Are the subsets $\mathrm{T}_{n}(\mathbb{K})$ and $\mathrm{T}_{n}^{+}(\mathbb{K})$ subalgebras of $\mathcal{M}_{n}(\mathbb{K})$?

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

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

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Show that there exists a vector $x$ in $E$ such that $u^{p-1}(x) \neq 0$.

Let $U \in \mathcal{M}_n(\mathbb{C})$ be a unitary matrix. Show that $\|Ux\|_2 = \|x\|_2$ for all $x \in \mathbb{C}^n$.

Are the subsets $S_{2}(\mathbb{K})$ and $A_{2}(\mathbb{K})$ subalgebras of $\mathcal{M}_{2}(\mathbb{K})$?

For any matrix $B \in \mathcal { M } _ { N } ( \mathbb { R } )$, we set $\| B \| = \sup _ { \| x \| = 1 } \| B x \|$.
After justifying the existence of $\| B \|$, show that $B \mapsto \| B \|$ is a norm on $\mathcal { M } _ { N } ( \mathbb { R } )$ satisfying $$\forall x \in \mathbb { R } ^ { N } \quad \| B x \| \leq \| B \| \| x \|$$

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 matrix
$$C_Q = \left(\begin{array}{cccccc} 0 & \cdots & \cdots & \cdots & 0 & -a_0 \\ 1 & 0 & \cdots & \cdots & 0 & -a_1 \\ 0 & 1 & \ddots & & \vdots & -a_2 \\ \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ \vdots & & \ddots & 1 & 0 & -a_{n-2} \\ 0 & \cdots & \cdots & 0 & 1 & -a_{n-1} \end{array}\right).$$
Determine as a function of $Q$ the characteristic polynomial of $C_Q$.

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 matrix
$$C_Q = \left(\begin{array}{cccccc} 0 & \cdots & \cdots & \cdots & 0 & -a_0 \\ 1 & 0 & \cdots & \cdots & 0 & -a_1 \\ 0 & 1 & \ddots & & \vdots & -a_2 \\ \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ \vdots & & \ddots & 1 & 0 & -a_{n-2} \\ 0 & \cdots & \cdots & 0 & 1 & -a_{n-1} \end{array}\right).$$
Determine as a function of $Q$ the characteristic polynomial of $C_Q$.

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Verify that the family $\left(u^{k}(x)\right)_{0 \leqslant k \leqslant p-1}$ is free. Deduce that $p = 2$.

If $D \in \mathcal{M}_n(\mathbb{C})$ is a diagonal matrix whose diagonal coefficients are $d_0, \ldots, d_{n-1}$, show that $\|D\| = \max_{0 \leq i \leq n-1} |d_i|$.

Suppose $n \geqslant 3$. Are the subsets $\mathrm{S}_{n}(\mathbb{K})$ and $\mathrm{A}_{n}(\mathbb{K})$ subalgebras of $\mathcal{M}_{n}(\mathbb{K})$?

Let $A \in \mathcal { S } _ { N } ( \mathbb { R } )$ be a matrix with eigenvalues (not necessarily distinct) $\lambda _ { 1 } , \ldots , \lambda _ { N }$. Show that $$\| A \| = \max _ { 1 \leq i \leq N } \left| \lambda _ { i } \right|$$

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Show that $\operatorname{Ker} u = \operatorname{Im} u$.

Let $A, B \in \mathcal{M}_n(\mathbb{C})$. Suppose that there exists a unitary matrix $U \in \mathcal{M}_n(\mathbb{C})$ such that $B = UAU^{-1}$. Show that $\|A\| = \|B\|$.

Let $F$ be a vector subspace of $E$ of dimension $p$ and $\mathcal{A}_{F}$ the set of endomorphisms of $E$ that stabilise $F$, that is $\mathcal{A}_{F} = \{ u \in \mathcal{L}(E) \mid u(F) \subset F \}$.
Show that $\mathcal{A}_{F}$ is a subalgebra of $\mathcal{L}(E)$.

Let $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$. For all $x \in \mathbb { R } ^ { N }$, we set $\| x \| _ { A } = \langle x , A x \rangle ^ { 1 / 2 }$.
a) Show that the map $x \mapsto \| x \| _ { A }$ is a norm on $\mathbb { R } ^ { N }$.
b) Show that there exist constants $C _ { 1 }$ and $C _ { 2 }$ strictly positive, which we will express in terms of the eigenvalues of $A$, such that $$\forall x \in \mathbb { R } ^ { N } \quad C _ { 1 } \| x \| \leq \| x \| _ { A } \leq C _ { 2 } \| x \| .$$

Show that $f$ is cyclic if and only if there exists a basis $\mathcal{B}$ of $E$ in which the matrix of $f$ is of the form $C_Q$, where $Q$ is a monic polynomial of degree $n$.

Show that $f$ is cyclic if and only if there exists a basis $\mathcal{B}$ of $E$ in which the matrix of $f$ is of the form $C_Q$, where $Q$ is a monic polynomial of degree $n$.

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Construct a basis of $E$ in which the matrix of $u$ is equal to $J_2$.

We fix a polynomial $f \in \mathbb{C}[X]$ of degree $n \geq 1$. We consider a complex number $z \in \overline{\mathbb{D}}$ and we define the matrices $M \in \mathcal{M}_{n+1}(\mathbb{C})$ and $P \in \mathcal{M}_{n+1,1}(\mathbb{C})$ by $$M = \left(\begin{array}{cccccc} z & 0 & 0 & \ldots & 0 & \sqrt{1-|z|^2} \\ \sqrt{1-|z|^2} & 0 & 0 & \ldots & 0 & -\bar{z} \\ 0 & & & & & 0 \\ 0 & & & & & 0 \\ \vdots & & I_{n-1} & & & \vdots \\ 0 & & & & & 0 \end{array}\right)$$ and $$P = \left(\begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \end{array}\right)$$ Show that $M$ is a unitary matrix.

Let $F$ be a vector subspace of $E$ of dimension $p$ and $\mathcal{A}_{F}$ the set of endomorphisms of $E$ that stabilise $F$, that is $\mathcal{A}_{F} = \{ u \in \mathcal{L}(E) \mid u(F) \subset F \}$.
Show that $\operatorname{dim} \mathcal{A}_{F} = n^{2} - pn + p^{2}$.
One may consider a basis of $E$ in which the matrix of every element of $\mathcal{A}_{F}$ is block triangular.

Let $A \in \mathcal { S } _ { N } ( \mathbb { R } )$ and let $P \in \mathbb { R } [ X ]$ be a polynomial. Show that $P ( A ) \in \mathcal { S } _ { N } ( \mathbb { R } )$ and specify the eigenvalues and eigenvectors of $P ( A )$ in terms of those of $A$.

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Deduce that the nilpotent matrices in $\mathcal{M}_2(\mathbb{C})$ are exactly the matrices with zero trace and zero determinant.

We fix a polynomial $f \in \mathbb{C}[X]$ of degree $n \geq 1$. We consider a complex number $z \in \overline{\mathbb{D}}$ and we define the matrices $M \in \mathcal{M}_{n+1}(\mathbb{C})$ and $P \in \mathcal{M}_{n+1,1}(\mathbb{C})$ by $$M = \left(\begin{array}{cccccc} z & 0 & 0 & \ldots & 0 & \sqrt{1-|z|^2} \\ \sqrt{1-|z|^2} & 0 & 0 & \ldots & 0 & -\bar{z} \\ 0 & & & & & 0 \\ 0 & & & & & 0 \\ \vdots & & I_{n-1} & & & \vdots \\ 0 & & & & & 0 \end{array}\right)$$ and $$P = \left(\begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \end{array}\right)$$ Show that $z^k = P^T M^k P$ for all integer $0 \leq k \leq n$.