Is the solution to a system of linear ordinary differential equations, the exponential function is just a 1-dimensional subcase.

Note that more generally, the matrix exponential can be defined on any ring.

The matrix exponential is of particular interest in the study of Lie groups, because in the case of the Lie algebra of a matrix Lie group, it provides the correct exponential map.

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