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?)
The complex eigenvalues of a real matrix come in conjugate-complex pairs, with conjugate-complex pairs of eigenvectors. But the converse is true: If you associate to conjugate-complex pairs of eigenvectors any conjugate-complex pairs of eigenvalues, the result will be real because it is the sum of two terms which are conjugate-complex. You can take any diagonal matrix $\Psi$ with $\Psi_{ii} =\overline{\Psi_{jj}}$ if $\Lambda_{ii} = \overline{\Lambda_{jj}}$, and $Q\Psi Q^{-1}$ will be orthogonal.
