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 $\varphi_A$. Thus we have $$\varphi_A(X) = (X - \lambda_1)^{m_1} \cdots (X - \lambda_\ell)^{m_\ell}$$ with $m = m_1 + \cdots + m_\ell$. Show that the map $$T : P \in \mathbb{C}_{m-1}[X] \mapsto \left(P(\lambda_1), P'(\lambda_1), \cdots, P^{(m_1-1)}(\lambda_1), \cdots, P(\lambda_\ell), P'(\lambda_\ell), \cdots, P^{(m_\ell-1)}(\lambda_\ell)\right) \in \mathbb{C}^m$$ is an isomorphism and deduce that there exists a unique polynomial $Q \in \mathbb{C}_{m-1}[X]$ such that $$\forall i \in \llbracket 1; \ell \rrbracket, \forall k \in \llbracket 0; m_i - 1 \rrbracket, Q^{(k)}(\lambda_i) = U^{(k)}(\lambda_i)$$
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 $\varphi_A$. Thus we have
$$\varphi_A(X) = (X - \lambda_1)^{m_1} \cdots (X - \lambda_\ell)^{m_\ell}$$
with $m = m_1 + \cdots + m_\ell$.
Show that the map
$$T : P \in \mathbb{C}_{m-1}[X] \mapsto \left(P(\lambda_1), P'(\lambda_1), \cdots, P^{(m_1-1)}(\lambda_1), \cdots, P(\lambda_\ell), P'(\lambda_\ell), \cdots, P^{(m_\ell-1)}(\lambda_\ell)\right) \in \mathbb{C}^m$$
is an isomorphism and deduce that there exists a unique polynomial $Q \in \mathbb{C}_{m-1}[X]$ such that
$$\forall i \in \llbracket 1; \ell \rrbracket, \forall k \in \llbracket 0; m_i - 1 \rrbracket, Q^{(k)}(\lambda_i) = U^{(k)}(\lambda_i)$$