Let $\mathcal { A }$ be a non-zero vector subspace of $\mathcal { M } ( n , \mathbb { K } )$ stable by bracket, and let $\mathcal { E }$ be the intersection of $\mathcal { A }$ and $\mathcal { D } ( n , \mathbb { K } )$. Let $H$ be an element of $\mathcal { E }$.
a) Calculate the image under $\Phi _ { H }$ of the canonical basis of $\mathcal { M } ( n , \mathbb { K } )$. Deduce that $\Phi _ { H }$ is a diagonalisable endomorphism of $\mathcal { M } ( n , \mathbb { K } )$.
b) Show that there exists a basis of $\mathcal { A }$ in which the matrices of the endomorphisms of $\mathcal { A }$ induced by the $\Phi _ { H }$, for $H \in \mathcal { E }$, are diagonal.

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, intersecting the $x$-axis. We denote by $L_1$ and $L_2$, with coordinates respectively $(\lambda_1, 0)$ and $(\lambda_2, 0)$, with $\lambda_1 < \lambda_2$, the two intersection points of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$. We keep the notations $E, F, G, H$ from III.D.
Show that $A$ is diagonalizable.

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, tangent to the $x$-axis. We call $L$, with coordinates $(\lambda, 0)$, the point of tangency of $\mathcal{C}(\Omega, r)$ with the $x$-axis. Let $A$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$.
Is the matrix $A$ diagonalizable? Is it trigonalizable?

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r \geqslant 0$ disjoint from the $x$-axis. We denote by $K$ the orthogonal projection of $\Omega$ onto the $x$-axis. Let $A$ be a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$.
Does there exist a matrix $P$ in $\mathrm{GL}_2(\mathbb{R})$ such that the matrix $P^{-1}AP$ is diagonal? Does there exist a matrix $P$ in $\mathrm{GL}_2(\mathbb{R})$ such that the matrix $P^{-1}AP$ is upper triangular?

Let $s$ be an endomorphism of $E$ such that $s \circ s = \operatorname{Id}_E$. We set $F = \operatorname{Ker}(s - \operatorname{Id}_E)$ and $G = \operatorname{Ker}(s + \operatorname{Id}_E)$. a) Show that $F$ and $G$ are two supplementary subspaces of $E$. b) Deduce that $s$ is a symmetry and specify its elements.

In this question, $E$ is a real vector space of dimension $n$ and $f$ is an endomorphism of $E$.
V.D.1) Show that if $f$ is diagonalisable then there exist $n$ hyperplanes of $E$, $(H_i)_{1 \leqslant i \leqslant n}$, all stable by $f$ such that $\bigcap_{i=1}^{n} H_i = \{0\}$.
V.D.2) Is an endomorphism $f$ of $E$ for which there exist $n$ hyperplanes of $E$ stable by $f$ and with intersection reduced to the zero vector necessarily diagonalisable?

Let $M$ and $N$ be in $S _ { n } ( \mathbb { R } )$. Show that there exists $U \in O _ { n } ( \mathbb { R } )$ such that $N = U M U ^ { - 1 }$, if and only if $\chi _ { M } = \chi _ { N }$.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Is the matrix $M$ defined in question I.A.3 and the matrix $M'$ of size $n+1$ given by $$M' = \left(\begin{array}{ccccc} 1 & 1 & 0 & \ldots & 0 \\ 0 & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & & \ddots & \ddots & 1 \\ 0 & \ldots & \ldots & 0 & 1 \end{array}\right)$$ similar?

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 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 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 $U Y = \overline{\lambda} Y$.

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.

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 $f$ 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 $\mathcal{M}_{2}(\mathbb{C})$.

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 $M$ 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: $$\left(\begin{array}{cc} a & b \\ -b & a \end{array}\right)$$

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.

Let $T$ be a delta endomorphism of $\mathbb{K}[X]$. For $n \in \mathbb{N}$, we denote by $T_n$ the restriction of $T$ to $\mathbb{K}_n[X]$.
Show that $T_n$ is an endomorphism of $\mathbb{K}_n[X]$. Is it diagonalizable?

Justify that an adjacency matrix of a non-empty graph is diagonalizable.

Let $V$ be a finite-dimensional vector space, let $h$ be an endomorphism of $V$ and let $W$ be a subspace stable by $h$. We denote by $h_W$ the endomorphism of $W$ induced by $h$, that is $h_W : W \rightarrow W$, $v \mapsto h(v)$. Prove that if $h$ is diagonalizable, then $h_W$ is also diagonalizable.