Existence of Matrix Logarithm

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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 $\lambda$.