I'm self-studying a bit of linear algebra and got stuck with the following problem:
Let $\tau$ be a normal operator on a complex finite-dimensional inner product space V or a self-adjoint operator on a real finite-dimensional inner-product space. I want to show that $\tau^*=p(\tau)$, where $p(x)\in\mathbb{C}[x]$.
This looks to me like using minimal or characteristic polynomials, but I don't manage to prove it. Then my idea was to use the orthogonal spectral resolution of $\tau$, i.e. $\tau=\lambda_{1}\rho_{1}$+...+$\lambda_{k}\rho_{k}$, where $im(\rho_{i})=E_{\lambda_{i}}$ ($\lambda_{i}$ eigenvalues, $E_{\lambda_{i}}$ corresponding Eigenspace), and $ker(\rho_{i})=\bigodot_{j\neq i} E_{\lambda_{j}}$. Then using properties of the adjoint I find. $\tau^*=\overline{\lambda_{1}}\rho_{1}$+...+$\overline{\lambda_{k}}\rho_{k}$.And then? Thank you for any suggestion.