Let S be a self adjoint operator defined on $\mathbb{C}^{n}$, with mutually distinct eigenvalues $\lambda_{1},...\lambda_{m}$, $m \leq n$. Then according to the spectral theorem one has: $$S = \sum_{j=1}^m \lambda_{j} \pi_{j}$$ Where $\pi_{j}$ is the orthogonal projection on the eigenspace of $\lambda_{j}$. Find polynomials $p$ and $q$ such that: $p(S) = 0$ and $q(S) = \text{arctan}(e^{S})$!
I think that $p$ has to be the characteristic polynomial of $S$, so $p(t) = (t-\lambda_{1})\cdots(t-\lambda_{m})$ right? But what about $q$? Can someone help me with this please?