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 $\lambda$.