Exhibit a matrix $M \in S _ { 2 } ( \mathbb { Q } )$ for which $\sqrt { 2 }$ is an eigenvalue.

We denote by $\mathscr { R }$ the set of totally real numbers and we admit that there exists a function $t : \mathscr { R } \rightarrow \mathbb { Q }$ satisfying the following two properties: (i) for $x , y \in \mathscr { R }$ and $\lambda , \mu \in \mathbb { Q }$, we have $t ( \lambda x + \mu y ) = \lambda t ( x ) + \mu t ( y )$ (ii) for $x$ totally positive, we have $t ( x ) \geqslant 0$ and the equality is strict if $x \neq 0$.
We consider a non-zero totally real number $z$. By definition, there exists a monic polynomial $Z ( X ) \in \mathbb { Q } [ X ]$ that annihilates $z$. We write $Z ( X )$ in the form: $$Z ( X ) = X ^ { d } - \left( a _ { d - 1 } X ^ { d - 1 } + \cdots + a _ { 1 } X + a _ { 0 } \right)$$ with $d \in \mathbb { N } ^ { \star }$ and $a _ { i } \in \mathbb { Q }$ for all $i \in \{ 0 , \ldots , d - 1 \}$. We further assume that $Z ( X )$ is chosen so that $d$ is minimal among the degrees of monic polynomials $P ( X ) \in \mathbb { Q } [ X ]$ such that $P ( z ) = 0$. We consider the matrix $S$ of size $d \times d$ whose coefficient $( i , j ) , 1 \leqslant i , j \leqslant d$, equals $t \left( z ^ { i + j } \right)$. For $X , Y \in \mathbb { R } ^ { d }$, we set $B ( X , Y ) = X ^ { T } S Y$.
13a. Show that $B ( X , X ) > 0$ for $X \in \mathbb { Q } ^ { d } , X \neq 0$.
13b. Deduce that the matrix $S$ is invertible.

We consider a non-zero totally real number $z$. We write $Z ( X ) = X ^ { d } - \left( a _ { d - 1 } X ^ { d - 1 } + \cdots + a _ { 1 } X + a _ { 0 } \right)$ with $d$ minimal and $a_i \in \mathbb{Q}$. We set: $$M = \left( \begin{array} { c c c c c } 0 & 0 & \cdots & 0 & a _ { 0 } \\ 1 & 0 & \ddots & 0 & a _ { 1 } \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & a _ { d - 2 } \\ 0 & \cdots & 0 & 1 & a _ { d - 1 } \end{array} \right)$$ Compute the characteristic polynomial of $M$.

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ Show that $x_0$ is an eigenvector of $u$.

4. Let $C_m \in \mathbb{R}[Y]$ be the characteristic polynomial of $M_m$. a. Verify that $C_1 = Y$, $C_2 = Y^2 - 2$ and $$C_m = Y C_{m-1} - m(m-1) C_{m-2}, \quad m \geq 3$$ b. Calculate the determinant of $M_m$. c. Prove that, if $e_m$ denotes the integer part of $m/2$, $$C_m = \sum_{k=0}^{e_m} (-1)^k c_{m,k} Y^{m-2k}$$ with $$c_{m,0} = 1; \quad c_{m,k} = \sum_{(a_1, \ldots, a_k) \in J_k(m)} a_1(a_1+1) a_2(a_2+1) \cdots a_k(a_k+1), \quad 1 \leq k \leq e_m;$$ where $J_k(m)$ denotes the set of $k$-tuples of integers from $\{1, \ldots, m-1\}$ such that $a_i + 2 \leq a_{i+1}$ for $1 \leq i \leq k-1$.

Let $H$ be the matrix of the inner product $\phi(P,Q) = \int_0^1 P(t)Q(t)\,\mathrm{d}t$ in the canonical basis of $\mathbb{R}_{n-1}[X]$, with general term $h_{i,j} = \phi(X^i, X^j)$. Show that $H$ belongs to $\mathcal{S}_n(\mathbb{R})$ and that its eigenvalues are strictly positive.

Show that, if $A$ is nilpotent, that is, if there exists $p \in \mathbb{N}^\star$ such that $A^p = 0_n$, then the spectral radius of $A$ is zero.

We denote $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. Show that $\rho(A) \leqslant \max_{U \in C} \left| U^\top A U \right|$.

Let $A \in \mathcal{S}_n(\mathbb{R})$ and $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. Prove that $\rho(A) = \max_{U \in C} \left| U^\top A U \right|$.

Let $A \in \mathcal{S}_n(\mathbb{R})$ and $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. We further assume that the eigenvalues of $A$ are all positive. Show then that $\rho(A) = \max_{U \in C} \left( U^\top A U \right)$.

We assume that $\Sigma_Y$ satisfies $$\forall i \in \llbracket 1, n \rrbracket, \quad \sigma_{i,i} = \sigma^2 \quad \text{and} \quad \forall (i,j) \in \llbracket 1, n \rrbracket^2, \quad i \neq j \Longrightarrow \sigma_{i,j} = \sigma^2 \gamma$$ where $\sigma$ and $\gamma$ are two strictly positive real numbers. We denote by $J \in \mathcal{M}_n(\mathbb{R})$ the matrix whose coefficients are all equal to 1.
Determine the eigenvalues of $J$ and the dimension of each associated eigenspace. Also determine an eigenvector associated with its eigenvalue of maximal modulus.

We assume that $\Sigma_Y$ satisfies $$\forall i \in \llbracket 1, n \rrbracket, \quad \sigma_{i,i} = \sigma^2 \quad \text{and} \quad \forall (i,j) \in \llbracket 1, n \rrbracket^2, \quad i \neq j \Longrightarrow \sigma_{i,j} = \sigma^2 \gamma$$ where $\sigma$ and $\gamma$ are two strictly positive real numbers. We denote by $J \in \mathcal{M}_n(\mathbb{R})$ the matrix whose coefficients are all equal to 1, and $U_0$ is a unit vector such that the variance of $Z = U_0^\top Y$ is maximal.
Calculate the percentage of total variance represented by $Z$, that is, the ratio $\dfrac{\mathbb{V}(Z)}{\mathbb{V}_T(Y)}$.

Prove that a matrix $M$ in $\mathcal { M } _ { 2 } ( \mathbb { R } )$ is nilpotent if and only if $\operatorname { det } ( M ) = \operatorname { tr } ( M ) = 0$.

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$ and $\varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM)$. Using the fact that $A^{-1}M$ is similar to a real symmetric matrix, deduce that $\varphi_\alpha''(0) \geq 0$.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\alpha \in ] - \frac { 1 } { n } , + \infty \left[ \backslash \{ 0 \} \right.$. With $\varphi _ { \alpha } ^ { \prime \prime } ( 0 ) = \operatorname { det } ^ { - \alpha } ( A ) \left( \alpha \operatorname { Tr } ^ { 2 } \left( A ^ { - 1 } M \right) + \operatorname { Tr } \left( \left( A ^ { - 1 } M \right) ^ { 2 } \right) \right)$, deduce that $\varphi _ { \alpha } ^ { \prime \prime } ( 0 ) \geq 0$.

We consider a Markov kernel $K$. We assume that 1 is a simple eigenvalue of $K$. We assume that there exists a probability $\pi \in \mathscr{M}_{1,N}(\mathbf{R})$ such that:
(a) For all $j \in \llbracket 1;N \rrbracket$, $\pi[j] \neq 0$.
(b) $\forall (i,j) \in \llbracket 1;N \rrbracket^2$, $\pi[i] K[i,j] = K[j,i] \pi[j]$; we say that $K$ is $\pi$-reversible. Show that $\pi K = \pi$.

We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, and the endomorphism $u : X \mapsto (I_N - K)X$ with $q_u(X) = (u(X) \mid X)$. Show that for all $X \in E$, $$q_u(X) = \frac{1}{2} \sum_{i=1}^{N} \sum_{j=1}^{N} (X[i] - X[j])^2 K[i,j] \pi[i]$$ What can be said about the eigenvalues of $u$?

Let $J \in M_{n}(\mathbb{C})$ be the matrix defined by
$$J = \left( \begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & 0 & 1 \\ 1 & 0 & \ldots & 0 & 0 \end{array} \right).$$
Calculate the eigenvalues of $J$.

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$ and we set $\chi = \operatorname { det } \left( X I _ { n } - M \right) \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$.
Show that $M _ { \mid t = 0 }$ admits a real eigenvalue.

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$ and we set $\chi = \operatorname { det } \left( X I _ { n } - M \right) \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$. By Theorem 1, there exists $\rho _ { 1 } \in \mathbb { R } _ { + } ^ { * }$, $\rho _ { 1 } \leqslant \rho$ such that $\chi$ factors in the form $\chi = F G$ with $F \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathbb { R } _ { d } [ X ] \right)$ and $G \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathbb { R } _ { n - d } [ X ] \right)$ and $F _ { \mid t = 0 } = ( X - \lambda ) ^ { d }$.
Only in this question, we assume that $d = n$. Show that there exists a symmetric matrix $M _ { 0 } \in \mathscr { D } _ { \rho _ { 1 } } \left( S _ { n } ( \mathbb { R } ) \right)$ such that $M = \lambda I _ { n } + t M _ { 0 }$ for all $t \in U _ { \rho _ { 1 } }$.

Let $\mathscr{P}$ be the set of row vectors of size $d$ with non-negative coefficients whose coordinate sum equals 1: $$\mathscr{P} = \left\{ u \in \mathscr{M}_{1,d}\left(\mathbb{R}_{+}\right) : \sum_{j=1}^{d} u_j = 1 \right\}.$$ We consider a square matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ such that for all $i \in \{1,\ldots,d\}$, $$\sum_{j=1}^{d} P_{i,j} = 1$$ We further assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$P_{i,j} \geqslant c\nu_j.$$
Show that there exists a unique element $\mu$ of $\mathscr{P}$ such that $\mu P = \mu$.

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, and $\Delta_n$ the set of EDM of order $n$.
Specify the sum $\displaystyle\sum_{i=1}^{n} \lambda_i$ of the eigenvalues of an EDM of order $n$.

Determine the characteristic polynomial of the double star with $d _ { 1 } + d _ { 2 } + 2$ vertices, consisting respectively of two disjoint stars with $d _ { 1 }$ and $d _ { 2 }$ branches, to which an additional edge has been added connecting the two centers of the two stars. What is the rank of the adjacency matrix of this double star?

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Show that if $A \in \mathscr{M}_n(\mathbb{R})$ then $\varphi_A$ has real coefficients (that is, $\varphi_A \in \mathbb{R}[X]$).

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 ${}^t Z Y = 0$.