We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. Prove that (II.2) admits a nonzero periodic solution of period $p$ if and only if 1 is an eigenvalue of $Q$.

We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. Deduce that (II.2) admits a nonzero periodic solution of period $p$ if and only if $\operatorname { tr } ( Q ) = 2$. Prove that in this case, either all solutions of (II.2) are periodic of period $p$, or (II.2) admits an unbounded solution.
One may prove that there exists a matrix $P \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ and a complex number $\alpha$ such that $Q = P \left( \begin{array} { c c } 1 & \alpha \\ 0 & 1 \end{array} \right) P ^ { - 1 }$ and, in the case where $\alpha \neq 0$, consider the solution of Sol(II.2) whose image by $\Psi$ is the vector $P \binom { 0 } { 1 }$.

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.

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 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), $\lim _ { k \rightarrow + \infty } \left\| Y _ { k } \right\| _ { \infty } = 0$.

We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. We assume that $E$ is of dimension 4 and that $u$ has no real eigenvalue. Let $\mathcal{B}$ be a basis of $E$ such that $\operatorname{Mat}_{\mathcal{B}}(\omega) = J_4$, and let $U \in \mathcal{M}_4(\mathbb{R})$ be the matrix of $u$ in $\mathcal{B}$.
Show that $U$ is diagonalizable over $\mathbb { C }$. Deduce that there exist $\lambda \in \mathbb { C } \backslash \mathbb { R }$ and vectors $Z$ and $Y$ of $\mathbb { C } ^ { 4 }$ linearly independent over $\mathbb { C }$ such that $U Z = \lambda Z$ and $U Y = \overline{\lambda} Y$.

We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. We assume that $E$ is of dimension 4 and that $u$ has no real eigenvalue. Let $\mathcal{B}$ be a basis of $E$ such that $\operatorname{Mat}_{\mathcal{B}}(\omega) = J_4$, and let $U \in \mathcal{M}_4(\mathbb{R})$ be the matrix of $u$ in $\mathcal{B}$. Let $\lambda \in \mathbb{C} \backslash \mathbb{R}$ and $Z, Y \in \mathbb{C}^4$ be as in question 15.
Let $Z _ { 1 } , Z _ { 2 } , Y _ { 1 } , Y _ { 2 }$ be vectors of $\mathbb { R } ^ { 4 }$ such that $Z = Z _ { 1 } + i Z _ { 2 }$ and $Y = Y _ { 1 } + i Y _ { 2 }$. Let $\left( z _ { 1 } , z _ { 2 } , y _ { 1 } , y _ { 2 } \right) \in E ^ { 4 }$ have coordinates respectively $Z _ { 1 } , Z _ { 2 } , Y _ { 1 } , Y _ { 2 }$ in the basis $\mathcal { B }$. Show that $\widetilde { \mathcal { B } } : = \left( z _ { 1 } , z _ { 2 } , y _ { 1 } , - y _ { 2 } \right)$ is a basis of $E$.

Let $V = {}^{ t } \left( v _ { 1 } , \ldots , v _ { n } \right)$ be an eigenvector of $A _ { n }$ associated with a complex eigenvalue $\lambda$, where $A_n$ is the square matrix of size $n$:
$$A _ { n } = \left( \begin{array} { c c c c c c } 2 & - 1 & 0 & \ldots & \ldots & 0 \\ - 1 & 2 & - 1 & \ddots & & \vdots \\ 0 & - 1 & 2 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & 0 \\ \vdots & & \ddots & \ddots & 2 & - 1 \\ 0 & \ldots & \ldots & 0 & - 1 & 2 \end{array} \right)$$
Show that $\lambda$ is necessarily real and that the components $v _ { i }$ of $V$ satisfy the relation:
$$v _ { i + 1 } - ( 2 - \lambda ) v _ { i } + v _ { i - 1 } = 0, \quad 1 \leq i \leq n$$
where we set $v _ { 0 } = v _ { n + 1 } = 0$.

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

Let $\lambda$ be an eigenvalue of $A _ { n }$.
(a) Show that the complex roots $r _ { 1 } , r _ { 2 }$ of the polynomial
$$P ( r ) = r ^ { 2 } - ( 2 - \lambda ) r + 1$$
are distinct and conjugate.
(b) We set $r _ { 1 } = \overline { r _ { 2 } } = \rho e ^ { i \theta }$ with $\rho > 0$ and $\theta \in \mathbb { R }$.
Show that we necessarily have $\sin ( ( n + 1 ) \theta ) = 0$ and $\rho = 1$.

Determine the set of eigenvalues of $A _ { n }$ and a basis of eigenvectors.

Conclude that $A_n(a,b,c)$ is diagonalizable and give its eigenvalues.

We set $M_n = \left(\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ 0 & & & \ddots & 1 \\ 1 & 0 & \cdots & \cdots & 0 \end{array}\right)$ and $\omega_n = \mathrm{e}^{2i\pi/n}$.
Justify that $M_n$ is diagonalizable. Specify its eigenvalues (expressed using $\omega_n$) and give a basis of eigenvectors of $M_n$.

Show that every circulant matrix is diagonalizable. Specify its eigenvalues and a basis of eigenvectors.

Let $M$ be in $\mathcal{M}_n(\mathbb{C})$. We assume that $f_M$ is diagonalizable. We denote by $(\lambda_1, \ldots, \lambda_n)$ its eigenvalues (not necessarily distinct) and by $(e_1, \ldots, e_n)$ a basis of eigenvectors associated with these eigenvalues. Let $u = \sum_{i=1}^{n} u_i e_i$ be a vector of $\mathbb{C}^n$ where $(u_1, \ldots, u_n)$ are $n$ complex numbers.
Give a necessary and sufficient condition on $(u_1, \ldots, u_n, \lambda_1, \ldots, \lambda_n)$ for $(u, f_M(u), \ldots, f_M^{n-1}(u))$ to be a basis of $\mathbb{C}^n$.

Deduce a necessary and sufficient condition for a diagonalizable endomorphism to be cyclic. Then characterize its cyclic vectors.

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 $\lambda = 2\cos\theta$.

Let $N = \left(\begin{array}{ccccc} 0 & 0 & \cdots & \cdots & 0 \\ 1 & 0 & & & \vdots \\ 0 & \ddots & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)$.
Give the eigenvalues of $N$ and the associated eigenspaces. Is it diagonalizable?

Let $\lambda$ be an eigenvalue of $B$ and $Y = \left(\begin{array}{c} y_{1} \\ \vdots \\ y_{q} \end{array}\right)$ an associated eigenvector. Show that, if we impose $y_{0} = y_{q+1} = 0$, then, for all $k \in \llbracket 1, q \rrbracket$, $y_{k-1} - \lambda y_{k} + y_{k+1} = 0$.

Using the results of Q30 and Q31, deduce that there exists $j \in \llbracket 1, q \rrbracket$ such that $\lambda = 2\cos\frac{j\pi}{q+1}$.

Determine the spectrum of $B$ (the square matrix of order $q$ with coefficient $(i,j)$ equal to 1 if $|i-j|=1$ and 0 otherwise) and a basis of eigenvectors of $B$.

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 $\left(a_0, a_1, \ldots, a_{n-1}\right) \in \mathbb{K}^n$ 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.