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 $u(A)$.
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 $u(A)$.