Are power series in a normal matrix themselves normal? Are (convergent) power series in a normal matrix themselves normal? I have looked around for this result, and not found it. How might we prove it?

Yes, because a matrix is normal if and only if it is unitarily diagonalizable, we can simultaneously diagonalize a matrix and analytic functions of that matrix, given that said function is analytic in a domain containing the spectrum of the matrix. More concretely, if $X = S \Lambda S^{-1}$, where $\Lambda$ is the diagonal matrix of eigenvalues, and if
$$ f(z) = \sum_{k=0}^{\infty} a_n z^n $$
is a function that is holomorphic in some domain $\Omega$ containing all the eigenvalues of $X$, then
$$ f(X) = \sum_{k=0}^{\infty} a_n (S\Lambda S^{-1})^n = \sum_{k=0}^{\infty} a_n S \Lambda^n S^{-1} = S f(\Lambda) S^{-1}$$
which makes $f(X)$ a matrix that is unitarily diagonalizable.
