ the characteristic polynomial of $C_Q$.

In this part, we assume that $\mathbb{K} = \mathbb{R}$ and that $E$ is a Euclidean space. We say that an endomorphism $f$ of $E$ is orthocyclic if there exists an orthonormal basis of $E$ in which the matrix of $f$ is of the form $C_Q$ (companion matrix).
Let $f \in \mathrm{O}(E)$. Deduce that $f$ is orthocyclic if and only if $\chi_f = X^n - 1$ or $\chi_f = X^n + 1$.

Let $M \in \mathcal{M}_n(\mathbb{K})$. Show that $M$ and $M^{\top}$ have the same spectrum.

Let $\left(a_0, a_1, \ldots, a_{n-1}\right) \in \mathbb{K}^n$ and $Q(X) = X^n + a_{n-1}X^{n-1} + \cdots + a_0$. We consider the matrix
$$C_Q = \left(\begin{array}{cccccc} 0 & \cdots & \cdots & \cdots & 0 & -a_0 \\ 1 & 0 & \cdots & \cdots & 0 & -a_1 \\ 0 & 1 & \ddots & & \vdots & -a_2 \\ \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ \vdots & & \ddots & 1 & 0 & -a_{n-2} \\ 0 & \cdots & \cdots & 0 & 1 & -a_{n-1} \end{array}\right).$$
Determine as a function of $Q$ the characteristic polynomial of $C_Q$.

In this part, we assume that $\mathbb{K} = \mathbb{R}$ and that $E$ is a Euclidean space. The inner product of two vectors $x, y$ of $E$ is denoted $(x \mid y)$ and we denote by $\mathrm{O}(E)$ the group of vector isometries of $E$. We say that an endomorphism $f$ of $E$ is orthocyclic if there exists an orthonormal basis of $E$ in which the matrix of $f$ is of the form $C_Q$ (companion matrix).
Let $f \in \mathrm{O}(E)$. Deduce that $f$ is orthocyclic if and only if $\chi_f = X^n - 1$ or $\chi_f = X^n + 1$.

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Deduce that the nilpotent matrices in $\mathcal{M}_2(\mathbb{C})$ are exactly the matrices with zero trace and zero determinant.

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
Show that, if $A$ is nilpotent, then 0 is the unique eigenvalue of $A$.

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
Show that a matrix is nilpotent if, and only if, its characteristic polynomial is equal to $X^n$.