Deduce that every matrix of $\mathcal{M}_{2}(\mathbb{C})$ is similar to a Toeplitz matrix.

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 $\omega_n = \mathrm{e}^{2i\pi/n}$.
We set $\Phi_n = \left(\omega_n^{(p-1)(q-1)}\right)_{1 \leqslant p,q \leqslant n} \in \mathcal{M}_n(\mathbb{C})$. Justify that $\Phi_n$ is invertible and give without calculation the value of the matrix $\Phi_n^{-1} M_n \Phi_n$.

Deduce a necessary and sufficient condition for a cyclic matrix to be diagonalizable.

Let $T$ be an upper triangular matrix, $A = N + T$ and $k \geqslant 0$. Show that $A$ is similar to a matrix $L$ whose diagonal coefficients of order $k$ are all equal and satisfying $\forall i \in \llbracket -1, k-1 \rrbracket, L^{(i)} = A^{(i)}$.

Deduce that every cyclic matrix is similar to a Toeplitz matrix.

Let $M \in \mathcal{M}_n(\mathbb{K})$. Show that $M^{\top}$ is diagonalisable if and only if $M$ is diagonalisable.

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
What are the matrices in $\mathcal{M}_n(\mathbb{C})$ that are both nilpotent and diagonalizable?

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
Show that an upper triangular matrix in $\mathcal{M}_n(\mathbb{C})$ with zero diagonal is nilpotent and that a nilpotent matrix is similar to an upper triangular matrix with zero diagonal.

We denote $A = \left(\begin{array}{ccc} 1 & 3 & -7 \\ 2 & 6 & -14 \\ 1 & 3 & -7 \end{array}\right)$ and $u$ the endomorphism of $\mathbb{C}^3$ canonically associated with $A$.
Prove that $A$ is similar to the matrix $\operatorname{diag}\left(J_2, J_1\right)$. Give the value of an invertible matrix $P$ such that $A = P \operatorname{diag}\left(J_2, J_1\right) P^{-1}$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$.
Show that there exist a partition $\sigma = \left(\alpha_1, \ldots, \alpha_k\right)$ of $n$ and a basis $\mathcal{B}$ of $E$ in which the matrix of $u$ is equal to the matrix $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis $\mathcal{B}$ is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
Assume that there exist a partition $\sigma'$ of the integer $n$ and a basis $\mathcal{B}'$ of $E$ such that the matrix of $u$ in $\mathcal{B}'$ is equal to $N_{\sigma'}$. Show that $\sigma = \sigma'$.

Let $A$ be the matrix $\left(\begin{array}{ccccc} 0 & -1 & 2 & -2 & -1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & -1 & 1 & 0 \end{array}\right)$ and $u$ the endomorphism canonically associated with $A$.
Determine the partition $\sigma$ of the integer 5 associated with $u$ and give the matrix $N_\sigma$.

Using the result of question 31, prove that if $M \in \mathcal{M}_n(\mathbb{C})$ is nilpotent, then $M$, $2M$ and $M^\top$ are similar.

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 $\Gamma(\mathbb{R})$ is not a diagonalisable subalgebra of $\mathcal{M}_{2}(\mathbb{R})$.

In this part, we assume $n \geqslant 2$. Let $J \in \mathcal{M}_{n}(\mathbb{R})$ be the matrix canonically associated with the endomorphism $\varphi \in \mathcal{L}(\mathbb{R}^{n})$ defined by $\varphi: e_{j} \mapsto e_{j+1}$ if $j \in \{1, \ldots, n-1\}$ and $\varphi(e_{n}) = e_{1}$, where $(e_{1}, \ldots, e_{n})$ is the canonical basis of $\mathbb{R}^{n}$.
Show that $J$ is diagonalisable in $\mathcal{M}_{n}(\mathbb{C})$.

In this part, we assume $n \geqslant 2$. Let $J \in \mathcal{M}_{n}(\mathbb{R})$ be the matrix canonically associated with the endomorphism $\varphi \in \mathcal{L}(\mathbb{R}^{n})$ defined by $\varphi: e_{j} \mapsto e_{j+1}$ if $j \in \{1, \ldots, n-1\}$ and $\varphi(e_{n}) = e_{1}$, where $(e_{1}, \ldots, e_{n})$ is the canonical basis of $\mathbb{R}^{n}$.
Is the matrix $J$ diagonalisable in $\mathcal{M}_{n}(\mathbb{R})$?

In this part, we assume $n \geqslant 2$. For all $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$, we set $$J(a_{0}, \ldots, a_{n-1}) = \left( \begin{array}{cccc} a_{0} & a_{n-1} & \cdots & a_{1} \\ a_{1} & a_{0} & \cdots & a_{2} \\ \vdots & \vdots & & \vdots \\ a_{n-1} & a_{n-2} & \cdots & a_{0} \end{array} \right)$$ Let $\mathcal{A}$ be the set of matrices of $\mathcal{M}_{n}(\mathbb{R})$ of the form $J(a_{0}, \ldots, a_{n-1})$ where $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$.
Is the subset $\mathcal{A}$ a subalgebra of $\mathcal{M}_{n}(\mathbb{C})$?

In this part, we assume $n \geqslant 2$. For all $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$, we set $$J(a_{0}, \ldots, a_{n-1}) = \left( \begin{array}{cccc} a_{0} & a_{n-1} & \cdots & a_{1} \\ a_{1} & a_{0} & \cdots & a_{2} \\ \vdots & \vdots & & \vdots \\ a_{n-1} & a_{n-2} & \cdots & a_{0} \end{array} \right)$$ Let $\mathcal{A}$ be the set of matrices of $\mathcal{M}_{n}(\mathbb{R})$ of the form $J(a_{0}, \ldots, a_{n-1})$ where $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$.
Show that there exists $P \in \mathrm{GL}_{n}(\mathbb{C})$ such that, for every matrix $A \in \mathcal{A}$, the matrix $P^{-1}AP$ is diagonal.

Let $E$ be a $\mathbb{C}$-vector space of finite dimension $n \geqslant 1$. Let $\mathcal{A}$ be a subalgebra of $\mathcal{L}(E)$ consisting of nilpotent endomorphisms. We assume that $n \geqslant 2$ and that the result is true for all natural integers $d \leqslant n-1$. We fix a vector subspace $V$ of $E$ distinct from $E$ and $\{0\}$ stable by all elements of $\mathcal{A}$, and denote by $r$ its dimension. Let also $s = n - r$. There exists a basis $\mathcal{B}$ of $E$ such that for all $u \in \mathcal{A}$, $$\operatorname{Mat}_{\mathcal{B}}(u) = \left( \begin{array}{cc} A(u) & B(u) \\ 0 & D(u) \end{array} \right)$$ where $A(u) \in \mathcal{M}_{r}(\mathbb{C})$, $B(u) \in \mathcal{M}_{r,s}(\mathbb{C})$ and $D(u) \in \mathcal{M}_{s}(\mathbb{C})$.
Show that $\mathcal{A}$ is trigonalisable.

Let $E$ be a $\mathbb{C}$-vector space of finite dimension $n \geqslant 1$. Let $\mathcal{A}$ be a subalgebra of $\mathcal{L}(E)$ consisting of nilpotent endomorphisms.
Show that there exists a basis of $E$ in which the matrices of elements of $\mathcal{A}$ belong to $\mathrm{T}_{n}^{+}(\mathbb{C})$.

For all permutations $\rho, \rho' \in \mathfrak{S}_n$, show that $P_{\rho\rho'} = P_\rho P_{\rho'}$. Deduce that, for all permutations $\sigma, \tau \in \mathfrak{S}_n$, if $\sigma$ and $\tau$ are conjugate then $P_\sigma$ and $P_\tau$ are similar.

Show that if $\sigma \in \mathfrak{S}_n$, then $\chi_\sigma(X) = \prod_{\ell=1}^{n} \left(X^\ell - 1\right)^{c_\ell(\sigma)}$.
One may justify that $P_\sigma$ is similar to a block diagonal matrix whose blocks are matrices of the form $\Gamma_\ell$ ($\ell \geq 1$), where $\Gamma_\ell$ is defined above if $\ell \geq 2$ and where $\Gamma_\ell = (1)$ if $\ell = 1$.

Deduce property (S): The permutation matrices $P_\sigma$ and $P_\tau$ are similar if and only if the permutations $\sigma$ and $\tau$ are conjugate.
One may compute $T_\sigma D$ where $T_\sigma$ is the cycle type of $\sigma$ and $D$ is the divisor matrix defined in I.D.

Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ be a permutation endomorphism of $E$. Show that $u$ is diagonalizable and that its trace belongs to $\llbracket 0, n \rrbracket$.

Let $A, B$ be two diagonalizable matrices of $\mathcal{M}_n(\mathbb{C})$. Show that $A$ and $B$ are similar if and only if they have the same characteristic polynomial.