How important is the exponential of a matrix in computational science?

Question

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?

Answer 1

The matrix exponential has uses in the theory of systems of linear ordinary differential equations and in various applications in control theory, but these all lie beyond what would be discussed in an introductory course in linear algebra, and the topic is not needed to develop the rest of introductory linear algebra, so it really doesn't make any sense to introduce the topic in such a course.

Answer 2

It's an important concept when developing numerical methods for ordinary differential equations, at least as long as they are linear. Consider, for simplicity, a homogenous system of the form $\dot x(t)=Ax(t), x(0)=x_0$, then the solution is $x(t)=e^{At}x_0$. This is easily extended to the case $\dot x(t)=Ax(t)+f(t)$.

The point now is not that one may actually want to compute, say $x(2)=e^{2A}x_0$. The reason is that computing $e^{2A}$ is typically rather difficult. But, in the typical form of time stepping schemes, we may want to compute $x(t+\Delta t)=e^{A\Delta t}x(t)$ for some small $\Delta t$, and this is of course a much more manageable undertaking since $e^{\text{something small}}$ can readily be approximated. It is not difficult to show that many of the usual time stepping schemes can in fact be derived this way.