Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. We assume throughout this part that $A$ is diagonalizable in $\mathscr{M}_n(\mathbb{C})$ and we denote by $\lambda_1, \cdots, \lambda_\ell$ its eigenvalues with $\lambda_i \neq \lambda_j$ if $i \neq j$. Let $B \in \mathscr{M}_n(\mathbb{C})$ be an invertible matrix. Show that $$u(BAB^{-1}) = B\, u(A)\, B^{-1}.$$

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. We assume throughout this part that $A$ is diagonalizable in $\mathscr{M}_n(\mathbb{C})$ and we denote by $\lambda_1, \cdots, \lambda_\ell$ its eigenvalues with $\lambda_i \neq \lambda_j$ if $i \neq j$. Let $D \in \mathscr{M}_n(\mathbb{C})$ be a diagonal matrix and $S \in \mathscr{M}_n(\mathbb{C})$ be an invertible matrix such that $A = SDS^{-1}$.
(a) Show that $u(D)$ is diagonal and that $$\forall i \in \llbracket 1; n \rrbracket, [u(D)]_{i,i} = U([D]_{i,i}).$$ (b) Deduce an expression for $u(A)$.

In this part, we assume that $n \geqslant 4$. Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ satisfying the condition $(C^\star)$: $R_u > 1$.
Let $H \in \mathscr{M}_n(\mathbb{C})$ be the matrix given by $$H = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & & \ddots & \ddots & 1 \\ 0 & \cdots & \cdots & 0 & 0 \end{pmatrix}.$$
(a) Determine the polynomial $\varphi_H$ in this case.
(b) Let $A = H + \alpha I_n$ where $\alpha \in \mathbb{C}$ is such that $|\alpha| < R_u$. Show that $$u(A) = \sum_{k=0}^{n-1} \frac{U^{(k)}(\alpha)}{k!} H^k$$ and deduce that $$u(A) = \begin{pmatrix} U(\alpha) & \frac{U^{(1)}(\alpha)}{1!} & \frac{U^{(2)}(\alpha)}{2!} & \cdots & \frac{U^{(n-1)}(\alpha)}{(n-1)!} \\ 0 & U(\alpha) & \frac{U^{(1)}(\alpha)}{1!} & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \frac{U^{(2)}(\alpha)}{2!} \\ \vdots & & \ddots & \ddots & \frac{U^{(1)}(\alpha)}{1!} \\ 0 & \cdots & \cdots & 0 & U(\alpha) \end{pmatrix}.$$

In this part, we assume that $n \geqslant 4$. Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ satisfying the condition $(C^\star)$: $R_u > 1$.
Let $G \in \mathscr{M}_n(\mathbb{C})$ be the matrix defined by $$G = Y\, {}^t Z$$ where $Y, Z \in \mathscr{M}_{n,1}(\mathbb{R})$ are two column vectors such that ${}^t Y Y = {}^t Z Z = 1$.
(a) Show that $G$ has rank 1 and give its image.
(b) Show that $0$ and ${}^t Z Y$ are the only eigenvalues of $G$.
(c) Deduce that $G \in \mathbb{M}_n(u)$.
(d) Determine $\varphi_G$ when ${}^t Z Y \neq 0$.
(e) Deduce that if ${}^t Z Y \neq 0$ then $$u(G) = U(0) I_n + \frac{U({}^t Z Y) - U(0)}{{}^t Z Y} G.$$ (f) Determine a simple expression for $u(G)$ when ${}^t Z Y = 0$.

In this part, we assume that $n \geqslant 4$. Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ satisfying the condition $(C^\star)$: $R_u > 1$.
Let $F \in \mathscr{M}_n(\mathbb{C})$ be the matrix defined by $$[F]_{k,j} = \frac{1}{\sqrt{n}} \omega^{(k-1)(j-1)} \text{ for all } (k,j) \in \llbracket 1; n \rrbracket^2,$$ where $\omega = e^{-2\pi i/n}$ (here $i$ denotes the usual complex number satisfying $i^2 = -1$).
(a) Show that $F$ is invertible and that $F^{-1} = \bar{F}$.
(b) Show that $F^2 \in \mathscr{M}_n(\mathbb{R})$.
(c) Deduce that $F^4 = I_n$ and that $F \in \mathbb{M}_n(u)$.
(d) Deduce that $$\begin{aligned} u(F) = & \frac{1}{4}\left(U(1)(F + I_n) - U(-1)(F - I_n)\right)(F^2 + I_n) \\ & + \frac{i}{4}\left(U(i)(F + iI_n) - U(-i)(F - iI_n)\right)(F^2 - I_n) \end{aligned}$$

Let $P \in \mathbf{C}[X]$ of degree $p$. We write $P = \sum_{k=0}^{p} a_k X^k$, where $a_0, \ldots, a_p$ are complex numbers, and $a_p \neq 0$. Show that $P$ is reciprocal if and only if for every integer $k$, $0 \leq k \leq p$, we have the equality $a_k = a_{p-k}$.

Let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n \backslash \{\mathbf{0}\}$. We set $M = \mathbf{u v}^T$. Show that $M$ is a square matrix of size $n \times n$, of rank 1.

Explain why the matrix $J_n$ is diagonalizable.

Let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n \backslash \{\mathbf{0}\}$. We set $M = \mathbf{u v}^T$. Show that $M$ is a square matrix of size $n \times n$, of rank 1.

Calculate with justification the rank of the following matrix $J \in \mathcal{M}_n(\mathbb{R})$: $$J = \left(\begin{array}{cccc} 1 & 1 & \cdots & 1 \\ 1 & 1 & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \cdots & 1 \end{array}\right)$$

Calculate with justification the rank of the following matrix $J \in \mathcal{M}_n(\mathbb{R})$: $$J = \left(\begin{array}{cccc} 1 & 1 & \cdots & 1 \\ 1 & 1 & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \cdots & 1 \end{array}\right).$$

Conversely, let $K \in \mathcal{M}_n(\mathbb{R})$ be a square matrix of rank 1. Show that there exist $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n \backslash \{\mathbf{0}\}$ such that $K = \mathbf{u v}^T$.

Conversely, let $K \in \mathcal{M}_n(\mathbb{R})$ be a square matrix of rank 1. Show that there exist $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n \backslash \{\mathbf{0}\}$ such that $K = \mathbf{u v}^T$.

Let $\mathbf{u}, \mathbf{v}, \mathbf{x}, \mathbf{y} \in \mathbb{R}^n \backslash \{0\}$. Show that $\mathbf{u v}^T = \mathbf{x y}^T$ if and only if there exists $\lambda \in \mathbb{R} \backslash \{0\}$ such that $$\mathbf{u} = \lambda \mathbf{x}, \quad \text{and} \quad \mathbf{v} = \frac{1}{\lambda} \mathbf{y}$$

Let $\mathbf{u}, \mathbf{v}, \mathbf{x}, \mathbf{y} \in \mathbb{R}^n \backslash \{\mathbf{0}\}$. Show that $\mathbf{u v}^T = \mathbf{x y}^T$ if and only if there exists $\lambda \in \mathbb{R} \backslash \{0\}$ such that $$\mathbf{u} = \lambda \mathbf{x}, \quad \text{and} \quad \mathbf{v} = \frac{1}{\lambda} \mathbf{y}.$$

Let $K \in \mathcal{M}_n(\mathbb{R})$ be a matrix of rank 1, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$ be such that $K = \mathbf{u v}^T$.
(a) Show that $\operatorname{Tr}(K) = \langle \mathbf{v}, \mathbf{u} \rangle$.
(b) Show that $K^2 = \operatorname{Tr}(K) K$.
(c) Deduce that $K$ is diagonalizable if and only if $\operatorname{Tr}(K) \neq 0$.

Let $K \in \mathcal{M}_n(\mathbb{R})$ be a matrix of rank 1, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$ be such that $K = \mathbf{u v}^T$.
(a) Show that $\operatorname{Tr}(K) = \langle \mathbf{v}, \mathbf{u} \rangle$.
(b) Show that $K^2 = \operatorname{Tr}(K) K$.
(c) Deduce that $K$ is diagonalizable if and only if $\operatorname{Tr}(K) \neq 0$.

Let $A$ be a matrix belonging to $\mathbf{GL}_n$. Let $x$ be a nonzero real number. Express $\det(x I_n - A)$ in terms of $x$, $\det A$ and $\det\left(\frac{1}{x} I_n - A^{-1}\right)$.

Let $P \in \mathcal{M}_n(\mathbb{R})$. Show that $P$ is an orthogonal projector of rank 1 if and only if there exists $\mathbf{y} \in \mathbb{R}^n$ with $\|\mathbf{y}\| = 1$ such that $P = \mathbf{y y}^T$.

Let $P \in \mathcal{M}_n(\mathbb{R})$. Show that $P$ is an orthogonal projector of rank 1 if and only if there exists $\mathbf{y} \in \mathbb{R}^n$ with $\|\mathbf{y}\| = 1$ such that $P = \mathbf{y y}^T$.

Let $A$ be a matrix belonging to $\mathbf{GL}_n$. Suppose in this question that $A$ is similar to its inverse. Specify the values that the determinant of $A$ can take, and deduce that $\chi_A$ is either reciprocal or antireciprocal.

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Calculate the block matrix product $$\left(\begin{array}{cc} \mathbb{I}_n & 0 \\ \mathbf{v}^T & 1 \end{array}\right) \left(\begin{array}{cc} \mathbb{I}_n + \mathbf{u}\mathbf{v}^T & \mathbf{u} \\ 0 & 1 \end{array}\right) \left(\begin{array}{cc} \mathbb{I}_n & 0 \\ -\mathbf{v}^T & 1 \end{array}\right)$$

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Calculate the block matrix product $$\left(\begin{array}{cc} \mathbb{I}_n & 0 \\ \mathbf{v}^T & 1 \end{array}\right) \left(\begin{array}{cc} \mathbb{I}_n + \mathbf{u v}^T & \mathbf{u} \\ 0 & 1 \end{array}\right) \left(\begin{array}{cc} \mathbb{I}_n & 0 \\ -\mathbf{v}^T & 1 \end{array}\right).$$

Let $B \in \mathbf{M}_n$ be a diagonalizable matrix. Suppose that the characteristic polynomial of $B$ is reciprocal or antireciprocal. Prove that $B$ is invertible and similar to its inverse.

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Show that $$\operatorname{det}\left(\mathbb{I}_n + \mathbf{u}\mathbf{v}^T\right) = 1 + \langle \mathbf{v}, \mathbf{u} \rangle$$