We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Deduce that $\operatorname{det}\left(A_{N}\right) = \operatorname{det}\left(N^{\top} A^{-1} N\right) \operatorname{det}(A)$.

We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Show that there exists $P \in \mathcal{G}_{n,2}(\mathbb{R})$ such that $\operatorname{det}\left(P^{\top} A^{-1} P\right) = 0$ if and only if there exists $P^{\prime} \in \mathcal{G}_{n,2}(\mathbb{R})$ such that $\operatorname{det}\left(P^{\prime\top} A P^{\prime}\right) = 0$.

We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Show that if $N^{\prime} = \left(\begin{array}{ll} N_{1}^{\prime} & N_{2}^{\prime} \end{array}\right)$ then $$\operatorname{det}\left(N^{\prime\top} A N^{\prime}\right) = \left(N_{1}^{\prime\top} A_{s} N_{1}^{\prime}\right)\left(N_{2}^{\prime\top} A_{s} N_{2}^{\prime}\right) - \left(N_{1}^{\prime\top} A_{s} N_{2}^{\prime}\right)^{2} + \left(N_{1}^{\prime\top} A_{a} N_{2}^{\prime}\right)^{2}$$

We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Deduce that if $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$, then $\operatorname{det}\left(N^{\top} A^{-1} N\right) > 0$.

We return to the example of subsection II.B with $\mu = 1$, i.e., $$A(1) = \left(\begin{array}{ccc} 1 & -1 & 1 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \end{array}\right)$$ How should we choose $N^{\prime} = \left(\begin{array}{ll} N_{1}^{\prime} & N_{2}^{\prime} \end{array}\right)$ so that $\operatorname{det}\left(N^{\prime\top} A N^{\prime}\right) = 0$?

We return to the example of subsection II.B with $\mu = 1$, i.e., $$A(1) = \left(\begin{array}{ccc} 1 & -1 & 1 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \end{array}\right)$$ Determine a vector subspace $F$ of $E_{3}$ such that $\dim F = 1$ and such that $A(1)$ is $F$-singular.

Let $F$ be a vector subspace of $E_{n}$ of dimension $n-p$, where $1 \leqslant p \leqslant n-1$. Show that $A$ is $F$-singular if $\operatorname{det}\left(N^{\prime\top} A N^{\prime}\right) = 0$ for a matrix $N^{\prime} \in \mathcal{G}_{n,p}(\mathbb{R})$ that one will define.

Let $F$ be a vector subspace of $E_{n}$ of dimension $n-p$, where $1 \leqslant p \leqslant n-1$. We now assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$. Show that if $X \in \mathcal{M}_{p,1}(\mathbb{R})$ is non-zero then $X^{\top} N^{\prime\top} A N^{\prime} X > 0$.

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 $F$ 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 $\operatorname{det}\left(N^{\prime\top} A N^{\prime}\right) > 0$.

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 $A$ is $F$-regular for every non-zero vector subspace $F$ of $E_{n}$.

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.

Give an example of a positively stable matrix $A$ such that $A_{s}$ is not positive definite.

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.

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 $$\forall M \in \mathcal{M}_{n}(\mathbb{R}), \quad \Phi(M) = A^{\top} M + MA$$
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 $\operatorname{det}(B) > 0$.

Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix. For all real $t$, we set $V(t) = \exp\left(-tA^{\top}\right) \exp(-tA)$ 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.

Verify the following property: $\forall ( A , B ) \in \left( \mathcal { M } _ { n } ( \mathbb { C } ) \right) ^ { 2 } \quad \| A B \| _ { 0 } \leqslant n \| A \| _ { 0 } \cdot \| B \| _ { 0 }$

Verify the following property: $\forall A \in \mathcal { M } _ { n } ( \mathbb { C } ) , \forall Y \in \mathbb { C } ^ { n } \quad \| A Y \| _ { \infty } \leqslant n \| A \| _ { 0 } \cdot \| Y \| _ { \infty }$

We assume that $p$ is an integer greater than or equal to 2, that $\left( a _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( b _ { k } \right) _ { k \in \mathbb { N } }$ are two sequences of real numbers that are $p$-periodic and that $\forall k \in \mathbb { N } , b _ { k } \neq 0$. We denote by Sol(II.2) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ that satisfy the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ Justify that the application $\left. \Psi : \left\lvert \, \begin{array} { l l l } \operatorname { Sol } ( \mathrm { II } .2 ) & \rightarrow & \mathbb { C } ^ { 2 } \\ \left( z _ { k } \right) _ { k \in \mathbb { N } } & \mapsto & \left( z _ { 0 } \right) \\ z _ { 1 } \end{array} \right. \right)$ is an isomorphism of $\mathbb { C }$-vector spaces.

We assume that $p$ is an integer greater than or equal to 2, that $\left( a _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( b _ { k } \right) _ { k \in \mathbb { N } }$ are two sequences of real numbers that are $p$-periodic and that $\forall k \in \mathbb { N } , b _ { k } \neq 0$. We denote by Sol(II.2) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ that satisfy the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ To any complex sequence $\left( z _ { k } \right) _ { k \in \mathbb { N } }$, we associate the sequence $\left( Z _ { k } \right) _ { k \in \mathbb { N } }$ of elements of $\mathbb { C } ^ { 2 }$ defined by $$\forall k \in \mathbb { N } , \quad Z _ { k } = \binom { z _ { k } } { z _ { k + 1 } }$$ Prove that the sequence $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ is a solution of (II.2) if and only if the sequence $\left( Z _ { k } \right) _ { k \in \mathbb { N } }$ is a solution of a system (II.3) of the form $$\forall k \in \mathbb { N } , \quad Z _ { k + 1 } = A _ { k } Z _ { k }$$ Specify the matrix $A _ { k } \in \mathcal { M } _ { 2 } ( \mathbb { C } )$.

We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. Prove that $\operatorname { det } Q = 1$.

We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. We fix a solution $\left( Z _ { k } \right) _ { k \in \mathbb { N } }$ of (II.3). Prove that, for any natural integer $k$ and any natural integer $r \in \llbracket 1 , p - 1 \rrbracket$, $$\left\{ \begin{array} { l } Z _ { k p } = Q ^ { k } Z _ { 0 } \\ Z _ { k p + r } = A _ { r - 1 } A _ { r - 2 } \cdots A _ { 0 } Q ^ { k } Z _ { 0 } \end{array} \right.$$

Let $n$ and $p$ be two integers greater than or equal to 2. We fix throughout this part a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which we assume to be $p$-periodic, that is such that $\forall k \in \mathbb { N } , A _ { k + p } = A _ { k }$. We denote by $\operatorname { Sol }$ (III.1) the set of sequences $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ of vectors of $\mathbb { C } ^ { n }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } , \quad Y _ { k + 1 } = A _ { k } Y _ { k }$$ Justify that we define a sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ by setting $\left\{ \begin{array} { l } \Phi _ { 0 } = I _ { n } \\ \Phi _ { k + 1 } = A _ { k } \Phi _ { k } \quad \forall k \in \mathbb { N } \end{array} \right.$ and that $\left( Y _ { k } \right) _ { k \in \mathbb { N } } \in \operatorname { Sol }$ (III.1) if and only if $\forall k \in \mathbb { N } , Y _ { k } = \Phi _ { k } Y _ { 0 }$.