Let $M \in S _ { n + 1 } ( \mathbb { R } )$ be a symmetric matrix. We write $M$ in block form $$M = \left[ \begin{array} { l l } A & y \\ { } ^ { t } y & a \end{array} \right]$$ with $a \in \mathbb { R } , y \in \mathcal { M } _ { n , 1 } ( \mathbb { R } )$ and $A \in S _ { n } ( \mathbb { R } )$.
(a) If the spectrum of $A$ is $\operatorname { Sp } ( A ) = \left( \mu _ { 1 } \geqslant \cdots \geqslant \mu _ { n } \right)$, show that there exist $U \in O _ { n + 1 } ( \mathbb { R } )$ and $z \in \mathcal { M } _ { n , 1 } ( \mathbb { R } )$ such that $$U M ^ { t } U = \left[ \begin{array} { c c } \Delta \left( \mu _ { 1 } , \ldots , \mu _ { n } \right) & z \\ t _ { z } & a \end{array} \right]$$ (b) Deduce that there exist non-negative real numbers $\alpha _ { j }$ (for $j = 1 , \ldots , n$ ) such that $$\chi _ { M } = ( X - a ) Q _ { 0 } - \sum _ { j = 1 } ^ { n } \alpha _ { j } \frac { Q _ { 0 } } { \left( X - \mu _ { j } \right) } , \quad \text { where } \quad Q _ { 0 } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) .$$ (c) Show that $\operatorname { Sp } ( M )$ and $\operatorname { Sp } ( A )$ are interlaced.

Let $(a_0, \ldots, a_{n-1}) \in \mathbb{C}^n$. Let $\lambda$ be a complex number. By discussing in $\mathbb{C}^n$ the system $C(a_0, \ldots, a_{n-1})X = \lambda X$, show that $\lambda$ is an eigenvalue of $C(a_0, \ldots, a_{n-1})$ if and only if $\lambda$ is a root of a polynomial of $\mathbb{C}[X]$ to be specified.

We say that a complex number $z$ is totally real (resp. totally positive) if there exists a non-zero polynomial $P ( X )$ with rational coefficients such that: (i) $z$ is a root of $P$, and (ii) all roots of $P$ are in $\mathbb { R }$ (resp. in $\mathbb { R } _ { + }$).
Let $M$ be a symmetric matrix with coefficients in $\mathbb { Q }$. Show that the eigenvalues of $M$ are totally real.

We say that a complex number $z$ is totally real (resp. totally positive) if there exists a non-zero polynomial $P ( X )$ with rational coefficients such that: (i) $z$ is a root of $P$, and (ii) all roots of $P$ are in $\mathbb { R }$ (resp. in $\mathbb { R } _ { + }$).
Let $M$ be a symmetric matrix with coefficients in $\mathbb { Q }$. Show that the eigenvalues of $M$ are totally real.

We consider two distinct complex numbers $\alpha$ and $\beta$. We assume that a matrix $A \in \mathcal{M}_3(\mathbf{C})$ has $\alpha$ as a simple eigenvalue and $\beta$ as a double eigenvalue.
$\mathbf{18}$ ▷ Show that $A$ is similar to a matrix of the form $$T = \left( \begin{array}{ccc} \alpha & 0 & 0 \\ 0 & \beta & a \\ 0 & 0 & \beta \end{array} \right)$$ where $a$ is a certain complex number. Compute $T^n$ for $n$ a natural integer, then $e^{tT}$ for $t$ real. Deduce from this a necessary and sufficient condition on $\alpha$ and $\beta$ for $\lim_{t \rightarrow +\infty} e^{tA} = 0_3$.

We assume in this part that all roots of $p$ are stable and have multiplicity 1 and we denote by $h = Xp'$ (where $p'$ is the derivative polynomial of $p$) and $h_0$ the reciprocal polynomial of $h$. We recall that, according to question 3, there exists a real number $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.
We admit that the map defined on $S_n(\mathbf{R})$ with values in $\mathbf{R}^n$ which associates to a symmetric matrix the $n$-tuple of its real eigenvalues counted with their multiplicities, arranged in decreasing order, is continuous.
Deduce that $\sigma(p) = n - 1 - \pi(J(p'))$.

8. Calculate the characteristic polynomial of the matrix $A$ (you may reason by induction on $d$).