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