Let $A \in \mathrm { GL } ( 3 , \mathbb { Q } )$ with $A ^ { t } A = I _ { 3 }$. Assume that $$A \left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right] = \lambda \left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right]$$ for some $\lambda \in \mathbb { C }$.
(A) Determine the possible values of $\lambda$.
(B) Determine $x + y + z$ where $x , y , z$ is given by $$\left[ \begin{array} { l } x \\ y \\ z \end{array} \right] = A \left[ \begin{array} { c } 1 \\ - 1 \\ 0 \end{array} \right]$$

Consider the real matrix
$$A = \left( \begin{array} { l l } \lambda & 2 \\ 3 & 5 \end{array} \right)$$
Assume that $-1$ is an eigenvalue of $A$. Which of the following are true?
(A) The other eigenvalue is in $\mathbb { C } \backslash \mathbb { R }$.
(B) $A + I _ { 2 }$ is singular.
(C) $\lambda = 1$.
(D) Trace of $A$ is 5.

Let $n \in \mathbb{N}^*$ and $A \in \mathcal{S}_n(\mathbb{R})$. Show that $A$ is positive if and only if all its eigenvalues are positive.

Let $n \in \mathbb{N}^*$ and $A \in \mathcal{S}_n(\mathbb{R})$. Show that $A$ is positive definite if and only if all its eigenvalues are strictly positive.

For $n \in \mathbb{N}^*$, $A \in \mathcal{S}_n(\mathbb{R})$ and $i \in \llbracket 1; n \rrbracket$, we denote by $A^{(i)}$ the square matrix of order $i$ extracted from $A$, consisting of the first $i$ rows and the first $i$ columns of $A$.
Let $A \in \mathcal{S}_n(\mathbb{R})$. We assume that $A$ is positive definite.
For all $i \in \llbracket 1; n \rrbracket$, show that the matrix $A^{(i)}$ is positive definite and deduce that $\operatorname{det}\left(A^{(i)}\right) > 0$.

We set $\xi_i^j = \begin{cases}0 & \text{if } i = j \\ 1 & \text{otherwise}\end{cases}$ and $N_k = (m_{ij} + k\xi_i^j)$ with $k$ a strictly positive real number.
Show that there exists a minimal real $k_0$ that we will specify as a function of the eigenvalues of $\Psi(M)$, such that the matrix $\Psi(N_{k_0})$ has non-negative eigenvalues.

We set $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ and $M_c = \left((d_{ij} + c\xi_i^j)^2\right)$ with $c > 0$. Let $\lambda_{\min}$ and $\mu_{\min}$ denote the respective minimal eigenvalues of $\Psi(M)$ and $\Psi(D)$.
Deduce that for $c = \widetilde{c} = -2\mu_{\min} + \sqrt{4\mu_{\min}^2 - 2\lambda_{\min}} > 0$, $\Psi(M_c)$ has non-negative eigenvalues and that for all $c > \widetilde{c}$ and for all non-zero vector $X \in \mathcal{H}$, ${}^t X \Psi(M_c) X > 0$.

We set $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ and $M_c = \left((d_{ij} + c\xi_i^j)^2\right)$ with $c > 0$. We seek the minimal constant $c^* > 0$ (if it exists) satisfying:

• $\Psi(M_{c^*})$ has non-negative eigenvalues,
• for all $c > c^*$ and for all non-zero vector $X \in \mathcal{H}$, ${}^t X \Psi(M_c) X > 0$.

We know that $c^*$ is bounded above by $\widetilde{c}$.
We consider $\mathcal{A} = \left\{X \in \mathcal{H} \mid \|X\| = 1 \text{ and } 4\left({}^t X \Psi(D) X\right)^2 - 2\, {}^t X \Psi(M) X \geqslant 0\right\}$ and we define the mapping $$\alpha: \begin{cases}\mathcal{A} \longrightarrow \mathbb{R} \\ X \longmapsto -2\, {}^t X \Psi(D) X + \sqrt{4\left({}^t X \Psi(D) X\right)^2 - 2\, {}^t X \Psi(M) X}\end{cases}$$
Show that there exists $X^* \in \mathcal{A}$ such that $\alpha(X^*) = \sup_{X \in \mathcal{A}} \alpha(X)$ and $\alpha(X^*) > 0$.

We set $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ and $M_c = \left((d_{ij} + c\xi_i^j)^2\right)$ with $c > 0$. Let $X^* \in \mathcal{A}$ be such that $\alpha(X^*) = \sup_{X \in \mathcal{A}} \alpha(X) > 0$, and denote $\alpha^* = \alpha(X^*)$.
Show that:

• ${}^t X^* \Psi(M_{\alpha^*}) X^* = 0$,
• $\Psi(M_{\alpha^*})$ has non-negative eigenvalues,
• for all $c > \alpha^*$ and for all non-zero vector $X \in \mathcal{H}$, ${}^t X \Psi(M_c) X > 0$.

Conclude that $c^* = \alpha^*$.

We set $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ and $M_c = \left((d_{ij} + c\xi_i^j)^2\right)$ with $c > 0$. We consider $\gamma$ a real eigenvalue of the matrix $\left(\begin{array}{cc}0 & 2\Psi(M) \\ -I_n & -4\Psi(D)\end{array}\right)$ and $\binom{X_1}{X_2}$ an associated eigenvector.
a) Show that ${}^t X_2 \Psi(M_\gamma) X_2 = 0$ and that $X_2 \neq 0$. Conclude that $\gamma \leqslant c^*$.
b) What conclusion do we draw from this on the calculation of the smallest additive constant $c^*$?

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

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.
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.

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 $A \in \mathcal{M}_{n}(\mathbb{R})$ 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 $\bar{X}^{\top}AX \in \mathcal{M}_{1}(\mathbb{C})$ with the complex number equal to its unique entry.
Show that, if $X \neq 0$, then $\operatorname{Re}(\bar{X}^{\top}AX) > 0$, where $\operatorname{Re}(z)$ denotes the real part of $z \in \mathbb{C}$.
b) Show that $A$ 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.
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. 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 $\operatorname{det}(B) > 0$.

We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. Show that if $| \operatorname { tr } Q | < 2$, then every solution of (II.2) is bounded.

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 $B ^ { p } = \Phi _ { p }$. Suppose that $B$ is diagonalizable and that all its eigenvalues have modulus strictly less than 1. Prove that for every solution $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ of (III.1), $\lim _ { k \rightarrow + \infty } \left\| Y _ { k } \right\| _ { \infty } = 0$.

With the same setup as Q27 (matrices $A$, $B$, $I_q$, $r$, $F_n$), show that the sequence $\left(F_{n}\right)_{n \in \mathbb{N}}$ is bounded regardless of the choice of $F_{0}$ if and only if the eigenvalues of $A$ belong to $[-1, 1]$.

Let $\lambda$ be an eigenvalue of $B$ (the square matrix of order $q$ with coefficient $(i,j)$ equal to 1 if $|i-j|=1$ and 0 otherwise) and let $Y = \left(\begin{array}{c} y_{1} \\ \vdots \\ y_{q} \end{array}\right)$ be an associated eigenvector. By considering a coefficient of $Y$ whose absolute value is maximal, show that $\lambda \in [-2, 2]$ and justify the existence of an element $\theta$ of $[0, \pi]$, such that $\lambda = 2\cos\theta$.

Show that every eigenvalue of $A _ { n }$ is in the interval $]0,4[$.

Let $A \in \mathcal { S } _ { N } ( \mathbb { R } )$. Show that $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$ if and only if the eigenvalues of $A$ are all strictly positive real numbers.

Let $M = \left( m _ { i , j } \right)$ be a stochastic matrix of $\mathcal { M } _ { n } ( \mathbb { R } )$ and $\lambda$ an eigenvalue (real or complex) of $M$. We denote by $\left( u _ { 1 } , \ldots , u _ { n } \right)$ the components (real or complex), in the canonical basis, of an eigenvector $u$ associated with $\lambda$.
Let $h \in \{ 1 , \ldots , n \}$ such that $\left| u _ { h } \right| = \max _ { 1 \leqslant i \leqslant n } \left| u _ { i } \right|$. Show that $\left| \lambda - m _ { h , h } \right| \leqslant 1 - m _ { h , h }$. Deduce that $| \lambda | \leqslant 1$.

Let $M = \left( m _ { i , j } \right)$ be a stochastic matrix of $\mathcal { M } _ { n } ( \mathbb { R } )$ and $\lambda$ an eigenvalue (real or complex) of $M$. We denote by $\left( u _ { 1 } , \ldots , u _ { n } \right)$ the components (real or complex), in the canonical basis, of an eigenvector $u$ associated with $\lambda$.
Let $\delta = \min _ { 1 \leqslant i \leqslant n } m _ { i , i }$. Show that $| \lambda - \delta | \leqslant 1 - \delta$. Give a geometric interpretation of this result and show that, if all diagonal terms of $M$ are strictly positive, then 1 is the only eigenvalue of $M$ with modulus 1.

Let $M \in \mathrm { GL } _ { n } ( \mathbb { R } )$. Show that the eigenvalues of $M ^ { \top } M$ are all strictly positive.
Deduce that there exists a symmetric matrix $S$ with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$.

Let $M \in \mathrm { GL } _ { n } ( \mathbb { R } )$. Show that the eigenvalues of $M ^ { \top } M$ are all strictly positive.
Deduce that there exists a symmetric matrix $S$ with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$.