For the positive matrix $A$ of $\mathcal{M}_n(\mathbb{R})$, show that the following conditions are equivalent:

• the matrix $A$ is irreducible;
• the matrix $B = I_n + A + A^2 + \cdots + A^{n-1}$ is strictly positive;
• the matrix $C = (I_n + A)^{n-1}$ is strictly positive.

Let $A$ be irreducible. Show that no row (and no column) of $A$ is identically zero.

In this question, $A$ is a given irreducible matrix.
Suppose that $\forall i \in \llbracket 1, n \rrbracket, a_{i,i} > 0$. Show that $A^{n-1} > 0$ (so $A$ is primitive). Reason in terms of paths in $A$.

In this question, $A$ is a given irreducible matrix.
Suppose that: $\exists i \in \llbracket 1, n \rrbracket, a_{i,i} > 0$. Show that $A$ is primitive.
For all $j$ and $k$ in $\llbracket 1, n \rrbracket$, one can show that there exists in $A$ a path from $j$ to $k$ passing through $i$, and consider the maximum $m$ of the lengths of the paths thus obtained. One will prove that $A^m > 0$.

Let $A$ be an imprimitiv matrix with coefficient of imprimitivity $p \geqslant 2$.
For any integer $m$ that is not a multiple of $p$, show that the diagonal of $A^m$ is identically zero. One can be interested in the trace of $A^m$.
Deduce that the result of question IV.B.3 no longer holds if $A$ is imprimitiv.

Let $A \geqslant 0$ in $\mathcal{M}_n(\mathbb{R})$, an irreducible matrix. We denote $r$ its spectral radius. Let $p \geqslant 1$ be the coefficient of imprimitivity of $A$ (reminder: by convention, $p = 1$ if $A$ is primitive). Let $\chi_A(X) = X^n + c_{k_1}X^{n-k_1} + c_{k_1}X^{n-k_2} + \cdots + c_{k_s}X^{n-k_s}$ be its characteristic polynomial, written according to decreasing powers and showing only the nonzero coefficients $c_k$.
We recall that the spectrum of $A$ is invariant under the map $z \mapsto \omega z$, where $\omega = \exp(2\mathrm{i}\pi/p)$.
Deduce that, for all $k \in \{k_1, k_2, \ldots, k_s\}$, the integer $k$ is divisible by $p$. Think of the elementary symmetric functions of the $\lambda_i$.

Let $A \geqslant 0$ in $\mathcal{M}_n(\mathbb{R})$, an irreducible matrix. We denote $r$ its spectral radius. Let $p \geqslant 1$ be the coefficient of imprimitivity of $A$. Let $\chi_A(X) = X^n + c_{k_1}X^{n-k_1} + c_{k_1}X^{n-k_2} + \cdots + c_{k_s}X^{n-k_s}$ be its characteristic polynomial, written according to decreasing powers and showing only the nonzero coefficients $c_k$. We will show that $p$ is the gcd of the integers $k_1, k_2, \ldots, k_s$.
Conversely, we assume by contradiction that the $k_j$ are all divisible by $qp$, with $q \geqslant 2$. We set $\beta = \mathrm{e}^{2\mathrm{i}\pi/(qp)}$ (so $\beta^q = \omega$). Show that $\beta r$ is an eigenvalue of $A$ and conclude.

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 $i$).

Specify the set of eigenvalues of $\tau$. Is the application $\tau$ diagonalizable?

Is the application $\tau$ bijective? If so, specify $\tau^{-1}$. Is the expression of $\tau^j$ found in question I.A.2 for $j \in \mathbb{N}$ valid for $j \in \mathbb{Z}$?

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

Give an example of a symmetric matrix $S$ in $\mathcal{S}_{2}(\mathbb{R})$ such that $\operatorname{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and for which there does not exist a matrix $A \in \mathrm{O}_{2}(\mathbb{R})$ satisfying $A_{s} = S$.

Let $S \in \mathcal{S}_{n}(\mathbb{R})$.
a) We assume that $\operatorname{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and that for every eigenvalue $\lambda$ of $S$ in $]-1,1[$, the eigenspace of $S$ associated with $\lambda$ has even dimension. Show that there exists $A \in \mathrm{O}_{n}(\mathbb{R})$ such that $A_{s} = S$.
b) Conversely, show that if there exists $A \in \mathrm{O}_{n}(\mathbb{R})$ such that $A_{s} = S$, then $\operatorname{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and for every eigenvalue $\lambda$ of $S$ in $]-1,1[$, the eigenspace of $S$ associated with $\lambda$ has even dimension.

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part.
Give an example of a positively stable matrix $A$ such that $A_{s}$ is not positive definite.

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part. Recall that for any matrix $M \in \mathcal{M}_{n}(\mathbb{C})$, $\exp(M) = \sum_{k=0}^{\infty} \frac{M^{k}}{k!}$.
Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix with complex eigenvalues $\lambda_{1}, \ldots, \lambda_{n}$ and let $\alpha$ be a real number such that $0 < \alpha < \min_{1 \leqslant j \leqslant n} \operatorname{Re}(\lambda_{j})$.
Show that the function $t \mapsto \mathrm{e}^{\alpha t}\exp(-tA)$ is bounded on $\mathbb{R}^{+}$.
One may apply question III.B.2 to an upper triangular matrix $T$ similar to $A - \alpha I_{n}$.

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part. 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.

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part. 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 $\forall M \in \mathcal{M}_{n}(\mathbb{R}), \Phi(M) = A^{\top}M + MA$.
a) Show that there exists a unique matrix $B \in \mathcal{M}_{n}(\mathbb{R})$ such that $A^{\top}B + BA = I_{n}$.
b) Show that $B$ is symmetric and that $\operatorname{det}(B) > 0$.

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part. Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix. Recall that $\exp(M) = \sum_{k=0}^{\infty} \frac{M^{k}}{k!}$ for any $M \in \mathcal{M}_{n}(\mathbb{C})$. For all real $t$, we set $V(t) = \exp(-tA^{\top})\exp(-tA)$ and $W(t) = \int_{0}^{t} V(s)\,\mathrm{d}s$.
a) Show that, for all real $t$, $V(t) \in \mathcal{S}_{n}^{++}(\mathbb{R})$ and that, if $t > 0$, $W(t) \in \mathcal{S}_{n}^{++}(\mathbb{R})$.
b) Show that, for all real $t$, $A^{\top}W(t) + W(t)A = I_{n} - V(t)$.
c) What do we obtain by letting $t$ tend to $+\infty$ in the previous equality? Deduce that the matrix $B$ of question III.C.2 is positive definite.

Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$ for all $k \in \mathbb{N}$. Prove that $\forall k \in \mathbb { N } , \Phi _ { k + p } = \Phi _ { k } \Phi _ { p }$.

Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$. The matrix $\Phi _ { p }$ is called the Floquet matrix of equation (III.1) and its complex eigenvalues are called the Floquet multipliers of (III.1). Let $\rho$ be a Floquet multiplier of (III.1).
a) Prove that there exists a nonzero solution $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ of (III.1) satisfying $\forall k \in \mathbb { N } , Y _ { k + p } = \rho Y _ { k }$.
b) Let $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ be such a solution, prove that, if $| \rho | < 1 , \lim _ { k \rightarrow + \infty } \left\| Y _ { k } \right\| _ { \infty } = 0$.

We still assume that $p$ is an integer greater than or equal to 2. We denote by $B$ a matrix in $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$. Deduce that $\Phi _ { p }$ is diagonalizable if and only if $B$ is diagonalizable.

We fix a natural number $p$ greater than or equal to 2. Prove that every invertible matrix in $\mathcal { M } _ { n } ( \mathbb { C } )$ admits at least one $p$-th root.
One may use the fact that every upper triangular and invertible matrix admits at least one upper triangular $p$-th root (as established in V.B.3).

Say, by briefly justifying the answer, whether the following assertions are correct for all $A , B \in M _ { n } ( \mathbb { C } ) , \mu \in \mathbb { C }$. i) $\rho ( \mu A ) = | \mu | \rho ( A )$. ii) $\rho ( A + B ) \leqslant \rho ( A ) + \rho ( B )$. iii) $\rho ( A B ) \leqslant \rho ( A ) \rho ( B )$. iv) For $P \in M _ { n } ( \mathbb { C } )$ invertible, $\rho \left( P ^ { - 1 } A P \right) = \rho ( A )$. v) $\rho \left( { } ^ { t } A \right) = \rho ( A )$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$. We consider the map $S$ from $E \times E$ to $\mathbb{R}$ defined by $$\forall (v,w) \in E^2, S(v,w) = (v \mid T(w)) + (T(v) \mid w)$$ and we denote by $G$ the set of elements $u \in E$ satisfying: (a) $u \in \operatorname{Im}(T)$, (b) $\forall v \in E, S(u,v) = 0$.
Show that $G$ is a vector subspace of $E$ and that $G \cap \operatorname{ker}(T) = \{0_E\}$.

Show that for any matrix $A \in M _ { n } ( \mathbb { C } )$, $$\rho ( A ) \leqslant \| A \| .$$