# Existence of Matrix Logarithm

## Theorem

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

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

- $(1): \quad T$ 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 $\lambda$.