Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$.
Show that there exists a polynomial $Q \in \mathbb{C}[X]$ such that $Q(A) = E(A)$.

Let $A \in M_p(\mathbb{C})$ and $n \in \mathbb{N}^*$. We denote $$P_n(X) = \left(1 + \frac{X}{n}\right)^n \in \mathbb{C}[X]$$ and $\chi_A$ the characteristic polynomial of $A$ defined by $$\chi_A(X) = \det\left(A - XI_p\right)$$
Show that there exists a unique pair $\left(Q_n, R_n\right) \in \mathbb{C}[X] \times \mathbb{C}_{p-1}[X]$ such that $$P_n = Q_n \chi_A + R_n$$

Let $A \in M_p(\mathbb{C})$. Let $k \in \mathbb{N}^*$ and $\lambda_1, \lambda_2, \ldots, \lambda_k$ be the roots of $\chi_A$ pairwise distinct, with respective multiplicities $n_1, n_2, \ldots, n_k$. For every integer $q$ between 1 and $p$, $J_q$ denotes the matrix of $M_q(\mathbb{C})$ whose coefficients are all zero except those just above the diagonal which equal 1.
Let $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ be the block diagonal matrix defined by $$B = \begin{pmatrix} \lambda_1 I_{n_1} + J_{n_1} & 0 & \ldots & 0 \\ 0 & \lambda_2 I_{n_2} + J_{n_2} & \cdots & 0 \\ \vdots & \ddots & \ddots & 0 \\ 0 & \ldots & 0 & \lambda_k I_{n_k} + J_{n_k} \end{pmatrix}$$
Show that $\chi_B = \chi_A$.

We assume that the conditions of questions 12 and 14 are satisfied and that $\lambda(0) \neq 0$. Let $C = (q - q^{-1}) E \circ F + q^{-1} H + q H^{-1}$ with $H^{-1}$ the inverse of $H$.
16a. Show that $C = (q - q^{-1}) F \circ E + q H + q^{-1} H^{-1}$.
16b. For $i \in \mathbf{Z}$, show that $v_i$ is an eigenvector of $C$.
16c. Deduce that $C$ is a homothety of $V$ and calculate its ratio $R(\lambda(0), \mu(0), q)$ in terms of $\lambda(0), \mu(0)$ and $q$.
16d. We fix $q$ and $\lambda(0)$. Show that the map $\mu(0) \mapsto R(\lambda(0), \mu(0), q)$ is a bijection from $\mathbf{C}$ to $\mathbf{C}$.
16e. We fix $q$ and $\mu(0)$. Show that the map $\lambda(0) \mapsto R(\lambda(0), \mu(0), q)$ is a surjection from $\mathbf{C}^*$ to $\mathbf{C}$ but not a bijection.

Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$, and $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ the unique algebra morphism such that $\Psi_a(\phi) \circ P_a = P_a \circ \phi$ for all $\phi \in \mathcal{U}_q$. Give a necessary and sufficient condition on $R(\lambda(0), \mu(0), q)$ for the operator $\Psi_a(F)$ to be nilpotent.

We are given a matrix $S \in \mathbf{S}_n$, an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ and integers $1 \leqslant p < q \leqslant n$ such that $s_{pq} \neq 0$. We define $S' = {}^t R_{p,q}(\theta) S R_{p,q}(\theta)$ and denote its coefficients by $s_{ij}'$.
We seek an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ for which we have $s_{pq}' = 0$.
(a) Show that $s_{pq}' = 0$ if and only if $t = \tan\theta$ satisfies the equation $$t^2 + \frac{s_{pp} - s_{qq}}{s_{pq}} t - 1 = 0 \tag{1}$$
(b) Show that this equation admits one solution $t_0 \in ]-1, 1]$ and another $t_1 \notin ]-1, 1]$. What is the relationship between the angles $\theta_0$ and $\theta_1$ that correspond to these roots?
(c) In all that follows, we choose one of the two roots $t$ of equation (1). We thus have $s_{pq}' = 0$. A more precise choice will be made starting from question 12. Verify that $s_{pp}' - s_{pp} = t s_{pq}$; establish an analogous formula for $s_{qq}' - s_{qq}$.
(d) We decompose $S$ in the form $S = D + E$ with $D$ diagonal and $E$ with zero diagonal. We similarly decompose $S' = D' + E'$. Calculate $\|E'\|^2$ in terms of $\|E\|^2$ and $\left(s_{pq}\right)^2$.
(e) By justifying that $\|S'\| = \|S\|$, deduce an expression for $\|D'\|^2$ in terms of $\|D\|^2$ and $\left(s_{pq}\right)^2$.

Let $\left(A^{(m)}\right)_{m \in \mathbb{N}}$ be a sequence in $\mathbf{M}_n$. We assume that for all $m \in \mathbb{N}$, the matrix $A^{(m+1)}$ is similar to $A^{(m)}$. We further assume that this sequence converges to a diagonal matrix $D$. If $P_m$ denotes the characteristic polynomial of $A^{(m)}$, show that the coefficients of $P_m$ converge to those of the characteristic polynomial of $D$ when $m \rightarrow +\infty$.
Deduce that the characteristic polynomial of $D$ is equal to that of $A^{(0)}$.

Let $\left(A^{(m)}\right)_{m \in \mathbb{N}}$ be a sequence in $\mathbf{M}_n$. We assume that for all $m \in \mathbb{N}$, the matrix $A^{(m+1)}$ is similar to $A^{(m)}$, and that this sequence converges to a diagonal matrix $D$.
Finally, show that the diagonal entries of $D$ are the eigenvalues of $A^{(0)}$. What can be said about their multiplicities?

In the optimal version of the Jacobi algorithm, we choose for each $m$ a pair $(p_m, q_m)$ such that the absolute value of the coefficient $\sigma_{ij}^{(m)}$ is maximal precisely when $(i,j) = (p_m, q_m)$. In other words, $$\forall i < j, \quad \left|\sigma_{ij}^{(m)}\right| \leqslant \left|\sigma_{p_m q_m}^{(m)}\right|$$
Show then that the diagonal coefficients of $D$ are the eigenvalues of $\Sigma$.

We fix a matrix $A \in \mathcal{M}_d(\mathbb{R})$, with characteristic polynomial $\chi_A(X) = \det(A - X \cdot I_d) = \sum_{k=0}^d a_k X^k$.
Deduce that $\chi_A(A)$ is the zero matrix. One may use cofactor matrices.

In this part $E$ is a real vector space of dimension $n$ equipped with a basis $\mathcal{B} = (\varepsilon_i)_{1 \leqslant i \leqslant n}$. We consider an endomorphism $f$ of $E$ and we denote by $A$ its matrix in the basis $\mathcal{B}$.
Determine thus the stable plane(s) of $f$ when $n = 3$ and $A$ is the matrix $A = \begin{pmatrix} 1 & -4 & 0 \\ 1 & -2 & -1 \\ 1 & 1 & 0 \end{pmatrix}$ considered in IV.F.

If $M \in \mathcal{S}_{n}(\mathbb{R})$, express $s^{\downarrow}(-M)$ as a function of the coordinates $\left(m_{1}, \ldots, m_{n}\right)$ of $s^{\downarrow}(M)$.

Let $M = \left(\begin{array}{cc}\lambda & h \\ h & \mu\end{array}\right)$ be a matrix of $\mathcal{S}_{2}(\mathbb{R})$. Calculate $s^{\downarrow}(M)$.

Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$ its ordered spectrum, and $\left(v_{1}, \ldots, v_{n}\right)$ an orthonormal basis from the spectral resolution of $M$. Let $j$ be an integer, $1 \leqslant j \leqslant n$. We denote by $\mathcal{V}_{j}$ the vector subspace of $\mathbb{R}^{n}$ spanned by $\left(v_{1}, \ldots, v_{j}\right)$, and by $\mathcal{W}_{j}$ the one spanned by $\left(v_{j}, v_{j+1}, \ldots, v_{n}\right)$. Show the equalities $$\inf_{x \in \mathcal{V}_{j},\, \|x\|=1} \langle x, Mx \rangle = \sup_{x \in \mathcal{W}_{j},\, \|x\|=1} \langle x, Mx \rangle = m_{j}.$$

For $T = \left( t _ { i j } \right) \in \mathcal { M } _ { n + 1 } ( \mathbb { R } )$, we denote by $T _ { \leqslant n }$ the extracted matrix of size $n$ whose coefficients are the $t _ { i j }$ for $1 \leqslant i , j \leqslant n$. Let $M \in S _ { n + 1 } ( \mathbb { R } )$. Show that the set $$\left\{ \operatorname { Sp } \left( \left( U M U ^ { - 1 } \right) _ { \leqslant n } \right) \in \mathbb { R } ^ { n } , \text { for } U \text { ranging over } O _ { n + 1 } ( \mathbb { R } ) \right\} ,$$ denoted $\mathcal { C } _ { M }$, is a compact subset of $\mathbb { R } ^ { n }$.

For $T = \left( t _ { i j } \right) \in \mathcal { M } _ { n + 1 } ( \mathbb { R } )$, we denote by $T _ { \leqslant n }$ the extracted matrix of size $n$ whose coefficients are the $t _ { i j }$ for $1 \leqslant i , j \leqslant n$. Let $M \in S _ { n + 1 } ( \mathbb { R } )$, and denote $\mathcal { C } _ { M } = \left\{ \operatorname { Sp } \left( \left( U M U ^ { - 1 } \right) _ { \leqslant n } \right) \in \mathbb { R } ^ { n } , \text { for } U \text { ranging over } O _ { n + 1 } ( \mathbb { R } ) \right\}$. We further assume that the eigenvalues of $M$ are distinct. We thus have $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } > \cdots > \lambda _ { n + 1 } \right)$.
(a) Let $\widehat { \mu } = \left( \mu _ { 1 } > \cdots > \mu _ { n } \right)$ such that $\operatorname { Sp } ( M )$ and $\widehat { \mu }$ are strictly interlaced. Show that $\widehat { \mu }$ belongs to $\mathcal { C } _ { M }$.
(b) Show that $$\mathcal { C } _ { M } = \left\{ \widehat { \mu } = \left( \mu _ { 1 } \geqslant \cdots \geqslant \mu _ { n } \right) , \text { such that } \operatorname { Sp } ( M ) \text { and } \widehat { \mu } \text { are interlaced } \right\} .$$

We consider the application $$\begin{array} { r c c c } \operatorname { Diag } _ { n } : & S _ { n } ( \mathbb { R } ) & \longrightarrow & \mathbb { R } ^ { n } \\ M = \left( m _ { i j } \right) & \longmapsto & \left( m _ { 11 } , m _ { 22 } , \ldots , m _ { n n } \right) \end{array}$$ Let $M \in S _ { n } ( \mathbb { R } )$. We study the set $$\mathcal { D } _ { M } = \left\{ \operatorname { Diag } _ { n } \left( U M U ^ { - 1 } \right) , \text { for } U \text { ranging over } O _ { n } ( \mathbb { R } ) \right\} .$$ We first study the case $n = 2$. We then denote $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } \geqslant \lambda _ { 2 } \right)$. Show that $\mathcal { D } _ { M }$ is the line segment in $\mathbb { R } ^ { 2 }$ whose endpoints are $( \lambda _ { 1 } , \lambda _ { 2 } )$ and $( \lambda _ { 2 } , \lambda _ { 1 } )$.

We consider the application $$\begin{array} { r c c c } \operatorname { Diag } _ { n } : & S _ { n } ( \mathbb { R } ) & \longrightarrow & \mathbb { R } ^ { n } \\ M = \left( m _ { i j } \right) & \longmapsto & \left( m _ { 11 } , m _ { 22 } , \ldots , m _ { n n } \right) \end{array}$$ Let $M = \left( m _ { i j } \right) \in S _ { n } ( \mathbb { R } )$. We denote $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } \geqslant \cdots \geqslant \lambda _ { n } \right) \in \mathbb { R } ^ { n }$. We propose to prove that, for all $s \in \{ 1 , \ldots , n \}$, we have: $$\sum _ { i = 1 } ^ { s } m _ { i i } \leqslant \sum _ { i = 1 } ^ { s } \lambda _ { i }$$ (a) What do you think of the case $s = n$ ?
(b) Express $\sum _ { i = 1 } ^ { n - 1 } m _ { i i }$ in terms of the eigenvalues of the matrix $M _ { \leqslant n - 1 }$ obtained by removing the last row and last column of $M$. Deduce inequality (3) when $s = n - 1$.
(c) By proceeding by induction on $n$, show inequality (3), for all $s \in \{ 1 , \ldots , n \}$.

We denote by $E$ the vector space $\mathbb { R } ^ { 2 }$ equipped with the standard inner product and the canonical basis $\mathcal { B } = \left\{ e _ { 1 } , e _ { 2 } \right\}$. We define a basis $\mathcal { C } = \left\{ \omega _ { 1 } , \omega _ { 2 } \right\}$ of $E$ by $\omega _ { 1 } = e _ { 1 }$ and $\omega _ { 2 } = \frac { 1 } { 2 } \left( e _ { 1 } + \sqrt { 3 } e _ { 2 } \right)$. Let $s_1$ be the orthogonal reflection with respect to $\mathbb{R}\omega_1$ and $s_2$ the orthogonal reflection with respect to $\mathbb{R}\omega_2$. Let $H$ be the set of vectors $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathbb { R } ^ { 3 }$ such that $m _ { 1 } + m _ { 2 } + m _ { 3 } = 0$, $H^+$ the subset with $m_1 \geqslant m_2 \geqslant m_3$, and $\varphi : H \to E$ defined by $\varphi(m_1,m_2,m_3) = (m_1-m_2)\omega_1 + (m_2-m_3)\omega_2$. We consider the application $$\begin{array} { r c c c } \operatorname { Diag } _ { n } : & S _ { n } ( \mathbb { R } ) & \longrightarrow & \mathbb { R } ^ { n } \\ M = \left( m _ { i j } \right) & \longmapsto & \left( m _ { 11 } , m _ { 22 } , \ldots , m _ { n n } \right) \end{array}$$ and $\mathcal { D } _ { M } = \left\{ \operatorname { Diag } _ { n } \left( U M U ^ { - 1 } \right) , \text { for } U \text { ranging over } O _ { n } ( \mathbb { R } ) \right\}$. Let $M \in S _ { 3 } ( \mathbb { R } )$ be a matrix with zero trace. We denote $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } \geqslant \lambda _ { 2 } \geqslant \lambda _ { 3 } \right)$. We propose to describe $\varphi \left( \mathcal { D } _ { M } \right)$.
(a) Let $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H$. Let $\sigma$ be a permutation of $\{ 1,2,3 \}$. Show that $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathcal { D } _ { M }$ if and only if $\left( m _ { \sigma ( 1 ) } , m _ { \sigma ( 2 ) } , m _ { \sigma ( 3 ) } \right) \in \mathcal { D } _ { M }$.
(b) Using question 13, show that the intersection $H ^ { + } \cap \mathcal { D } _ { M }$ is contained in $\mathcal { Q } _ { \widehat { \lambda } }$.
(c) Let $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathcal { D } _ { M }$. Show that the segment of $H$ whose vertices are $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right)$ and $\left( m _ { 2 } , m _ { 1 } , m _ { 3 } \right)$ is contained in $\mathcal { D } _ { M }$. One may use question 12. Similarly, show that the segment of $H$ whose vertices are $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right)$ and $\left( m _ { 1 } , m _ { 3 } , m _ { 2 } \right)$ is contained in $\mathcal { D } _ { M }$.
(d) Show that $\mathcal { D } _ { M }$ contains $\mathcal { Q } _ { \widehat { \lambda } }$.
(e) Show that if $\lambda _ { 1 } > \lambda _ { 2 } > \lambda _ { 3 }$ then $\varphi \left( \mathcal { D } _ { M } \right)$ is a hexagon, whose vertices will be determined.

Let $M \in \mathcal{Y}_n$ and $\lambda$ a complex eigenvalue of $M$. Prove that $|\lambda| \leqslant n$ and give an explicit example where equality holds.

Determine the eigenvectors common to all elements of $\mathcal{P}_n$ in the cases $n = 2$ and $n = 3$.

Show that the spectral radius of an irreducible matrix is strictly positive.

We define the matrix $Z_n = (z_{i,j}) \in \mathcal{M}_n(\mathbb{R})$ by $z_{i,j} = \begin{cases} 1 & \text{if } 1 \leqslant i < n \text{ and } j = i+1 \\ 1 & \text{if } (i,j) \in \{(n-1,1),(n,2)\} \\ 0 & \text{in all other cases} \end{cases}$
Show that the characteristic polynomial of $Z_n$ is $X(X^{n-1} - 2)$.
Deduce that $Z_n$ is imprimitiv and specify its coefficient of imprimitivity.

Let $A \geqslant 0$ in $\mathcal{M}_n(\mathbb{R})$, an irreducible matrix. We denote $r$ its spectral radius. Let $p \geqslant 1$ be the coefficient of imprimitivity of $A$ (reminder: by convention, $p = 1$ if $A$ is primitive). Let $\chi_A(X) = X^n + c_{k_1}X^{n-k_1} + c_{k_1}X^{n-k_2} + \cdots + c_{k_s}X^{n-k_s}$ be its characteristic polynomial, written according to decreasing powers and showing only the nonzero coefficients $c_k$.
We recall that the spectrum of $A$ is invariant under the map $z \mapsto \omega z$, where $\omega = \exp(2\mathrm{i}\pi/p)$.
Deduce that, for all $k \in \{k_1, k_2, \ldots, k_s\}$, the integer $k$ is divisible by $p$. Think of the elementary symmetric functions of the $\lambda_i$.

Let $A \geqslant 0$ in $\mathcal{M}_n(\mathbb{R})$, an irreducible matrix. We denote $r$ its spectral radius. Let $p \geqslant 1$ be the coefficient of imprimitivity of $A$. Let $\chi_A(X) = X^n + c_{k_1}X^{n-k_1} + c_{k_1}X^{n-k_2} + \cdots + c_{k_s}X^{n-k_s}$ be its characteristic polynomial, written according to decreasing powers and showing only the nonzero coefficients $c_k$. We will show that $p$ is the gcd of the integers $k_1, k_2, \ldots, k_s$.
Conversely, we assume by contradiction that the $k_j$ are all divisible by $qp$, with $q \geqslant 2$. We set $\beta = \mathrm{e}^{2\mathrm{i}\pi/(qp)}$ (so $\beta^q = \omega$). Show that $\beta r$ is an eigenvalue of $A$ and conclude.