This might be a basic question, nonetheless I cannot give a proof.

Given an orthogonal matrix $A$ with eigendecomposition $A = Q \Lambda Q^{-1}$. Given also a diagonal real matrix $\Phi$ and matrix power definition $[\Lambda^\Phi]_{ii} := \Lambda_{ii}^{\Phi_{ii}}$.

Why is $Q \Lambda^\Phi Q^{-1}$ orthogonal? (Unitarity is simple, but why is it real?)