I am having a hard time coming up with an example such that an invertible $ T \in L(R^n, R^n)$ such that there is no $S \in L(R^n,R^n)$ with $e^S = T.$
2026-03-25 23:42:21.1774482141
Invertible $ T \in L(R^n, R^n) $ such that there is no $S \in L(R^n,R^n)$ with $e^S = T.$
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Any matrix $T$ with negative determinant will do, since you have $$ \det (e^S) = e^{\text{tr}(S)}>0$$ This however is only because you require $S$ to be a real metrix. In the space of complex matrices you can always calculate $S=\ln T$ for invertible $T$.