I'm trying to calculate the Chebychev polynomial $T_n(A)$ of a matrix $A$. Since $A$ is symmetric, my idea is to diagonalise $A$ like $X=U^\dagger A U$, then use the explicite expressions mentioned here:
$ T_n(x) = \begin{cases} \cos(n\arccos(x)), & \ x \in [-1,1] \\ \cosh(n \, \mathrm{arccosh}(x)), & \ x \ge 1 \\ (-1)^n \cosh(n \, \mathrm{arccosh}(-x)), & \ x \le -1 \\ \end{cases} \,\! $
where $x$ runs over all eigenvalues of $A$ and then transform back like $UT_n(X)U^\dagger$. While this work?