Existence of Matrix Logarithm

Theorem
Let $T$ be a square matrix of order $n$.

Then there exists a real matrix $S$ such that $e^S = T$ :


 * $(1): \quad T$ is not a singular matrix

and:
 * $(2): \quad $for every negative eigenvalue $\lambda$ of $T$ and for every positive integer $k$, the Jordan form of $T$ has an even number of $k \times k$ blocks associated with $\lambda$.