Show that the dimension of the vector space $E ^ { * }$ equals $n$.

Show that for $A , B \in M _ { n } ( \mathbb { C } ) , \| A B \| \leqslant \| A \| \| B \|$.

Show that $\omega ( x , x ) = 0$ for all $\omega \in \mathrm { A } ( E )$ and for all $x \in E$.

Let $A \in M _ { n } ( \mathbb { C } )$. We denote by $a _ { i , j }$ the coefficient of $A$ with row index $i$ and column index $j$. Show that $$\| A \| = \max _ { 1 \leqslant j \leqslant n } \left( \sum _ { i = 1 } ^ { n } \left| a _ { i , j } \right| \right)$$

Let $\omega \in \mathrm { A } ( E )$ and $\mathcal { B } = \left( b _ { 1 } , \ldots , b _ { n } \right)$ a basis of $E$.
(a) Show that there exists a unique matrix $M \in \mathcal { M } _ { n } ( \mathbb { R } )$, whose coefficients we shall specify, such that for all $( x , y ) \in E ^ { 2 } , \omega ( x , y ) = { } ^ { t } X M Y$ where $X , Y \in \mathbb { R } ^ { n }$ are the column matrices representing respectively $x$ and $y$ in the basis $\mathcal { B }$: $$X = \left( \begin{array} { c } x _ { 1 } \\ \vdots \\ x _ { n } \end{array} \right) , \quad Y = \left( \begin{array} { c } y _ { 1 } \\ \vdots \\ y _ { n } \end{array} \right) , \quad \begin{aligned} & x = x _ { 1 } b _ { 1 } + \cdots + x _ { n } b _ { n } \\ & y = y _ { 1 } b _ { 1 } + \cdots + y _ { n } b _ { n } . \end{aligned}$$ We then denote $M = \operatorname { Mat } _ { \mathcal { B } } ( \omega )$.
(b) Show that $M$ is antisymmetric, that is, ${ } ^ { t } M = - M$.
(c) Show that the vector space $\mathrm { A } ( E )$ is of dimension 1 when $E$ is of dimension 2.
(d) Show the equivalence between the three following statements.
$\left( \mathcal { E } _ { 1 } \right) : \quad \omega$ is a symplectic form on $E$.
$\left( \mathcal { E } _ { 2 } \right) : \quad$ For all $x \in E \backslash \{ 0 \}$, there exists $y \in E$ such that $\omega ( x , y ) \neq 0$.
$\left( \mathcal { E } _ { 3 } \right) : \quad \operatorname { Mat } _ { \mathcal { B } } ( \omega )$ is invertible.

We say that a sequence $\left( A ^ { ( k ) } \right) _ { k \in \mathbb { N } }$ of matrices in $M _ { n } ( \mathbb { C } )$ converges to a matrix $B \in M _ { n } ( \mathbb { C } )$ when $$\forall i \in \llbracket 1 , n \rrbracket , \forall j \in \llbracket 1 , n \rrbracket , \lim _ { k \rightarrow + \infty } \left( a _ { i , j } \right) ^ { ( k ) } = b _ { i , j }$$ Show that the sequence $( A ^ { ( k ) } )$ converges to $B$ if and only if $\lim _ { k \rightarrow + \infty } \left\| A ^ { ( k ) } - B \right\| = 0$.

Show that, if there exists a symplectic form on $E$, then $E$ is of even dimension.

We consider in this question a matrix $A \in M _ { n } ( \mathbb { C } )$ that is upper triangular, $$A = \left( \begin{array} { c c c c c } a _ { 1,1 } & a _ { 1,2 } & \ldots & \ldots & a _ { 1 , n } \\ 0 & a _ { 2,2 } & \ldots & \ldots & a _ { 2 , n } \\ \vdots & \ddots & \ddots & & \vdots \\ \vdots & & \ddots & \ddots & \vdots \\ 0 & \ldots & \ldots & 0 & a _ { n , n } \end{array} \right)$$ We assume that $$\forall i \in \llbracket 1 , n \rrbracket , \left| a _ { i , i } \right| < 1$$ For any real $b > 0$, we set $P _ { b } = \operatorname { diag } \left( 1 , b , b ^ { 2 } , \ldots , b ^ { n - 1 } \right) \in M _ { n } ( \mathbb { R } )$. a) Compute $P _ { b } ^ { - 1 } A P _ { b }$. What happens when $b$ tends to 0? b) Show that there exists $b > 0$ such that $$\left\| P _ { b } ^ { - 1 } A P _ { b } \right\| < 1$$ c) Deduce that the sequence $\left( A ^ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ converges to 0.

Determine the spectral radius of the following matrices $$\left( \begin{array} { l l } 0 & 0 \\ 0 & 1 \end{array} \right) , \quad \left( \begin{array} { l l } 0 & 0 \\ 1 & 0 \end{array} \right) , \quad \left( \begin{array} { l l } 1 & 0 \\ 0 & 0 \end{array} \right) , \quad \left( \begin{array} { c c } 0 & - 1 \\ 2 & 0 \end{array} \right) , \quad \left( \begin{array} { l l } 3 & 2 \\ 1 & 2 \end{array} \right)$$

We keep all the notations from Parts I and II and assume hypotheses (H1)–(H5). Let $\mathcal{B} = (z_1, \ldots, z_\ell)$, where $\ell = 2m-2$, be a basis of $G$. For any element $u$ of $G$, we denote by $U$ (capital letter) the column vector containing the coordinates of $u$ with respect to the basis $\mathcal{B}$. We denote by $A = [a_{i,j}]_{1 \leq i,j \leq \ell}$ and $B = [b_{i,j}]_{1 \leq i,j \leq \ell}$ the two square matrices whose coefficients are defined by $$\forall 1 \leq i,j \leq \ell, \quad a_{i,j} = (z_i \mid z_j), \quad b_{i,j} = (T(z_i) \mid T(z_j))$$
(a) Let $u, v \in G$. Show that $$(u \mid v) = {}^t U A V, \quad (T(u) \mid T(v)) = {}^t U B V$$ and deduce that $A$ and $B$ are invertible.
(b) Let $\lambda \in \mathbb{R}$. Show that an element $u \in G$ is a solution of $(\mathcal{P}_\lambda)$ if and only if $$(A - \lambda B) U = 0$$ Deduce that $(\mathcal{P}_\lambda)$ admits a non-zero solution if and only if $\operatorname{det}(A - \lambda B) = 0$.
(c) We define the function $\psi$ on $\mathbb{R}$ by $$\forall t \in \mathbb{R}, \psi(t) = \frac{\operatorname{det}(A - tB)}{\operatorname{det}(B)}$$ Show that this function $\psi$ is independent of the choice of basis $\mathcal{B}$.
(d) Justify why we can choose the basis $\mathcal{B}$ so that $B = I_\ell$. Deduce that $\psi$ is a polynomial function and specify its degree.
(e) Show that the polynomial $\psi$ is split over $\mathbb{R}[X]$ and that its roots are either simple or double.
(f) Show that $$\psi(X) = \frac{1}{S(w_1, T^{2m-1}(w_1)) S(w_2, T^{2m-1}(w_2))} Q_1(X) Q_2(X)$$ (justify why necessarily the denominator is non-zero). Deduce that $Q_1$ and $Q_2$ are split over $\mathbb{R}[X]$ and have simple roots.

Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$. By applying the previous results to the matrix ${ } ^ { t } A$, we obtain the existence of $w _ { 0 } \in \mathbb { R } ^ { n }$, whose all components are strictly positive, such that ${ } ^ { t } A w _ { 0 } = \rho ( A ) w _ { 0 }$. We set $$F = \left\{ x \in \mathbb { C } ^ { n } \mid { } ^ { t } x w _ { 0 } = 0 \right\}$$ a) Show that $F$ is a vector subspace of $\mathbb { C } ^ { n }$ stable by $\varphi _ { A }$, and that $$\mathbb { C } ^ { n } = F \oplus \mathbb { C } v _ { 0 }$$ b) Show that if $v$ is an eigenvector of $A$ associated with an eigenvalue $\mu \neq \rho ( A )$, then $v \in F$. Deduce property (iii): if $v$ is an eigenvector of $A$ whose all components are positive, then $v \in \operatorname { ker } ( A - \rho ( A ) I _ { n } )$.

Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$. We use the notation from question 17: $w_0$, $v_0$, $F = \left\{ x \in \mathbb { C } ^ { n } \mid { } ^ { t } x w _ { 0 } = 0 \right\}$, and $\mathbb { C } ^ { n } = F \oplus \mathbb { C } v _ { 0 }$. a) We denote by $\psi$ the endomorphism of $F$ defined as the restriction of $\varphi _ { A }$ to $F$. Show that all eigenvalues of $\psi$ have modulus strictly less than $\rho ( A )$. Deduce that $\rho ( A )$ is a simple root of the characteristic polynomial of $A$ and that $$\operatorname { ker } \left( A - \rho ( A ) I _ { n } \right) = \mathbb { C } v _ { 0 }$$ b) Show that if $x \in F , \lim _ { k \rightarrow + \infty } \frac { A ^ { k } x } { \rho ( A ) ^ { k } } = 0$. c) Let $x$ be a positive non-zero vector. Determine the limit of $\frac { A ^ { k } x } { \rho ( A ) ^ { k } }$ when $k$ tends to $+ \infty$.

What is the cardinality of $\mathcal{M}_{n}(\{-1,1\})$? Is this set a vector subspace of $\mathcal{M}_{n}(\mathbb{R})$?

Show that for every matrix $A$ in $\mathcal{M}_{n}(\{-1,1\})$, the set $S(A)$ is included in $\{-n^{2}, \ldots, n^{2}\}$. Show that the inclusion is strict (one may think of a parity argument), and show that $S(A)$ is a symmetric set, in the sense that an integer $k$ is in $S(A)$ if and only if $-k$ is in $S(A)$.

Let $A$ and $B$ be in $\mathcal{M}_{n}(\{-1,1\})$. Suppose that there exist diagonal matrices $C$ and $D$ containing only 1s and $-1$s on the diagonal, such that $B = CAD$. Show that $S(A) = S(B)$.

In this question only, we assume $n = 2$, and we denote $$I = \left(\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right) \quad \text{and} \quad J = \left(\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right)$$ Calculate $S(I)$ and $S(J)$, and deduce $S(A)$ for all $A \in \mathcal{M}_{2}(\{-1,1\})$.

For $A = (a_{i,j})_{1 \leqslant i,j \leqslant n} \in \mathcal{M}_{n}(\{-1,1\})$ and $Y = (y_{i})_{1 \leqslant i \leqslant n} \in \{-1,1\}^{n}$, we denote $$g_{A}(Y) = \max\left\{{}^{t}X A Y \mid X \in \{-1,1\}^{n}\right\}.$$
Show that the function $g_{A}$ can be rewritten as $$g_{A}(Y) = \sum_{i=1}^{n} \left|\sum_{j=1}^{n} a_{i,j} y_{j}\right|.$$

Show that $\operatorname{Toep}_{n}(\mathbb{C})$ is a vector subspace of $\mathcal{M}_{n}(\mathbb{C})$. Give a basis for it and specify its dimension.

Give the expression of $\langle A , B \rangle _ { F }$ as a function of the coefficients of $A$ and $B$.

Show that if two matrices $A$ and $B$ commute $(AB = BA)$ and if $P$ and $Q$ are two polynomials of $\mathbb{C}[X]$, then $P(A)$ and $Q(B)$ commute.

Let $u \in \mathbb { R } ^ { p }$. Show that $\| A u \| _ { 2 } \leqslant \| A \| _ { F } \| u \| _ { 2 }$.

Let $A = \left(\begin{array}{cc} a & b \\ c & a \end{array}\right)$ be a Toeplitz matrix of size $2 \times 2$, where $(a, b, c)$ are complex numbers. Give the characteristic polynomial of $A$.

Show that $\| A C \| _ { F } \leqslant \| A \| _ { F } \| C \| _ { F }$.

Let $A = \left(\begin{array}{cc} a & b \\ c & a \end{array}\right)$ be a Toeplitz matrix of size $2 \times 2$, where $(a, b, c)$ are complex numbers. Discuss, depending on the values of $(a, b, c)$, the diagonalizability of $A$.

We consider three strictly positive integers $n , p$ and $k$ such that $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ is non-empty. Let $A$ be a matrix of $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$.
Show that $k \leqslant \min ( n , p )$ and that for all $\lambda \in \mathbb { R } ^ { * } , \lambda A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$.