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

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?

• I've only seen it in when doing control in state space. As in en.wikipedia.org/wiki/State-transition_matrix . You can't really do state space analysis without it. But I do not know if this can be considered computational or not. You can decide on that. Oct 10, 2013 at 22:59
• There is a 1978 classic paper, "Nineteen Dubious Ways to Compute the Exponential of a Matrix" by van Loan and Moler you should read. And also this recent blog post by Moler: A balancing act for the matrix exponential Oct 13, 2013 at 22:32
• Oct 13, 2013 at 22:47
• I think that the 1978 paper is not free, but the sequel, freely available as pointed out by Damien and by Cleve Moler on his blog, should be sufficient. Oct 14, 2013 at 18:53
• @StefanoM : Thanks for encouraging me to check out Moler's blog (I just did for the first time now). I think his link to the sequel paper is the same one Google gave me, and since that link is on Moler's blog, I'm going to assume it is legal. It appears Moler has spent a fair amount of time on this problem, so the problem is probably important. Oct 15, 2013 at 1:32

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.