= The matrix exponential is not surjective
{synonym}

https://en.wikipedia.org/wiki/Logarithm_of_a_matrix\#Existence mentions it always exists for all <invertible> <complex> matrices. But the <real> condition is more complicated. Notable counter example: -1 cannot be reached by any real $e^{tk}$.

The <Lie algebra exponential covering problem> can be seen as a generalized version of this problem, because
* <Lie algebra> of <GL(n)> is just the entire <M_n>
* we can immediately exclude non-invertible matrices from being the result of the exponential, because $e^{tM}$ has inverse $e^{-tM}$, so we already know that non-invertible matrices are not reachable


Discussion on "Existence of the matrix logarithm" by Ciro Santilli 32