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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.
Video 1. How (and why) to raise e to the power of a matrix by 3Blue1Brown (2021) Source.