In quantum information theory, calculating the log of a density operator is essential for things like the Von Neumann entropy or the entropy of entanglement. Unfortunately, this topic is considered a bit blasé and I haven't been able to find a source where the calculation is rigorously derived.
Here's what I know so far:
By the eigendecomposition rule, a density matrix $\rho$ can be written as:
$$\rho = VDV^{-1}$$
Where $D$ is a diagonal matrix with eigenvalues of $\rho$ as elements, and where $V$ is a square matrix whose columns are corresponding eigenvectors of $\rho$.
Apparently, taking the log of $\rho$ is equivalent to taking the log of each element in $D$ -- that is:
$$\log(\rho) = V\log(D)V^{-1}$$
Clearly for this to be true it must also be true that
$$ \log(VDV^{-1}) = V\log(D)V^{-1}$$
But It's not obvious to me why this should be the case. I have the feeling that I need to make use of the Taylor series expansion of $\log(\rho)$ but I'm not so confident in my ability to do this correctly. Any help is greatly appreciated.