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 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 $\mathbb{C}[A]$.

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1. We denote by $\Omega_n$ the set of symmetric positive matrices of order $n$ such that $M \cdot \mathbf{e} = 0$. We denote by $K$ the application from $\Omega_n$ to $\mathcal{M}_n(\mathbb{R})$ which associates to a matrix $A$ $$K(A) = \mathbf{e} \cdot \mathbf{a}^T + \mathbf{a} \cdot \mathbf{e}^T - 2A$$ where $\mathbf{a}$ is the column matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are the diagonal coefficients of $A$.
Show that for every matrix $A$ of $\Omega_n$ we have $K(A) \in \Delta_n$.

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Show that if $A \in \mathscr{M}_n(\mathbb{R})$ then $\varphi_A$ has real coefficients (that is, $\varphi_A \in \mathbb{R}[X]$).

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 $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Show that $\exp(\mathbb{C}[A])$ is a closed set of $(\mathbb{C}[A])^*$.

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, $\Delta_n$ the set of EDM of order $n$, and $\Omega_n$ the set of symmetric positive matrices of order $n$ such that $M \cdot \mathbf{e} = 0$. The application $T: \Delta_n \to \mathcal{M}_n(\mathbb{R})$ associates to $D$ the matrix $T(D) = -\frac{1}{2}PDP$, and the application $K: \Omega_n \to \mathcal{M}_n(\mathbb{R})$ associates to $A$ the matrix $K(A) = \mathbf{e}\cdot\mathbf{a}^T + \mathbf{a}\cdot\mathbf{e}^T - 2A$ where $\mathbf{a}$ is the column of diagonal coefficients of $A$.
Show that the applications $T: \Delta_n \rightarrow \Omega_n$ and $K: \Omega_n \rightarrow \Delta_n$ satisfy: $$T \circ K = \operatorname{Id}_{\Omega_n}.$$

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

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, and $\Delta_n$ the set of EDM of order $n$.
Specify the sum $\displaystyle\sum_{i=1}^{n} \lambda_i$ of the eigenvalues of an EDM of order $n$.

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 $U(\beta)$.

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, and $\Delta_n$ the set of EDM of order $n$.
Let $D$ be a non-zero EDM of order $n$. Show that for all $x \in \operatorname{Vect}(\mathbf{e})^\perp$, we have $$x^T D x \leqslant 0.$$

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 $Z \in \mathscr{M}_{d}(\mathbb{R})$ be an invertible matrix with $\mathrm{S} = Z^{T}Z$, and let $(\lambda_{i})_{1 \leqslant i \leqslant d}$ be the decreasing family of strictly positive eigenvalues of $S$ with associated orthonormal basis $(u_{1}, \ldots, u_{d})$. We consider $v_{i} = \frac{1}{\sqrt{\lambda_{i}}} Z u_{i}$ for all $1 \leqslant i \leqslant d$.

• [(a)] Show that $(v_{1}, \ldots, v_{d})$ is an orthonormal basis of $\mathbb{R}^{d}$.
• [(b)] Verify that if $U = (u_{1} | \ldots | u_{d})$, $V = (v_{1} | \ldots | v_{d})$ and $D = \operatorname{Diag}(\sqrt{\lambda_{1}}, \ldots, \sqrt{\lambda_{d}})$ then $Z = VDU^{T}$.

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 $A \in \mathbb{M}_n(u \star v)$).

Express in the form $Z = VDU^{T}$ (specifying your choices of $U$, $V$ and $D$) for the matrices $$Z_{1} = \begin{pmatrix} 0 & 1 \\ 2 & 0 \end{pmatrix} \quad \text{and} \quad Z_{2} = \begin{pmatrix} 0 & 1 \\ -2 & 0 \end{pmatrix}.$$

Let $Z \in \mathscr{M}_{d}(\mathbb{R})$ be an invertible matrix with singular value decomposition $Z = VDU^{T}$ where $U = (u_{1}|\ldots|u_{d})$, $V = (v_{1}|\ldots|v_{d})$ are orthogonal matrices and $D = \operatorname{Diag}(\sqrt{\lambda_{1}}, \ldots, \sqrt{\lambda_{d}})$. We assume that $\operatorname{det}(Z) > 0$.

• [(a)] Show that if $R \in \mathrm{SO}_{d}(\mathbb{R})$ then $V^{T}RU \in \mathrm{SO}_{d}(\mathbb{R})$.
• [(b)] Show that $$\sup_{R \in \mathrm{SO}_{d}(\mathbb{R})} \langle Z, R \rangle = \sup_{R \in \mathrm{SO}_{d}(\mathbb{R})} \langle D, R \rangle$$

Let $H$ be a Hadamard matrix of order $n$ with first row constant equal to 1. Let $\lambda_1, \ldots, \lambda_n$ be real numbers such that $$\lambda_1 > 0 \geq \lambda_2 \geq \ldots \geq \lambda_n$$ and $$\sum_{i=1}^{n} \lambda_i = 0.$$ We denote by $U$ the matrix $\frac{1}{\sqrt{n}} H$ and $\Lambda$ the diagonal matrix whose diagonal coefficients are the $\lambda_i$. We finally denote by $D = U^T \Lambda U$.
Show that $D$ is EDM.

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 $$\sum_{k=1}^{\ell} Q_k^A(A) = I_n.$$

Give the value of $\delta(\boldsymbol{x}, \boldsymbol{y})$ as a function of $V_{n}(\boldsymbol{x})$, $V_{n}(\boldsymbol{y})$ and the singular values of $Z(\boldsymbol{x}, \boldsymbol{y})$ in the case where $\operatorname{det}(Z(\boldsymbol{x}, \boldsymbol{y})) > 0$.

Let $H$ be a Hadamard matrix of order $n$ with first row constant equal to 1. Let $\lambda_1, \ldots, \lambda_n$ be real numbers such that $$\lambda_1 > 0 \geq \lambda_2 \geq \ldots \geq \lambda_n$$ and $$\sum_{i=1}^{n} \lambda_i = 0.$$ We denote by $U$ the matrix $\frac{1}{\sqrt{n}} H$ and $\Lambda$ the diagonal matrix whose diagonal coefficients are the $\lambda_i$. We finally denote by $D = U^T \Lambda U$.
Give a Euclidean distance matrix of order 4 such that its spectrum is $\{5, -1, -2, -2\}$.

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 $i \neq j$. Let $B \in \mathscr{M}_n(\mathbb{C})$ be an invertible matrix. Show that $$u(BAB^{-1}) = B\, u(A)\, B^{-1}.$$

We consider $R \in \mathrm{O}_{d}(\mathbb{R})$.

• [(a)] Show that if $\lambda$ is an eigenvalue of $R$ then $\lambda \in \{+1, -1\}$.
• [(b)] Show that $\operatorname{det}(R + I) = \operatorname{det}(R) \operatorname{det}(I + R^{T})$.
• [(c)] Deduce that if $\operatorname{det}(R) = -1$ then $\operatorname{det}(R + I) = 0$.

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 $i \neq j$. Let $D \in \mathscr{M}_n(\mathbb{C})$ be a diagonal matrix and $S \in \mathscr{M}_n(\mathbb{C})$ be an invertible matrix such that $A = SDS^{-1}$.
(a) Show that $u(D)$ is diagonal and that $$\forall i \in \llbracket 1; n \rrbracket, [u(D)]_{i,i} = U([D]_{i,i}).$$ (b) Deduce an expression for $u(A)$.

We consider $R \in \mathrm{O}_{d}(\mathbb{R})$ with $\operatorname{det}(R) = -1$.

• [(a)] Show that there exists an orthonormal basis $(u_{1}, \ldots, u_{d})$ of $\mathbb{R}^{d}$ such that $Ru_{d} = -u_{d}$ and $u_{d}^{T} Rx = 0$ for all $x \in E_{1}$ where $E_{1} = \operatorname{Vect}(u_{1}, \ldots, u_{d-1})$.
• [(b)] Deduce that $R(E_{1}) \subset E_{1}$ and then that $R(E_{1}) = E_{1}$.