Is this true: "$P$ is diagonal if and only if there exists a diagonal $\log P$"? This is the matrix logarithm. I just want to make sure of my reasoning.
If $\log P$ is diagonal, then $P$ is diagonal because \begin{align*} P &= \exp(\log P) \\ &= I + \log P + \frac{1}{2}(\log P)^2 + \cdots. \end{align*}
On the other hand if $P$ is diagonal, then $\log P$ is diagonal because $$ \log P = \left[ \begin{array}{ccc} \log P_{11} & 0 & 0\\ 0 & \ddots & 0\\ 0 & 0 & \log P_{nn} \end{array} \right]. $$
On the second one, is the logarithm unique? What happens if I change $P$ to orthogonal? What about invertible?
If $\log P$ is diagonal, clearly $P$ must also be diagonal, because $P=\exp(\log P)$ is a power series in $\log P$.
But then each diagonal entry of $P$ is the exponential of its counterpart $\log P$. Therefore, $P$ has a diagonal matrix logarithm if and only if $P$ is a diagonal matrix with an entrywise nonzero diagonal.