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 $A$.

Let $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$. We are given $b \in \mathbb { R } ^ { N }$ and we denote by $\tilde { x } \in \mathbb { R } ^ { N }$ the unique vector satisfying $A \tilde { x } = b$. We are given a vector $x _ { 0 } \in \mathbb { R } ^ { N }$, different from $\tilde { x }$, and we denote by $r _ { 0 } = b - A x _ { 0 }$. We set $H _ { 0 } = \{ 0 \}$ and for $k \geq 1$, $$H _ { k } = \left\{ P ( A ) r _ { 0 } \mid P \in \mathbb { R } [ X ] , \operatorname { deg } ( P ) \leq k - 1 \right\}$$ We denote by $m$ the smallest integer $k$ such that $H _ { k + 1 } = H _ { k }$. We denote by $d$ the number of distinct eigenvalues of $A$.
a) In the special case where $r _ { 0 }$ is an eigenvector of $A$, show that the integer $m$ is equal to 1.
b) In the general case, show that $m$ is less than or equal to $d$.
c) For any integer $n$ between 1 and $d$, construct an $x _ { 0 }$ such that the integer $m$ is equal to $n$.
d) Show that the set of $x _ { 0 }$ for which the dimension $m$ is exactly equal to $d$ is the complement of a finite union of sets of the form $\tilde { x } + E$, where $E$ is a vector space of dimension less than or equal to $N - 1$.

Let $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$. We are given $b \in \mathbb { R } ^ { N }$ and we denote by $\tilde { x } \in \mathbb { R } ^ { N }$ the unique vector satisfying $A \tilde { x } = b$. We are given a vector $x _ { 0 } \in \mathbb { R } ^ { N }$, different from $\tilde { x }$, and we denote by $r _ { 0 } = b - A x _ { 0 }$. We set $H _ { 0 } = \{ 0 \}$ and for $k \geq 1$, $$H _ { k } = \left\{ P ( A ) r _ { 0 } \mid P \in \mathbb { R } [ X ] , \operatorname { deg } ( P ) \leq k - 1 \right\}$$ We denote by $m$ the smallest integer $k$ such that $H _ { k + 1 } = H _ { k }$.
Show that there exists a polynomial $Q$ of degree $m$ such that $Q ( A ) e _ { 0 } = 0$, where $e _ { 0 } = x _ { 0 } - \tilde { x }$.

Let $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$. We are given $b \in \mathbb { R } ^ { N }$ and we denote by $\tilde { x } \in \mathbb { R } ^ { N }$ the unique vector satisfying $A \tilde { x } = b$. We are given a vector $x _ { 0 } \in \mathbb { R } ^ { N }$, different from $\tilde { x }$, and we denote by $r _ { 0 } = b - A x _ { 0 }$. We denote by $m$ the smallest integer $k$ such that $H _ { k + 1 } = H _ { k }$, and $Q$ is the polynomial of degree $m$ from question 9 such that $Q ( A ) e _ { 0 } = 0$ where $e _ { 0 } = x _ { 0 } - \tilde { x }$.
Show that the polynomial $Q$ satisfies $Q ( 0 ) \neq 0$.

By reasoning on the multiplicity of the roots of $\chi_\sigma$ and $\chi_\tau$, show that if $P_\sigma$ and $P_\tau$ are similar, then, for all $q \in \llbracket 1, n \rrbracket$,
$$\sum_{\substack{\ell=1 \\ q \mid \ell}}^{n} c_\ell(\sigma) = \sum_{\substack{\ell=1 \\ q \mid \ell}}^{n} c_\ell(\tau)$$
(We sum over the values of $\ell$ that are multiples of $q$ and belong to $\llbracket 1, n \rrbracket$.)

Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ and $v$ be two endomorphisms of $E$ such that, for all $k \in \mathbb{N}, \operatorname{Tr}\left(u^k\right) = \operatorname{Tr}\left(v^k\right)$. Show that $u$ and $v$ have the same characteristic polynomial.

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Deduce that $\rho(A) > 0$ then show that $\rho\left(\frac{A}{\rho(A)}\right) = 1$.

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. We consider an eigenvalue $\lambda \in \mathbb{C}$ of $A$ with modulus 1 and $x$ an eigenvector associated with $\lambda$. Show that $|x| \leqslant A|x|$.

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. We consider an eigenvalue $\lambda \in \mathbb{C}$ of $A$ with modulus 1 and $x$ an eigenvector associated with $\lambda$. Conclude (that 1 is an eigenvalue of $A$).

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. Show that $A$ admits a strictly positive eigenvector associated with the eigenvalue 1.

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. Show that 1 is the only eigenvalue of $A$ with modulus 1.
One may admit without proof that if $z_1, z_2, \ldots, z_k$ are non-zero complex numbers such that $|z_1 + \cdots + z_k| = |z_1| + \cdots + |z_k|$, then $\forall j \in \llbracket 1, k \rrbracket, \exists \lambda_j \in \mathbb{R}^+$ such that $z_j = \lambda_j z_1$.

Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. Show that $\dim\left(\ker\left(A - I_n\right)\right) = 1$.

By combining the results of sub-parts II.B and II.C, justify that we have proved Proposition 1: If $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix, then $\rho(A)$ is a dominant eigenvalue of $A$. The associated eigenspace $\ker\left(A - \rho(A) I_n\right)$ is one-dimensional and is spanned by a strictly positive eigenvector.

Throughout part III, $N$ is a fixed non-zero natural integer and $(X_n)_{n \in \mathbb{N}}$ is a homogeneous Markov chain on $\llbracket 0, N \rrbracket$ with transition matrix $Q$ where $q_{i,j} = P(X_{n+1} = j \mid X_n = i) > 0$. We define $a_{i,j}(t) = q_{i,j} \mathrm{e}^{jt}$ and $A(t) = \left(a_{i,j}(t)\right)_{0 \leqslant i,j \leqslant N} \in \mathcal{M}_{N+1}(\mathbb{R})$. Justify that $A(t)$ possesses a dominant eigenvalue $\gamma(t) > 0$.

Let $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Show that if $\lambda$ is an eigenvalue of $M$, then $1/\lambda$ is also an eigenvalue of $M$. Give an eigenvector associated with it.

Let $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Let $\lambda \in \mathrm{sp}_{\mathbb{R}}(M)$ and $p = \dim E_{\lambda}$. Let $(X_{1}, \ldots, X_{p})$ be a basis of $E_{\lambda}$. Show that $(J_{n} X_{1}, \ldots, J_{n} X_{p})$ is a basis of $E_{1/\lambda}$ and that $$\dim(E_{\lambda}) = \dim(E_{1/\lambda}).$$

Let $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. In this question $\lambda = 1$. Show that $E_{1}$ has even dimension and that there exists a basis of $E_{1}$ that is orthonormal and of the form $(X_{1}, \ldots, X_{p}, J_{n} X_{1}, \ldots, J_{n} X_{p})$ where $2p$ is the dimension of $E_{1}$.

Let $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. What about $E_{-1}$? (i.e., does $E_{-1}$ have even dimension and does there exist an orthonormal basis of $E_{-1}$ of the form $(X_{1}, \ldots, X_{p}, J_{n} X_{1}, \ldots, J_{n} X_{p})$?)

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Let $m$ be the linear map canonically associated with $M$. Show the equality $\mathrm{sp}_{\mathbb{R}}(M) = \emptyset$.

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Let $X$ denote an eigenvector of $M^{2}$ of norm 1 associated with a certain eigenvalue $\lambda$. Show that $MX$, $J_{n} X$ and $J_{n} MX$ are eigenvectors of $M^{2}$ and give the eigenvalues associated with each of these vectors.

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Show that all eigenvalues of $M^{2}$ are strictly negative.

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 $d \in \mathbb{N} ^ { * }$ 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 $t ( z ^ { i + j } )$. For $X , Y \in \mathbb { R } ^ { d }$, we set $B ( X , Y ) = X ^ { T } S Y$.
15a. Show that there exists a basis $(e _ { 1 } , \ldots , e _ { d })$ of $\mathbb { R } ^ { d }$ with $e _ { i } \in \mathbb { Q } ^ { d }$ for all $i$ and $B ( e _ { i } , e _ { j } ) = 0$ for $i \neq j$.
15b. Deduce that there exist $P \in \mathrm { GL } _ { d } ( \mathbb { Q } )$ and $q _ { 1 } , \ldots , q _ { d } \in \mathbb { Q }$, $q _ { i } > 0$, such that: $$S = P ^ { T } \cdot \operatorname { Diag } \left( q _ { 1 } , \ldots , q _ { d } \right) \cdot P$$

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 $d \in \mathbb{N} ^ { * }$ 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 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 $M$.

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 $d \in \mathbb{N} ^ { * }$ 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 $t ( z ^ { i + j } )$. For $X , Y \in \mathbb { R } ^ { d }$, we set $B ( X , Y ) = X ^ { T } S Y$.
We set: $$M = \left( \begin{array} { c c c c c } 0 & 0 & \cdots & 0 & a _ { 0 } \\ 1 & 0 & \ddots & 0 & a _ { 1 } \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & a _ { d - 2 } \\ 0 & \cdots & 0 & 1 & a _ { d - 1 } \end{array} \right)$$
There exist $P \in \mathrm { GL } _ { d } ( \mathbb { Q } )$ and $q _ { 1 } , \ldots , q _ { d } \in \mathbb { Q }$, $q _ { i } > 0$, such that $S = P ^ { T } \cdot \operatorname { Diag } \left( q _ { 1 } , \ldots , q _ { d } \right) \cdot P$.
17a. Verify that the matrix $S M$ is symmetric.
17b. Deduce that the matrix $R M R ^ { - 1 }$ is symmetric where $R = \operatorname { Diag } \left( \sqrt { q _ { 1 } } , \ldots , \sqrt { q _ { d } } \right) \cdot P$.