A matrix has a real logarithm if it has a positive spectrum.

Question

The title is a proposition I read in my notes that's left with no proof. Where can I read one?

Answer 1

This is almost trivial. Consider a $k\times k$ Jordan block $J_k(\lambda)$, where $\lambda>0$. Then $J_k(\lambda)$ is the Jordan form of $e^{J_k(\log\lambda)}$ and hence $J_k(\lambda)=Pe^{J_k(\log\lambda)}P^{-1}=e^{PJ_k(\log\lambda)P^{-1}}$ for some real invertible $P$. That is, $PJ_k(\log\lambda)P^{-1}$ is a real logarithm of $J_k(\lambda)$.

If you really need a reference, see

Walter J. Culver, On the existence and uniqueness of the real logarithm of a matrix, Proceedings of the American Mathematical Society, 17(5): 1146-1151, 1966,

although the main focus of this paper is placed on some more interesting cases than yours. You may also see Nick Higham's Functions of Matrices: Theory and Computation or Horn and Johnson's Topics in Matrix Analysis. There is a good chance that these reference books would contain the theorems you need.