Let $B \in M_3(\mathbb{R})$ be antisymmetric.
Show that $E(B)$ exists and is a rotation matrix. Specify the value of its unoriented angle as a function of $\|B\|_2$.

Let $D \in M_p(\mathbb{K})$ be a diagonal matrix.
Show that $E(D)$ exists and that $E(D) \in GL_p(\mathbb{C})$.

Let $D \in M_p(\mathbb{K})$ be a diagonal matrix.
Show that there exists a polynomial $Q \in \mathbb{C}[X]$ such that $Q(D) = E(D)$.

Let $A \in M_p(\mathbb{K})$ be a diagonalizable matrix.
Show that $E(A)$ exists.

Let $A \in M_p(\mathbb{K})$ be a diagonalizable matrix.
Show that $\det(E(A)) = e^{\operatorname{tr}(A)}$.

Let $A \in M_p(\mathbb{K})$ be a diagonalizable matrix. Let $x \in \mathbb{C}$.
Show that $E\left(xI_p + A\right)$ exists and that $$E\left(xI_p + A\right) = e^x E(A)$$

Let $A, B \in M_p(\mathbb{K})$ be two diagonalizable matrices. We assume that $A$ and $B$ commute.
Show that there exists $P \in GL_p(\mathbb{C})$ such that $P^{-1}AP$ and $P^{-1}BP$ are diagonal.
(We shall study the restrictions of $u_B$ to the eigenspaces of $u_A$.)

Let $A, B \in M_p(\mathbb{K})$ be two diagonalizable matrices. We assume that $A$ and $B$ commute.
Deduce that $E(A + B)$ exists and that $E(A + B) = E(A)E(B) = E(B)E(A)$.

Let $A \in M_p(\mathbb{C})$ and $k \in \mathbb{N}^*$ such that $A^k = 0$ and $A^{k-1} \neq 0$ (we say that $A$ is nilpotent of order $k$).
Show that, for every integer $j$ such that $1 \leqslant j \leqslant k$, $\operatorname{Ker} A^{j-1}$ is strictly included in $\operatorname{Ker} A^j$.

Let $A \in M_p(\mathbb{C})$ and $k \in \mathbb{N}^*$ such that $A^k = 0$ and $A^{k-1} \neq 0$ (we say that $A$ is nilpotent of order $k$).
Deduce that $k \leqslant p$.

Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$.
Show that $E(A)$ exists. Propose a Maple or Mathematica procedure taking as input a strictly upper triangular matrix $A$ and returning the value of $E(A)$.

Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$.
Show that there exists a polynomial $Q \in \mathbb{C}[X]$ such that $Q(A) = E(A)$.

Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$. Let $B \in M_p(\mathbb{C})$. We assume that $A$ and $B$ commute and that $E(B)$ exists.
We admit that, for every integer $i$ between 1 and $p$, $$\lim_{n \rightarrow \infty} \left(I_p + \frac{1}{n}B\right)^n = \lim_{n \rightarrow \infty} \left(I_p + \frac{1}{n}B\right)^{n-i}$$
Show that $E(A + B)$ exists and that $E(A + B) = E(A)E(B)$.

Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$. Let $x \in \mathbb{C}$.
Show that $E\left(xI_p + A\right)$ exists and that $E\left(xI_p + A\right) = e^x E(A)$.

Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$.
Show that $E(A) - I_p$ is nilpotent.

Let $A \in M_p(\mathbb{C})$ and $n \in \mathbb{N}^*$. We denote $$P_n(X) = \left(1 + \frac{X}{n}\right)^n \in \mathbb{C}[X]$$ and $\chi_A$ the characteristic polynomial of $A$ defined by $$\chi_A(X) = \det\left(A - XI_p\right)$$
Show that there exists a unique pair $\left(Q_n, R_n\right) \in \mathbb{C}[X] \times \mathbb{C}_{p-1}[X]$ such that $$P_n = Q_n \chi_A + R_n$$

Let $A \in M_p(\mathbb{C})$ and $n \in \mathbb{N}^*$. We denote $$P_n(X) = \left(1 + \frac{X}{n}\right)^n \in \mathbb{C}[X]$$ and $\chi_A$ the characteristic polynomial of $A$, with the Euclidean division $P_n = Q_n \chi_A + R_n$.
Show that $E(A)$ exists if and only if $\lim_{n \rightarrow \infty} R_n(A)$ exists.

Let $A \in M_p(\mathbb{C})$. Let $k \in \mathbb{N}^*$ and $\lambda_1, \lambda_2, \ldots, \lambda_k$ be the roots of $\chi_A$ pairwise distinct, whose respective multiplicities we denote by $n_1, n_2, \ldots, n_k$.
For every integer $q$ between 1 and $p$, we denote by $J_q$ the matrix of $M_q(\mathbb{C})$ whose coefficients are all zero except those located just above the diagonal which equal 1.
Show that, for every $x \in \mathbb{C}$, for every integer $q$ between 1 and $p$, the family $\left\{\left(xI_q + J_q\right)^i,\ 0 \leqslant i \leqslant q-1\right\}$ is free.

Let $A \in M_p(\mathbb{C})$. Let $k \in \mathbb{N}^*$ and $\lambda_1, \lambda_2, \ldots, \lambda_k$ be the roots of $\chi_A$ pairwise distinct, with respective multiplicities $n_1, n_2, \ldots, n_k$. For every integer $q$ between 1 and $p$, $J_q$ denotes the matrix of $M_q(\mathbb{C})$ whose coefficients are all zero except those just above the diagonal which equal 1.
Let $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ be the block diagonal matrix defined by $$B = \begin{pmatrix} \lambda_1 I_{n_1} + J_{n_1} & 0 & \ldots & 0 \\ 0 & \lambda_2 I_{n_2} + J_{n_2} & \cdots & 0 \\ \vdots & \ddots & \ddots & 0 \\ 0 & \ldots & 0 & \lambda_k I_{n_k} + J_{n_k} \end{pmatrix}$$
Show that $\chi_B = \chi_A$.

Let $A \in M_p(\mathbb{C})$ and $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ as defined in V.A.4. Let $i$ be an integer $\geqslant 1$.
Show that $$B^i = \begin{pmatrix} \left(\lambda_1 I_{n_1} + J_{n_1}\right)^i & 0 & \cdots & 0 \\ 0 & \left(\lambda_2 I_{n_2} + J_{n_2}\right)^i & \cdots & 0 \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & \left(\lambda_k I_{n_k} + J_{n_k}\right)^i \end{pmatrix}$$

Let $A \in M_p(\mathbb{C})$ and $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ as defined in V.A.4. Let $P$ be a non-zero annihilating polynomial of the matrix $B$.
a) Show that the degree of $P$ is $\geqslant p$.
b) Deduce that the family $\left\{B^i,\ 0 \leqslant i \leqslant p-1\right\}$ is free.

Let $A \in M_p(\mathbb{C})$ and $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ as defined in V.A.4. We denote $P_n(X) = \left(1 + \frac{X}{n}\right)^n$.
Show that $\lim_{n \rightarrow \infty} P_n(B)$ exists.

Let $A \in M_p(\mathbb{C})$ and $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ as defined in V.A.4.
Deduce that $E(A)$ exists.

Calculate the product ${}^t\left(R_{p,q}(\theta)\right) R_{p,q}(\theta)$. What property of $R_{p,q}(\theta)$ is recognized?

We are given $S \in \mathbf{S}_n$ and $R \in \mathbf{O}_n$. Verify that ${}^t R S R$ is symmetric and that it is similar to $S$.