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 companion matrix
$$C_Q = \left(\begin{array}{cccccc} 0 & \cdots & \cdots & \cdots & 0 & -a_0 \\ 1 & 0 & \cdots & \cdots & 0 & -a_1 \\ 0 & 1 & \ddots & & \vdots & -a_2 \\ \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ \vdots & & \ddots & 1 & 0 & -a_{n-2} \\ 0 & \cdots & \cdots & 0 & 1 & -a_{n-1} \end{array}\right).$$
Let $\lambda$ be an eigenvalue of $C_Q^{\top}$. Determine the dimension and a basis of the associated eigenspace.

Let $f$ be a cyclic endomorphism. Show that $f$ is diagonalisable if and only if $\chi_f$ is split over $\mathbb{K}$ and has all its roots simple.

Let $f$ be a cyclic endomorphism. Show that $f$ is diagonalisable if and only if $\chi_f$ is split over $\mathbb{K}$ and has all its roots simple.

Let $\Gamma(\mathbb{K})$ be the subset of $\mathcal{M}_{2}(\mathbb{K})$ consisting of matrices of the form $\left( \begin{array}{cc} a & -b \\ b & a \end{array} \right)$ where $(a, b) \in \mathbb{K}^{2}$.
Show that $\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right)$ is diagonalisable over $\mathbb{C}$. Deduce that $\Gamma(\mathbb{C})$ is a diagonalisable subalgebra of $\mathcal{M}_{2}(\mathbb{C})$.

The purpose of this question is to show that $\sqrt [ 3 ] { 2 }$ is not an eigenvalue of a symmetric matrix with coefficients in $\mathbb { Q }$. We reason by contradiction, assuming the existence of a matrix $M \in S _ { n } ( \mathbb { Q } )$ (for some integer $n$) for which $\sqrt [ 3 ] { 2 }$ is an eigenvalue.
4a. Show that $X ^ { 3 } - 2$ divides the characteristic polynomial of $M$. (One may begin by proving that $\sqrt [ 3 ] { 2 } \notin \mathbb { Q }$.)
4b. Conclude.

The purpose of this question is to show that $\sqrt [ 3 ] { 2 }$ is not an eigenvalue of a symmetric matrix with coefficients in $\mathbb { Q }$. We reason by contradiction, assuming the existence of a matrix $M \in S _ { n } ( \mathbb { Q } )$ (for some integer $n$) for which $\sqrt [ 3 ] { 2 }$ is an eigenvalue.
4a. Show that $X ^ { 3 } - 2$ divides the characteristic polynomial of $M$. (One may begin by proving that $\sqrt [ 3 ] { 2 } \notin \mathbb { Q }$.)
4b. Conclude.

For $n \in \mathbb { N } ^ { * }$, construct a matrix $M \in S _ { n } ( \mathbb { Q } )$ for which $\cos \left( \frac { 2 \pi } { n } \right)$ is an eigenvalue. (One may begin by constructing an orthogonal matrix with coefficients in $\mathbb { Q }$ that admits $e ^ { 2 i \pi / n }$ as an eigenvalue.)

For $n \in \mathbb { N } ^ { \star }$, construct a matrix $M \in S _ { n } ( \mathbb { Q } )$ for which $\cos \left( \frac { 2 \pi } { n } \right)$ is an eigenvalue. (One may begin by constructing an orthogonal matrix with coefficients in $\mathbb { Q }$ that admits $e ^ { 2 i \pi / n }$ as an eigenvalue.)

We define the Redheffer matrix $H_n$ and its characteristic polynomial $\chi_n$. We denote $\log_2$ the logarithm function in base 2.
Finally, show that $H_n$ has 1 as an eigenvalue and that its multiplicity is exactly
$$n - \lfloor \log_2 n \rfloor - 1.$$

We define the Redheffer matrix by $H_n = \left(h_{ij}\right)_{(i,j) \in \llbracket 1,n \rrbracket^2}$ where
$$h_{ij} = \begin{cases} 1 & \text{if } j = 1 \\ 1 & \text{if } i \text{ divides } j \text{ and } j \neq 1 \\ 0 & \text{otherwise.} \end{cases}$$
Using the expression
$$\chi_n(\lambda) = (\lambda - 1)^n - \sum_{k=1}^{\left\lfloor \log_2 n \right\rfloor} (\lambda - 1)^{n-k-1} S_k(n),$$
finally show that $H_n$ has 1 as an eigenvalue and that its multiplicity is exactly
$$n - \left\lfloor \log_2 n \right\rfloor - 1.$$

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 $h \in \{ 1 , \ldots , n \}$ 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.

We set $A_1 = \left(\begin{array}{ccc} 3 & -2 & 4 \\ -2 & 6 & 2 \\ 4 & 2 & 3 \end{array}\right)$.
By observing the first and last column of $A_1$, determine an eigenvector of $A_1$ and the associated eigenvalue $\lambda_1$.

Let $B$ be the matrix in $M_{2}(\mathbf{R})$ defined by: $$B = \left(\begin{array}{cc} 3 & 2 \\ -5 & 1 \end{array}\right)$$ Prove that $B$ is semi-simple and deduce the existence of an invertible matrix $Q$ in $M_{2}(\mathbf{R})$ and two real numbers $a$ and $b$ to be determined such that: $$B = Q \left(\begin{array}{cc} a & b \\ -b & a \end{array}\right) Q^{-1}$$ Hint: for an eigenvector $V$ of $B$, one may introduce the vectors $W_{1} = \operatorname{Re}(V)$ and $W_{2} = \operatorname{Im}(V)$.

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 $S ^ { 2 } = M ^ { \top } M$.

We set $A_1 = \left(\begin{array}{ccc} 3 & -2 & 4 \\ -2 & 6 & 2 \\ 4 & 2 & 3 \end{array}\right)$.
Determine the eigenspace of $A_1$ associated with the eigenvalue $\lambda_1$ and deduce the spectrum of $A_1$.

Let $M$ be a matrix in $M_{2}(\mathbf{R})$. We assume that $M$ has two complex eigenvalues $\mu = a + ib$ and $\bar{\mu} = a - ib$ with $a \in \mathbf{R}$ and $b \in \mathbf{R}^{*}$. Prove that $M$ is semi-simple and similar in $M_{2}(\mathbf{R})$ to the matrix: $$\left(\begin{array}{cc} a & b \\ -b & a \end{array}\right)$$

We denote by $G_0$ the subgroup of $G$ formed by elements $g$ such that $g(\mathcal{H})=\mathcal{H}$. For all $w\in V$ such that $B(w,w)>0$, we define the linear map $$s_w : v \mapsto v - 2\frac{B(v,w)}{B(w,w)}w.$$ Show that $s_w^2 = \mathrm{Id}_V$, and determine the eigenvalues and eigenspaces of $s_w$.

We set $A_1 = \left(\begin{array}{ccc} 3 & -2 & 4 \\ -2 & 6 & 2 \\ 4 & 2 & 3 \end{array}\right)$.
Orthodiagonalize $A_1$.

Let $M$ be a matrix in $M_{2}(\mathbf{R})$. Prove that $M$ is semi-simple if and only if one of the following conditions is satisfied:

1. [i)] $M$ is diagonalizable in $M_{2}(\mathbf{R})$;
2. [ii)] $\chi_{M}$ has two complex conjugate roots with non-zero imaginary part.

Show that, if $A \in \mathcal { M } _ { n } ( \mathbb { R } )$ is nilpotent, then 0 is an eigenvalue of $A$ and that it is the only complex eigenvalue of $A$.

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 $$\lim_{t \rightarrow +\infty} e^{tA} X = 0 \Longleftrightarrow X = 0.$$

In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. We introduce the following polynomials: $$\begin{aligned} & P_s(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) < 0}} (X - \lambda)^{m_\lambda}, \\ & P_i(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) > 0}} (X - \lambda)^{m_\lambda}, \\ & P_n(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) = 0}} (X - \lambda)^{m_\lambda}, \end{aligned}$$ and the subspaces $E_s = \operatorname{Ker}(P_s(A))$, $E_i = \operatorname{Ker}(P_i(A))$ and $E_n = \operatorname{Ker}(P_n(A))$ of $E = \mathbf{C}^n$.
$\mathbf{25}$ ▷ After justifying that $E = E_s \oplus E_i \oplus E_n$, show that $$E_s = \left\{ X \in E \mid \lim_{t \rightarrow +\infty} e^{tA} X = 0 \right\}.$$

In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$.
Show that there exist a non-zero natural number $r$, distinct complex numbers $\lambda _ { 1 } , \lambda _ { 2 } , \ldots$, $\lambda _ { r }$, and non-zero natural numbers $m _ { 1 } , m _ { 2 } , \ldots , m _ { r }$, such that: $$\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$$ where for $i \in \llbracket 1 ; r \rrbracket , E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$.