# 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. – Nasser Oct 10 '13 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 – Stefano M Oct 13 '13 at 22:32
• – Damien Oct 13 '13 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. – Stefano M Oct 14 '13 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. – Stefan Smith Oct 15 '13 at 1:32

## 2 Answers

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.

• In your opinion, how important are these uses you mention? That's my real question. The students in my class had a fair amount of matrix algebra background going into the class, and I had good reasons for teaching the matrix exponential that I'd rather not go into now. The book I'm complaining about covers much more than a semester's worth of material so I think the absence of the matrix exponential may be an important omission. – Stefan Smith Oct 11 '13 at 0:55
• There are many other topics that are more central to linear algebra than you could possibly cover in a one semester introductory course in linear algebra. This is a relatively unimportant topic that isn't really deserving of your time in such a course. – Brian Borchers Oct 11 '13 at 1:29
• I didn't want to get into this, but I am supposed to teach "matrix methods applied to systems of ODE", and the students have already learned how to compute eigenpairs and use them to solve systems of linear ODE with constant coefficients. I don't want to teach them something they've done already, so I decided to teach the matrix exponential and how to use it to solve such systems. Strang does this in his intro linear algebra book, and his book is shorter. And as I said before, the students have a fair amount of matrix algebra going into the course. – Stefan Smith Oct 11 '13 at 1:44
• In my experience there's really very little use for the matrix exponential in computational practice. In most practical situations you can diagonalize and then use the eigenvalues and eigenvectors without the matrix exponential. – Brian Borchers Oct 11 '13 at 2:16
• For me, very important. However, I have two special cases: The exponential of a "rotation vector" to a rotation matrix (which actually has a closed form expression); and the aforementioned case in control theory. Just because it's not common, doesn't mean it's not important to someone, somewhere :) – Damien Oct 13 '13 at 22:53

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.