Let $n \in \mathbf{N}$ and $P : \mathbf{R} \rightarrow \mathbf{R}$ defined by $P(x) = \sum_{k=0}^{n} a_k x^k$ where $a_k \geq 0$ for all $k \in \llbracket 0, n \rrbracket$ a polynomial with non-negative coefficients.
(a) Verify that $P[A] = \sum_{k=0}^{n} a_k A^{(k)}$ for all matrices $A \in \mathcal{M}_p(\mathbf{R})$.
(b) Show that if $A \in \operatorname{Sym}^+(p)$ then $P[A] \in \operatorname{Sym}^+(p)$.