The Petz-Renyi relative entropy is defined for positive semi-definite operators $\rho$ and $\sigma$ on a finite-dimensional Hilbert space as \begin{equation} D_\alpha(\rho||\sigma) = \frac{1}{\alpha - 1}\log \frac{\hbox{Tr}[\rho^{\alpha} \sigma^{1-\alpha}]}{\hbox{Tr}[\rho]} \end{equation} where $\alpha \in (0,1)\cup (1,\infty)$. (I found this as Eq. (3) while reading this paper.)
Since $\rho$ and $\sigma$ are positive semi-definite they can be diagonalized. Let $\sigma = UDU^\dagger$ with $U$ unitary and $D$ diagonal. Should I understand that $\sigma^{1-\alpha} = U D^{1-\alpha} U^\dagger$ in the relative entropy equation?
(Motivation for the question: My initial concern upon reading the equation above was how it applied to pure states. If $\alpha = 2$, $\sigma^{1-\alpha} = \sigma ^{-1}$. But $\sigma$ may not be full rank and may not have an inverse.)