We denote by $\lambda_1, \cdots, \lambda_\ell$ the eigenvalues of $A$, with $\lambda_i \neq \lambda_j$ if $i \neq j$. We denote by $m_1 \geqslant 1, \cdots, m_\ell \geqslant 1$ the multiplicities of $\lambda_1, \cdots, \lambda_\ell$ respectively as roots of $\varphi_A$. We set $u(A) = Q(A)$ where $Q$ is the unique polynomial in $\mathbb{C}_{m-1}[X]$ satisfying $\forall i \in \llbracket 1; \ell \rrbracket, \forall k \in \llbracket 0; m_i - 1 \rrbracket, Q^{(k)}(\lambda_i) = U^{(k)}(\lambda_i)$. Let $P \in \mathbb{C}[X]$. Show that $u(A) = P(A)$ if and only if $$\forall i \in \llbracket 1; \ell \rrbracket, \forall k \in \llbracket 0; m_i - 1 \rrbracket, P^{(k)}(\lambda_i) = U^{(k)}(\lambda_i)$$
We denote by $\lambda_1, \cdots, \lambda_\ell$ the eigenvalues of $A$, with $\lambda_i \neq \lambda_j$ if $i \neq j$. We denote by $m_1 \geqslant 1, \cdots, m_\ell \geqslant 1$ the multiplicities of $\lambda_1, \cdots, \lambda_\ell$ respectively as roots of $\varphi_A$. We set $u(A) = Q(A)$ where $Q$ is the unique polynomial in $\mathbb{C}_{m-1}[X]$ satisfying $\forall i \in \llbracket 1; \ell \rrbracket, \forall k \in \llbracket 0; m_i - 1 \rrbracket, Q^{(k)}(\lambda_i) = U^{(k)}(\lambda_i)$.
Let $P \in \mathbb{C}[X]$. Show that $u(A) = P(A)$ if and only if
$$\forall i \in \llbracket 1; \ell \rrbracket, \forall k \in \llbracket 0; m_i - 1 \rrbracket, P^{(k)}(\lambda_i) = U^{(k)}(\lambda_i)$$