CS people:
The title is the question, as I will explain. As everyone reading this probably knows, if $A$ is a square matrix of real or complex numbers, then $e^A$, or $\exp(A)$, is the matrix of the same dimensions as $A$ defined by the obvious power series (the only thing that might not be obvious is that $A^0 =I$, but one would have to be very contrary not to accept this definition). One could use fields, rings, or other algebraic structures (such as quaterions) rather than $\mathbb{R}$ or $\mathbb{C}$, but I would guess that the vast majority of people who work with matrices use real or complex matrices.
This spring I taught a linear algebra class using a textbook that did not define the exponential of a matrix or mention it at all. I taught the topic anyway, using another text as a resource. The textbook lacking the matrix exponential seems like a popular, mainstream textbook, it contains a lot of material at least as difficult as the matrix exponential, and the author probably knows a lot more about linear algebra than I do.
The total absence of the exponential of a matrix seemed to me like a major flaw of this book. In pure mathematics, any definition that is nontrivial and does not lead to a contradiction is important to someone. I know that some matrix Lie groups are important in quantum mechanics. So my question is, how important is the matrix exponential in computational science?