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?)

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

Given an orthogonal matrix $A$ with eigendecomposition $A = Q \Lambda Q^{-1}$ with only non-real eigenvalues. Given also a diagonal real matrix $\Phi$ with $\Phi_{ii} = \Phi_{jj}$ if $\Lambda_{ii} = \overline{\Lambda_{jj}}$. The following matrix power is defined as $[\Lambda^\Phi]_{ii} := \Lambda_{ii}^{\Phi_{ii}}$.

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