We fix a matrix $A \in \mathcal{M}_d(\mathbb{R})$.
Show that, for every $n \in \mathbb{N}^*$ and every $R$ large enough, the matrix $$\frac{1}{2\pi} \int_0^{2\pi} \left(R\mathrm{e}^{\mathrm{i}\theta}\right)^n \left(R\mathrm{e}^{\mathrm{i}\theta} I_d - A\right)^{-1} \mathrm{~d}\theta$$ equals $A^{n-1}$.

We fix a matrix $A \in \mathcal{M}_d(\mathbb{R})$. We consider the characteristic polynomial $$\chi_A(X) = \det\left(A - X \cdot I_d\right) = \sum_{k=0}^d a_k X^k$$ Show that for $R$ large enough: $$\chi_A(A) = \frac{1}{2\pi} \int_0^{2\pi} \left(R\mathrm{e}^{\mathrm{i}\theta}\right) \chi_A\left(R\mathrm{e}^{\mathrm{i}\theta}\right) \left(R\mathrm{e}^{\mathrm{i}\theta} I_d - A\right)^{-1} \mathrm{~d}\theta$$

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.

Let $r$ and $s$ be two non-zero natural integers. Let $A \in \mathcal { M } _ { r } ( \mathbb { R } ) , B \in \mathcal { M } _ { r , s } ( \mathbb { R } ) , C \in \mathcal { M } _ { s , r } ( \mathbb { R } )$ and $D \in \mathcal { M } _ { s } ( \mathbb { R } )$. We further assume that $A$ is invertible. We consider the matrix $M \in \mathcal { M } _ { r + s } ( \mathbb { R } )$ having the following block form $$M = \left[ \begin{array} { l l } A & B \\ C & D \end{array} \right]$$ Find two matrices $U \in \mathcal { M } _ { r , s } ( \mathbb { R } )$ and $V \in \mathcal { M } _ { s } ( \mathbb { R } )$ such that $$M = \left[ \begin{array} { c c } A & 0 \\ C & I _ { s } \end{array} \right] \cdot \left[ \begin{array} { c c } I _ { r } & U \\ 0 & V \end{array} \right]$$ and deduce that $$\operatorname { det } ( M ) = \operatorname { det } ( A ) \cdot \operatorname { det } \left( D - C A ^ { - 1 } B \right)$$

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.

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$
For $t \in \mathbb{R}$ and $n \in \mathbb{N}^{*}$, we introduce the matrix $$F_{n}(t) = I_{3} + \sum_{k=1}^{n} \frac{t^{k} \mathcal{M}^{k}}{k!} \in \mathcal{M}_{n}(\mathbb{K})$$
Show that the sequence $\left(F_{n}(t)\right)_{n \in \mathbb{N}}$ converges to a limit denoted $F(t) \in \mathcal{M}_{3}(\mathbb{R})$. Express $F(t)$ in terms of $R(\theta)$, where $\theta$ depends on $m$ and $t$.

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 $$\mathcal { C } _ { M } = \left\{ \widehat { \mu } = \left( \mu _ { 1 } \geqslant \cdots \geqslant \mu _ { n } \right) , \text { such that } \operatorname { Sp } ( M ) \text { and } \widehat { \mu } \text { are interlaced } \right\} .$$

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$ and $F(t)$ denotes the limit of $F_n(t) = I_3 + \sum_{k=1}^n \frac{t^k \mathcal{M}^k}{k!}$ as defined in question 9.
Show that $F \in C^{1}(\mathbb{R}, \mathcal{M}_{3}(\mathbb{R}))$ and that for all $t \in \mathbb{R}$, we have $F^{\prime}(t) = F(t)\mathcal{M}$.

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 $( \lambda _ { 2 } , \lambda _ { 1 } )$.

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$
Show that for $X \in \mathbb{R}^{3}$, $\mathcal{M}X \cdot X = 0$. Geometrically interpret the application $\mathbb{R}^{3} \rightarrow \mathbb{R}^{3},\, X \mapsto (I_{3} + \mathcal{M})X$.

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 $s \in \{ 1 , \ldots , n \}$.

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$
Give a necessary and sufficient condition on $m$ for the sequence $\left((I_{3} + \mathcal{M})^{n}\right)_{n \in \mathbb{N}}$ to converge in $\mathcal{M}_{3}(\mathbb{R})$.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Is the matrix $M$ defined in question I.A.3 and the matrix $M'$ of size $n+1$ given by $$M' = \left(\begin{array}{ccccc} 1 & 1 & 0 & \ldots & 0 \\ 0 & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & & \ddots & \ddots & 1 \\ 0 & \ldots & \ldots & 0 & 1 \end{array}\right)$$ similar?

Let $u \in \mathcal{N}(E)$. Show that $\operatorname{tr} u^{k} = 0$ for every $k \in \mathbf{N}^{*}$.

We fix a basis $\mathbf{B}$ of $E$. We denote by $\mathcal{N}_{\mathbf{B}}$ the set of endomorphisms of $E$ whose matrix in $\mathbf{B}$ is strictly upper triangular. Justify that $\mathcal{N}_{\mathbf{B}}$ is a nilpotent vector subspace of $\mathcal{L}(E)$ and that its dimension equals $\frac{n(n-1)}{2}$.

For a triangular matrix $T = \left( \begin{array} { l l } \lambda & a \\ 0 & \mu \end{array} \right) \in \mathbf { M } _ { 2 } ( \mathbb { C } )$, explicitly compute the successive powers $T ^ { n }$ for $n$ a strictly positive integer.

Let $A \in \mathbf { M } _ { 2 } ( \mathbf { C } )$ be a matrix and let $\epsilon > 0$ be a real number.
(a) Show the existence of a real number $\alpha > 0$ such that for every non-negative integer $n$ the absolute values of the coefficients of $A ^ { n }$ are bounded by $\alpha ( \rho ( A ) + \epsilon ) ^ { n }$.
(b) Deduce the existence of a real number $\beta > 0$ such that for every non-negative integer $n$ and every $x \in \mathbb { C } ^ { 2 }$ we have $$\left\| A ^ { n } x \right\| \leqslant \beta ( \rho ( A ) + \epsilon ) ^ { n } \| x \|$$

Let $J \in M_{n}(\mathbb{C})$ be the matrix defined by
$$J = \left( \begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & 0 & 1 \\ 1 & 0 & \ldots & 0 & 0 \end{array} \right).$$
Deduce the value of the determinant
$$\left| \begin{array}{ccccc} a_{0} & a_{1} & \ldots & a_{n-2} & a_{n-1} \\ a_{n-1} & a_{0} & \ddots & & a_{n-2} \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ a_{2} & & \ddots & a_{0} & a_{1} \\ a_{1} & a_{2} & \cdots & a_{n-1} & a_{0} \end{array} \right|$$
where $a_{0}, \ldots, a_{n-1}$ are arbitrary complex numbers.

Let $A$ be a matrix belonging to $\mathbf{GL}_n$. Let $x$ be a nonzero real number. Express $\det(x I_n - A)$ in terms of $x$, $\det A$ and $\det\left(\frac{1}{x} I_n - A^{-1}\right)$.

6. Application: norm and spectral radius. a. Let $T \in \mathrm { M } _ { d } ( \mathbb { C } )$ be an upper triangular matrix. Show that for all $\varepsilon > 0$, there exists a constant $C$ such that for all $n$ we have $\left\| T ^ { n } \right\| \leqslant C ( \sigma ( T ) + \varepsilon ) ^ { n }$. b. Show that $\lim _ { n \rightarrow \infty } \left\| T ^ { n } \right\| ^ { 1 / n } = \sigma ( T )$. c. Let now $A \in \mathrm { M } _ { d } ( \mathbb { C } )$ be an arbitrary matrix. Show that $\lim _ { n \rightarrow \infty } \left\| A ^ { n } \right\| ^ { 1 / n } = \sigma ( A )$. d. Show the equivalence
$$A ^ { n } \underset { n \rightarrow \infty } { \longrightarrow } 0 \Leftrightarrow \sigma ( A ) < 1 .$$

Part 2: Linear recurrent sequences with constant coefficients

We consider in this part 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.

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.

Let $B \in \mathbf{M}_n$ be a diagonalizable matrix. We assume that the characteristic polynomial of $B$ is reciprocal or antireciprocal. Prove that $B$ is invertible and similar to its inverse.

Show that the matrix $B = \left(\begin{array}{cccc} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & \frac{1}{2} & 1 \\ 0 & 0 & 0 & \frac{1}{2} \end{array}\right)$ is not similar to its inverse (although its characteristic polynomial $\left(X-2\right)^2\left(X-\frac{1}{2}\right)^2$ is reciprocal). One may determine the eigenspaces of $B$ and $B^{-1}$ for the eigenvalue 2.

We say that a matrix $S \in \mathbf{M}_n$ is a symmetry matrix if $S^2 = I_n$. Prove that if $S_1$ and $S_2$ are two symmetry matrices, the product matrix $A = S_1 S_2$ is invertible and similar to its inverse.