We consider $R \in \mathrm{O}_{d}(\mathbb{R})$.

• [(a)] Show that if $\lambda$ is an eigenvalue of $R$ then $\lambda \in \{+1, -1\}$.
• [(b)] Show that $\operatorname{det}(R + I) = \operatorname{det}(R) \operatorname{det}(I + R^{T})$.
• [(c)] Deduce that if $\operatorname{det}(R) = -1$ then $\operatorname{det}(R + I) = 0$.

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 $A$ be a matrix belonging to $\mathbf{GL}_n$. We assume 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.

Prove the Schur-Cohn criterion: If $J(p)$ is invertible then $p$ has no stable root and $\sigma(p) = \pi(J(p))$.

Problem 2, Part 2: Linear recurrence sequences with constant coefficients
We consider a sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ of complex numbers defined by the data of $u _ { 0 } , \ldots , u _ { d }$ and the linear recurrence relation $$u _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } a _ { i } u _ { n + i } + b ,$$ where the $a _ { i }$ and $b$ are complex numbers. We define $P \in \mathbb { C } [ X ]$ by $P ( X ) = X ^ { d } - \sum _ { i = 0 } ^ { d - 1 } a _ { i } X ^ { i }$ and we assume that all complex roots of $P$ have modulus strictly less than 1. The matrix $A \in \mathrm{M}_d(\mathbb{C})$ is as defined in question 7.
Calculate the characteristic polynomial of the matrix $A$ (one may reason by induction on $d$).

Properties of $h$ In this part, we are given: a finite-dimensional vector space $V$; a nilpotent endomorphism $u$ of $V$; a nonzero natural integer $N$ and the complex number $\zeta = \exp\frac{2\mathrm{i}\pi}{N}$; an invertible endomorphism $h$ of $V$ such that $h^N = \operatorname{id}_V$ and $h \circ u \circ h^{-1} = \zeta u$.
a) Prove that $h$ is diagonalizable.
b) Let $j$ be a natural integer strictly less than $N$. By denoting $V_j = \ker(h - \zeta^j \operatorname{id}_V)$ and $V_N = V_0$, verify that $u(V_j) \subset V_{j+1}$.
c) Calculate, for $k$ relative integer, $h^k \circ u \circ h^{-k}$ and, for $l$ natural integer, $h \circ u^l \circ h^{-1}$.

Two linear maps: decomposition We fix two nonzero natural integers $m$ and $n$, matrices $(A,B) \in \mathcal{M}_{n,m}(\mathbb{C}) \times \mathcal{M}_{m,n}(\mathbb{C})$, and denote $M = M_{A,B} = \begin{pmatrix} 0_m & B \\ A & 0_n \end{pmatrix}$ and $H = \begin{pmatrix} \mathrm{I}_m & 0_{m,n} \\ 0_{n,m} & -\mathrm{I}_n \end{pmatrix}$.
a) Calculate $H^2$, $HMH^{-1}$ and, for a polynomial $P$ of $\mathbb{C}[X]$, calculate $HP(M)H^{-1}$.
b) Prove that if a complex number $\lambda$ is an eigenvalue of $M$, then $-\lambda$ is also an eigenvalue of $M$ with the same multiplicity.
c) Let $\chi_M$ be the characteristic polynomial of $M$. We write it as $\chi_M = X^r Q$ where $r$ is an integer and $Q$ is a polynomial whose constant coefficient is nonzero. Briefly justify that $$\mathbb{C}^{m+n} = \ker M^r \oplus \ker Q(M)$$ and verify that these subspaces are stable under $H$.

Let $\lambda$ be an eigenvalue of $A$ with multiplicity $m \geqslant 2$. We set $E = \operatorname{Ker}\left(A - \lambda \mathbb{I}_n\right)$.
(a) Show that $\operatorname{dim}\left(E \cap \{\mathbf{u}\}^\perp\right) \geqslant m - 1$.
(b) Deduce that $\lambda$ is an eigenvalue of $B$ with multiplicity at least $m - 1$.

Let $J = \left\{k \in \{1, 2, \ldots, n\}, \left\langle \mathbf{w}_k, \mathbf{u} \right\rangle \neq 0\right\}$ be the set of indices $k$ such that $\left\langle \mathbf{w}_k, \mathbf{u} \right\rangle \neq 0$.
(a) Show that $J \neq \varnothing$.
(b) Let $\ell \notin J$. Show that $\lambda_\ell$ is an eigenvalue of $B$.
(c) Suppose that $J = \{j\}$ for some $j \in \{1, 2, \ldots, n\}$. Show that the eigenvalues of $B$ are $$\left(\lambda_1, \lambda_2, \ldots, \lambda_{j-1}, \lambda_j + 1, \lambda_{j+1}, \ldots, \lambda_n\right).$$

Suppose in this question that $\lambda_1 < \lambda_2 < \cdots < \lambda_n$, and that $J = \{1, 2, \ldots, n\}$. For $x \in \mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$ we set $$f(x) = \sum_{k=1}^n \frac{\left\langle \mathbf{w}_k, \mathbf{u} \right\rangle^2}{x - \lambda_k}$$ (a) Show that $f$ is of class $C^\infty$ on $\mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$, and calculate its derivative $f'(x)$.
(b) Show that the equation $f(x) = 1$ has a unique solution in each interval $]\lambda_\ell, \lambda_{\ell+1}[$ for all $\ell \in \{1, 2, \ldots, n-1\}$, and in $]\lambda_n, +\infty[$.
(c) We denote by $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ the eigenvalues of $B$. Show that $$\lambda_1 < \mu_1 < \lambda_2 < \mu_2 < \cdots < \lambda_n < \mu_n$$

We denote by $\lambda_{\text{max}}$ the largest of the eigenvalues of $J_n$ and $\lambda_{\text{min}}$ the smallest. Show that $$\forall x \in \Lambda_n, \quad n\lambda_{\min} \leqslant \sum_{1 \leqslant i,j \leqslant n} J_n(i,j) x_i x_j \leqslant n\lambda_{\max}$$

We denote by $J_n^{(\mathrm{C})}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ whose coefficients all equal $\frac{1}{n}$, except for its diagonal coefficients, which are zero. Each particle thus interacts in the same way with all other particles.
Determine the spectrum of $U_n = nJ_n^{(\mathrm{C})} + I_n$, then that of $J_n^{(\mathrm{C})}$.

We set $N = n^2$ and $$J_N^{(2)} = I_n \otimes J_n^{(1)} + J_n^{(1)} \otimes I_n \in \mathcal{M}_N(\mathbb{R})$$
Show that the eigenvalues of $J_N^{(2)}$ are the $\lambda_j + \lambda_k$, for $(j,k) \in \llbracket 1,n \rrbracket^2$.

We consider $A \in \mathcal{S}_n(\mathbb{R})$ symmetric with $B = A + \mathbf{u u}^T$ where $\|\mathbf{u}\| = 1$. Let $\lambda$ be an eigenvalue of $A$ with multiplicity $m \geqslant 2$. We set $E = \operatorname{Ker}\left(A - \lambda \mathbb{I}_n\right)$.
(a) Show that $\operatorname{dim}\left(E \cap \{\mathbf{u}\}^\perp\right) \geqslant m - 1$.
(b) Deduce that $\lambda$ is an eigenvalue of $B$ with multiplicity at least $m - 1$.

We consider $A \in \mathcal{S}_n(\mathbb{R})$ symmetric with eigenvalues $\lambda_1 \leqslant \cdots \leqslant \lambda_n$ and corresponding orthonormal basis of eigenvectors $\left(\mathbf{w}_1, \ldots, \mathbf{w}_n\right)$. We set $B = A + \mathbf{u u}^T$ with $\|\mathbf{u}\| = 1$. Let $J = \left\{k \in \{1,2,\ldots,n\}, \left\langle \mathbf{w}_k, \mathbf{u} \right\rangle \neq 0\right\}$ be the set of indices $k$ such that $\left\langle \mathbf{w}_k, \mathbf{u} \right\rangle \neq 0$.
(a) Show that $J \neq \varnothing$.
(b) Let $\ell \notin J$. Show that $\lambda_\ell$ is an eigenvalue of $B$.
(c) Suppose that $J = \{j\}$ for some $j \in \{1,2,\ldots,n\}$. Show that the eigenvalues of $B$ are $$\left(\lambda_1, \lambda_2, \ldots, \lambda_{j-1}, \lambda_j + 1, \lambda_{j+1}, \ldots, \lambda_n\right).$$

We consider $A \in \mathcal{S}_n(\mathbb{R})$ symmetric with eigenvalues $\lambda_1 \leqslant \cdots \leqslant \lambda_n$ and corresponding orthonormal basis of eigenvectors $\left(\mathbf{w}_1, \ldots, \mathbf{w}_n\right)$. We set $B = A + \mathbf{u u}^T$ with $\|\mathbf{u}\| = 1$. Suppose in this question that $\lambda_1 < \lambda_2 < \cdots < \lambda_n$, and that $J = \{1,2,\ldots,n\}$. For $x \in \mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$ we set $$f(x) = \sum_{k=1}^n \frac{\left\langle \mathbf{w}_k, \mathbf{u} \right\rangle^2}{x - \lambda_k}.$$
(a) Show that $f$ is of class $C^\infty$ on $\mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$, and calculate its derivative $f'(x)$.
(b) Show that the equation $f(x) = 1$ has a unique solution in each interval $]\lambda_\ell, \lambda_{\ell+1}[$ for all $\ell \in \{1,2,\ldots,n-1\}$, and in $]\lambda_n, +\infty[$.
(c) We denote by $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ the eigenvalues of $B$. Show that $$\lambda_1 < \mu_1 < \lambda_2 < \mu_2 < \cdots < \lambda_n < \mu_n.$$

In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. We consider an arbitrary orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, following the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$. Prove that there exists $x \in \mathbb{R}$ such that $\mathbb{E}\left[\chi_B(x)\right] \neq 0$.

We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$. Let $M$ be a nonzero real symmetric matrix of size $d \times d$ such that $\forall x \in \mathbb{R}^d,\, \langle x, Mx \rangle \geq 0$. We define $f(x) := -\frac{1}{2}\langle x, Mx \rangle$. Describe the set of minimizers of $f$ on $C$.

Let $n$ be the index of $u$, that is, the integer such that $u^{n-1} \neq 0$ and $u^n = 0$. Prove that there exists a vector $v$ such that $v$ is an eigenvector of $h$ and $u^{n-1}(v) \neq 0$.

Prove that if a complex number $\lambda$ is an eigenvalue of $M$, then $-\lambda$ is also an eigenvalue of $M$ with the same multiplicity.

Let $x \in \mathbb{R}$ and let $$P = \left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 3\end{array}\right], \quad Q = \left[\begin{array}{ccc}2 & x & x \\ 0 & 4 & 0 \\ x & x & 6\end{array}\right] \text{ and } R = PQP^{-1}$$
Then which of the following options is/are correct?
(A) There exists a real number $x$ such that $PQ = QP$
(B) $\det R = \det\left[\begin{array}{lll}2 & x & x \\ 0 & 4 & 0 \\ x & x & 5\end{array}\right] + 8$, for all $x \in \mathbb{R}$
(C) For $x = 0$, if $R\left[\begin{array}{l}1 \\ a \\ b\end{array}\right] = 6\left[\begin{array}{l}1 \\ a \\ b\end{array}\right]$, then $a + b = 5$
(D) For $x = 1$, there exists a unit vector $\alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$ for which $R\left[\begin{array}{l}\alpha \\ \beta \\ \gamma\end{array}\right] = \left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$

Let $A = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$ and $B = \left[ \begin{array} { l } \alpha \\ \beta \end{array} \right] \neq \left[ \begin{array} { l } 0 \\ 0 \end{array} \right]$ such that $A B = B$ and $a + d = 2021$, then the value of $a d - b c$ is equal to $\_\_\_\_$.

Let $A$ be a $2 \times 2$ real matrix and $I$ be the identity matrix of order 2 . If the roots of the equation $| A - x I | = 0$ be - 1 and 3 , then the sum of the diagonal elements of the matrix $A ^ { 2 }$ is $\_\_\_\_$ .

Let $A$ be a $2 \times 2$ symmetric matrix such that $A \left[ \begin{array} { l } 1 \\ 1 \end{array} \right] = \left[ \begin{array} { l } 3 \\ 7 \end{array} \right]$ and the determinant of $A$ be 1 . If $A ^ { - 1 } = \alpha A + \beta I$, where $I$ is an identity matrix of order $2 \times 2$, then $\alpha + \beta$ equals $\_\_\_\_$

Let $B = \left[ \begin{array} { l l } 1 & 3 \\ 1 & 5 \end{array} \right]$ and $A$ be a $2 \times 2$ matrix such that $A B ^ { - 1 } = A ^ { - 1 }$. If $B C B ^ { - 1 } = A$ and $C ^ { 4 } + \alpha C ^ { 2 } + \beta I = O$, then $2 \beta - \alpha$ is equal to
(1) 16
(2) 2
(3) 8
(4) 10