For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$
Prove that $H_n$ is invertible, then that $\operatorname{det}\left(H_n^{-1}\right)$ is an integer.

Let $A, B \in \mathcal{M}_n(\mathbb{R})$ and $P \in \mathrm{GL}_n(\mathbb{R})$ such that $B = P^{-1}AP$. Show that $f_A$ and $f_B$ have the same eigenvalues.

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. 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 $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ 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 $| \rho | < 1 , \lim _ { k \rightarrow + \infty } \left\| Y _ { k } \right\| _ { \infty } = 0$.

In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$.
$\mathbf{24}$ ▷ We assume, in this question only, that all eigenvalues of the matrix $A$ have real parts that are positive or zero. Show that, if $X \in \mathbf{C}^n$, we have $$\lim_{t \rightarrow +\infty} e^{tA} X = 0 \Longleftrightarrow X = 0.$$

Problem 2, Part 1: Adapted norms
We denote by $\mathrm { M } _ { d } ( \mathbb { C } )$ the space of $d \times d$ square matrices with complex coefficients and we identify $\mathbb { C } ^ { d }$ with the space of column vectors of size $d$. For a vector $x = \left( x _ { 1 } , \ldots , x _ { d } \right) \in \mathbb { C } ^ { d }$, we define $\| x \| _ { \infty } = \max _ { 1 \leqslant i \leqslant d } \left| x _ { i } \right|$. If $A$ is a matrix in $\mathrm { M } _ { d } ( \mathbb { C } )$ we denote by $\operatorname { Sp } ( A )$ the spectrum of $A$ and we define the spectral radius $\sigma ( A ) = \max \{ | \lambda | , \lambda \in \operatorname { Sp } ( A ) \}$.
Let $T = \left( t _ { i , j } \right) _ { 1 \leqslant i , j \leqslant d }$ be an upper triangular matrix. Show that for all $\varepsilon > 0$, there exists a norm $\| \cdot \| ^ { \prime }$ on $\mathbb { C } ^ { d }$ such that for all $x \in \mathbb { C } ^ { d }$ we have $$\| T x \| ^ { \prime } \leqslant ( \sigma ( T ) + \varepsilon ) \| x \| ^ { \prime }$$ (one may choose $\| \cdot \| ^ { \prime }$ in the form $\| x \| ^ { \prime } = \| P x \| _ { \infty }$ for a suitably chosen matrix $P$).

Problem 2, Part 1: Adapted norms
We denote by $\mathrm { M } _ { d } ( \mathbb { C } )$ the space of $d \times d$ square matrices with complex coefficients. If $A$ is a matrix in $\mathrm { M } _ { d } ( \mathbb { C } )$ we denote by $\operatorname { Sp } ( A )$ the spectrum of $A$ and we define the spectral radius $\sigma ( A ) = \max \{ | \lambda | , \lambda \in \operatorname { Sp } ( A ) \}$. We define $\| A \| = \sup _ { \| x \| _ { \infty } \leqslant 1 } \| A x \| _ { \infty }$.
Application: norm and spectral radius. a. Let $T \in \mathrm { M } _ { d } ( \mathbb { C } )$ be an upper triangular matrix. Show that for all $\varepsilon > 0$, there exists a constant $C$ such that for all $n$ we have $\left\| T ^ { n } \right\| \leqslant C ( \sigma ( T ) + \varepsilon ) ^ { n }$. b. Show that $\lim _ { n \rightarrow \infty } \left\| T ^ { n } \right\| ^ { 1 / n } = \sigma ( T )$. c. Let now $A \in \mathrm { M } _ { d } ( \mathbb { C } )$ be an arbitrary matrix. Show that $\lim _ { n \rightarrow \infty } \left\| A ^ { n } \right\| ^ { 1 / n } = \sigma ( A )$. d. Show the equivalence $$A ^ { n } \underset { n \rightarrow \infty } { \longrightarrow } 0 \Leftrightarrow \sigma ( A ) < 1 .$$

In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV.
We set $A = \begin{pmatrix} \mathrm{e}^{\beta - h} & \mathrm{e}^{-\beta - h} \\ \mathrm{e}^{-\beta + h} & \mathrm{e}^{\beta + h} \end{pmatrix}$.
Show then that $$\psi_n(h) \underset{n \rightarrow +\infty}{\longrightarrow} \ln\left(\mathrm{e}^{\beta} \operatorname{ch}(h) + \sqrt{\mathrm{e}^{2\beta} \operatorname{ch}^2(h) - 2\operatorname{sh}(2\beta)}\right).$$

We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$. Let $M$ be a nonzero real symmetric matrix of size $d \times d$ such that $\forall x \in \mathbb{R}^d,\, \langle x, Mx \rangle \geq 0$. We define $f(x) := -\frac{1}{2}\langle x, Mx \rangle$, so that $\nabla f(x) = -Mx$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $$x_{n+1} := P_C(x_n - \tau \nabla f(x_n)), \quad \text{with} \quad P_C(x) := \begin{cases} x & \text{if } \|x\| \leq 1, \\ x/\|x\| & \text{otherwise.} \end{cases}$$ Suppose in this question that $\|x_0\| \geq 1$. a) Show that $$\forall n \in \mathbb{N} \setminus \{0\},\, x_n = \frac{(\mathrm{I}_d + \tau M)^n x_0}{\|(\mathrm{I}_d + \tau M)^n x_0\|}$$ b) Calculate $\lim_{n \rightarrow \infty} x_n$. Hint. Decompose $x_0 = \sum_{1 \leq i \leq d} \alpha_i e_i$ in an orthonormal basis of eigenvectors $(e_1, \cdots, e_d)$, associated with the eigenvalues $\lambda_1, \cdots, \lambda_d$ of $M$. Introduce the set of indices $I := \{i \in \llbracket 1, d \rrbracket \mid \alpha_i \neq 0\}$, the eigenvalue $\lambda := \max_{i \in I} \lambda_i$, and the vector $x_0' := \sum_{i \in I'} \alpha_i e_i$ where $I' := \{i \in I \mid \lambda_i = \lambda\}$.

We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$. Let $M$ be a nonzero real symmetric matrix of size $d \times d$ such that $\forall x \in \mathbb{R}^d,\, \langle x, Mx \rangle \geq 0$. We define $f(x) := -\frac{1}{2}\langle x, Mx \rangle$, so that $\nabla f(x) = -Mx$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $$x_{n+1} := P_C(x_n - \tau \nabla f(x_n)), \quad \text{with} \quad P_C(x) := \begin{cases} x & \text{if } \|x\| \leq 1, \\ x/\|x\| & \text{otherwise.} \end{cases}$$ How does the sequence behave when $\|x_0\| < 1$?

I. Suppose that $\lambda$ is an eigenvalue of a regular matrix $\boldsymbol { P }$, prove that:

1. $\lambda$ is not zero.
2. $\lambda ^ { - 1 }$ is an eigenvalue of $\boldsymbol { P } ^ { - 1 }$ and $\lambda ^ { n }$ is an eigenvalue of $\boldsymbol { P } ^ { n }$, where $n$ is a positive integer.

II. Suppose $\boldsymbol { P }$ is an orthogonal matrix. When the following symmetric matrix $\boldsymbol { A }$ can be diagonalized by $\boldsymbol { P }$, find the matrix $\boldsymbol { P }$ and obtain the diagonalized matrix.
$$A = \left( \begin{array} { c c c } 2 & - 1 & 1 \\ - 1 & 2 & - 1 \\ 1 & - 1 & 2 \end{array} \right)$$
III. When a matrix $\boldsymbol { P }$, and vectors $\boldsymbol { r }$ and $\boldsymbol { x }$ are given as
$$\boldsymbol { P } = \left( \begin{array} { c c c } 1 & 1 & 1 \\ p & p ^ { 2 } & p ^ { 3 } \\ q & q ^ { 2 } & q ^ { 3 } \end{array} \right) , \quad \boldsymbol { r } = \left( \begin{array} { c } r \\ r ^ { 2 } \\ r ^ { 3 } \end{array} \right) , \quad \boldsymbol { x } = \left( \begin{array} { c } x \\ y \\ z \end{array} \right) ,$$
where $p , q$, and $r$ are non-zero real numbers that differ from each other.

1. Find the condition that $p$ and $q$ must satisfy in order for $\boldsymbol { P }$ to be a regular matrix.
2. When $\boldsymbol { P } ^ { \mathrm { T } } \boldsymbol { x } = \boldsymbol { r }$ has a single solution, obtain $\boldsymbol { x }$. Here, $\boldsymbol { P } ^ { \mathrm { T } }$ is the transposed matrix of $\boldsymbol { P }$.

IV. The matrix $\boldsymbol { P } _ { n }$ is an $n$-th order square matrix ( $n \geq 2$ ), as shown below, where $p$ and $q$ are real numbers that differ from each other.
$$\boldsymbol { P } _ { n } = \left( \begin{array} { c c c c c c } p + q & q & 0 & \cdots & 0 & 0 \\ p & p + q & \ddots & \ddots & \vdots & \vdots \\ 0 & p & \ddots & \ddots & 0 & \vdots \\ \vdots & 0 & \ddots & \ddots & q & 0 \\ \vdots & \vdots & \ddots & \ddots & p + q & q \\ 0 & 0 & \cdots & 0 & p & p + q \end{array} \right)$$

1. Obtain the recurrence formula satisfied by the determinant of $\boldsymbol { P } _ { n }$, $\left| \boldsymbol { P } _ { n } \right|$.
2. Express the determinant $\left| \boldsymbol { P } _ { n } \right|$ in terms of $p , q$, and $n$, using the recurrence formula in Question IV.1.

Problem 2

For a square matrix $\boldsymbol { A } , e ^ { \boldsymbol { A } }$ is defined as:
$$e ^ { A } = \boldsymbol { E } + \sum _ { k = 1 } ^ { \infty } \frac { 1 } { k ! } \boldsymbol { A } ^ { k } ,$$
where $\boldsymbol { E }$ is the identity matrix and $e$ is the base of natural logarithm.
I. Let $\boldsymbol { A }$ be a $3 \times 3$ square matrix which can be diagonalized by a regular matrix $\boldsymbol { P }$, i.e., $\boldsymbol { A } = \boldsymbol { P } \boldsymbol { D } \boldsymbol { P } ^ { - 1 }$, where $\boldsymbol { D }$ is a diagonal matrix:
$$\boldsymbol { D } = \left( \begin{array} { c c c } \lambda _ { 1 } & 0 & 0 \\ 0 & \lambda _ { 2 } & 0 \\ 0 & 0 & \lambda _ { 3 } \end{array} \right)$$
Here, $\lambda _ { 1 } , \lambda _ { 2 }$, and $\lambda _ { 3 }$ are complex numbers. Prove the following equation:
$$e ^ { \boldsymbol { A } } = \boldsymbol { P } \left( \begin{array} { c c c } e ^ { \lambda _ { 1 } } & 0 & 0 \\ 0 & e ^ { \lambda _ { 2 } } & 0 \\ 0 & 0 & e ^ { \lambda _ { 3 } } \end{array} \right) \boldsymbol { P } ^ { - 1 }$$
II. Let $\boldsymbol { A } = \left( \begin{array} { c c c } - 1 & 4 & 4 \\ - 5 & 8 & 10 \\ 3 & - 3 & - 5 \end{array} \right)$.

1. Find the regular matrix $\boldsymbol { P }$ and the diagonal matrix $\boldsymbol { D }$ such that $\boldsymbol { A } = \boldsymbol { P } \boldsymbol { D } \boldsymbol { P } ^ { - 1 }$.
2. Calculate $e ^ { \boldsymbol { A } }$.

III. Consider $\boldsymbol { A } = \left( \begin{array} { c c c } 0 & - x & 0 \\ x & 0 & 0 \\ 0 & 0 & 1 \end{array} \right) , \boldsymbol { B } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 1 \end{array} \right)$ and $\boldsymbol { a } = \left( \begin{array} { l } 1 \\ 1 \\ e \end{array} \right)$, where $x$ is a real number. In the following, the transpose of a vector $\boldsymbol { v }$ is denoted by $\boldsymbol { v } ^ { T }$.

1. Express the sum of the eigenvalues of $e ^ { \boldsymbol { A } }$ using $e$ and $x$.
2. Let $\boldsymbol { C } = \boldsymbol { B } e ^ { \boldsymbol { A } }$. Find the minimum and maximum values of $\frac { \boldsymbol { y } ^ { T } \boldsymbol { C } \boldsymbol { y } } { \boldsymbol { y } ^ { T } \boldsymbol { y } }$ for a real three-dimensional vector $\boldsymbol { y } ( \boldsymbol { y } \neq \mathbf { 0 } )$.
3. Let $f ( \boldsymbol { z } ) = \frac { 1 } { 2 } \boldsymbol { z } ^ { T } \boldsymbol { C } \boldsymbol { z } - \boldsymbol { a } ^ { T } \boldsymbol { z }$ for a real three-dimensional vector $\boldsymbol { z } = \left( \begin{array} { c } z _ { 1 } \\ z _ { 2 } \\ z _ { 3 } \end{array} \right)$ and $\boldsymbol { C }$ in III.2. Find $\sqrt { z _ { 1 } ^ { 2 } + z _ { 2 } ^ { 2 } + z _ { 3 } ^ { 2 } }$ for $\boldsymbol { z }$ such that $\frac { \partial f ( \boldsymbol { z } ) } { \partial z _ { 1 } } = \frac { \partial f ( \boldsymbol { z } ) } { \partial z _ { 2 } } = \frac { \partial f ( \boldsymbol { z } ) } { \partial z _ { 3 } } = 0$.

Let $M = \left[ \begin{array} { r r } 1 & 1 \\ - 2 & 4 \end{array} \right]$ and $X = \left[ \begin{array} { l } 1 \\ 2 \end{array} \right]$ such that
$$\begin{aligned} & \mathrm { M } \cdot \mathrm { X } = \mathrm { aX } \\ & \mathrm { M } ^ { - 1 } \cdot \mathrm { X } = \mathrm { bX } \end{aligned}$$
For real numbers a and b satisfying these equalities, what is the sum $a + b$?
A) $\frac { 1 } { 3 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 5 } { 3 }$
D) $\frac { 8 } { 3 }$
E) $\frac { 10 } { 3 }$