Give examples of Hadamard matrices of order 1 and 2.

Show that if $H$ is a Hadamard matrix then any matrix obtained by multiplying a row or column of $H$ by $-1$ or by exchanging two rows or two columns of $H$ is still a Hadamard matrix.

Show that $\mathbb{M}_n(u) \neq \{0_n\}$.

Show that if $H$ is a Hadamard matrix of order $n$ then there exists a Hadamard matrix of order $n$ whose coefficients of the first row are all equal to 1. Deduce that if $n \geq 2$ then $n$ is even.

Show that the following three assertions are equivalent
(i) $R_u = +\infty$,
(ii) $\mathbb{M}_n(u) = \mathscr{M}_n(\mathbb{C})$,
(iii) $\mathbb{M}_n(u) \neq \emptyset$ and $\forall A \in \mathbb{M}_n(u), \forall B \in \mathbb{M}_n(u), A + B \in \mathbb{M}_n(u)$, and give an example of a sequence $u$ satisfying these three assertions and such that $u_k \neq 0$ for every $k \in \mathbb{N}$.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Show that the transpose of $M$ is the matrix of the linear map identity $$\begin{array}{ccc} \mathbb{R}_{m}[X] & \longrightarrow & \mathbb{R}_{m}[X] \\ P & \longmapsto & P \end{array}$$ in the bases $\left(1, X, \ldots, X^{m}\right)$ at the start and $\left(1, (X-1), \ldots, (X-1)^{m}\right)$ at the end.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Establish that $M$ is invertible and explicitly determine its inverse.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Deduce that for all $\left(u_{0}, \ldots, u_{m}\right), \left(v_{0}, \ldots, v_{m}\right) \in \mathbb{R}^{m+1}$, $$\text{if} \quad \forall k \leqslant m, \quad u_{k} = \sum_{\ell=0}^{k} \binom{k}{\ell} v_{\ell}, \quad \text{then} \quad \forall k \leqslant m, \quad v_{k} = \sum_{\ell=0}^{k} (-1)^{k-\ell} \binom{k}{\ell} u_{\ell}$$

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Justify that the families $\left(1, X, \ldots, X^m\right)$ and $\left(1, (X-1), \ldots, (X-1)^m\right)$ are bases of $\mathbb{R}_m[X]$.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Show that the transpose of $M$ is the matrix of the linear map identity $$\begin{array}{ccc} \mathbb{R}_m[X] & \longrightarrow & \mathbb{R}_m[X] \\ P & \longmapsto & P \end{array}$$ in the bases $\left(1, X, \ldots, X^m\right)$ at the start and $\left(1, (X-1), \ldots, (X-1)^m\right)$ at the end.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Establish that $M$ is invertible and explicitly determine its inverse.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Deduce that for all $\left(u_0, \ldots, u_m\right), \left(v_0, \ldots, v_m\right) \in \mathbb{R}^{m+1}$, $$\text{if} \quad \forall k \leqslant m, \quad u_k = \sum_{\ell=0}^{k} \binom{k}{\ell} v_\ell, \quad \text{then} \quad \forall k \leqslant m, \quad v_k = \sum_{\ell=0}^{k} (-1)^{k-\ell} \binom{k}{\ell} u_\ell.$$

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ Show that $\exp(\mathbb{C}[A]) \subset (\mathbb{C}[A])^*$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ For $a \in \mathbb{R}$, we define the application $$\begin{array}{ccc} Z_a : [0,1] & \longrightarrow & \mathbb{C} \\ t & \longmapsto & t + iat(1-t) \end{array}$$ Let $M_1$ and $M_2$ be two elements of $(\mathbb{C}[A])^*$. Show that there exists $a \in \mathbb{R}$ such that $$\forall t \in [0,1], \quad M(t) = Z_a(t) M_1 + \left(1 - Z_a(t)\right) M_2 \in (\mathbb{C}[A])^*.$$

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ For $a \in \mathbb{R}$, we define the application $$\begin{array}{ccc} Z_a : [0,1] & \longrightarrow & \mathbb{C} \\ t & \longmapsto & t + iat(1-t) \end{array}$$ Using the result of question 10b, deduce that $(\mathbb{C}[A])^*$ is path-connected.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ Show that there exists an open set $U$ of $\mathbb{C}[A]$ containing $0$ and an open set $V$ of $\mathbb{C}[A]$ containing the identity matrix $I_n$ such that the exponential function induces a continuous bijection from $U \subset \mathbb{C}[A]$ to $V$ whose inverse is a continuous function on $V$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ Using the result of question 11a, deduce that $\exp(\mathbb{C}[A])$ is an open set of $\mathbb{C}[A]$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ Show that $\exp(\mathbb{C}[A])$ is a closed set of $(\mathbb{C}[A])^*$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ We want to show that $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$. We suppose that $\exp(\mathbb{C}[A]) \neq (\mathbb{C}[A])^*$ and we fix $M_1, M_2 \in (\mathbb{C}[A])^*$ such that $M_1 \in \exp(\mathbb{C}[A])$ and $M_2 \notin \exp(\mathbb{C}[A])$.
Using the result of question 13a and the path-connectedness of $(\mathbb{C}[A])^*$, conclude that $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$.

Let $\alpha \in \mathbb{C}$ such that $|\alpha| < R_u$. Show that $$u(\alpha I_n) = U(\alpha) I_n$$

We assume in this question only that $n = 2$. Determine $u(A)$ in the following case: $$A = \begin{pmatrix} \alpha & \gamma \\ 0 & \beta \end{pmatrix}$$ where $\alpha, \beta$ and $\gamma$ are fixed real numbers with $\alpha \neq \beta$ and $\{\alpha, \beta\} \subset D_u$. We will express the coefficients of $u(A)$ in terms of $\alpha, \beta$ and $\gamma, U(\alpha)$ and $U(\beta)$.

Let $B \in \mathbb{M}_n(u)$.
(a) Show that there exists a polynomial $R \in \mathbb{C}[X]$ such that $$u(A) = R(A) \text{ and } u(B) = R(B).$$ (b) We assume that $AB \in \mathbb{M}_n(u)$ and $BA \in \mathbb{M}_n(u)$. Show that $$A\, u(BA) = u(AB)\, A$$

Let $v = (v_k)_{k \geqslant 0}$ be another sequence of $\mathbb{C}$ such that $A \in \mathbb{M}_n(v)$. We assume in this question only that the values $\lambda_1, \cdots, \lambda_\ell$ are real. Show that $$(u \star v)(A) = u(A)\, v(A)$$ (after having justified that $A \in \mathbb{M}_n(u \star v)$).

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. We assume throughout this part that $A$ is diagonalizable in $\mathscr{M}_n(\mathbb{C})$ and we denote by $\lambda_1, \cdots, \lambda_\ell$ its eigenvalues with $\lambda_i \neq \lambda_j$ if $i \neq j$. Show that $$\varphi_A(X) = (X - \lambda_1) \cdots (X - \lambda_\ell).$$

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. We assume throughout this part that $A$ is diagonalizable in $\mathscr{M}_n(\mathbb{C})$ and we denote by $\lambda_1, \cdots, \lambda_\ell$ its eigenvalues with $\lambda_i \neq \lambda_j$ if $i \neq j$. For every $k \in \llbracket 1; \ell \rrbracket$ we define the polynomial: $$Q_k^A(X) = \prod_{j=1, j\neq k}^{\ell} \frac{X - \lambda_j}{\lambda_k - \lambda_j}$$
(a) Show that $$u(A) = \sum_{k=1}^{\ell} U(\lambda_k) Q_k^A(A).$$
(b) Show that for every $k \in \llbracket 1; \ell \rrbracket$, $Q_k^A(A)$ is a projection whose image and kernel we will specify.
(c) Deduce that $$\sum_{k=1}^{\ell} Q_k^A(A) = I_n.$$