The Set Tour, Part 7: Matrices

I feel a bit odd about this week’s guest in the Set Tour. I’ve been mostly concentrating on sets that get used as the domains or ranges for functions a lot. The ones I want to talk about here don’t tend to serve the role of domain or range. But they are used a great deal in some interesting functions. So I loosen my rule about what to talk about.

R^{m x n} and C^{m x n}

R^{m x n} might explain itself by this point. If it doesn’t, then this may help: the “x” here is the multiplication symbol. “m” and “n” are positive whole numbers. They might be the same number; they might be different. So, are we done here?

Maybe not quite. I was fibbing a little when I said “x” was the multiplication symbol. R^{2 x 3} is not a longer way of saying R⁶, an ordered collection of six real-valued numbers. The x does represent a kind of product, though. What we mean by R^{2 x 3} is an ordered collection, two rows by three columns, of real-valued numbers. Say the “x” here aloud as “by” and you’re pronouncing it correctly.

What we get is called a “matrix”. If we put into it only real-valued numbers, it’s a “real matrix”, or a “matrix of reals”. Sometimes mathematical terminology isn’t so hard to follow. Just as with vectors, Rⁿ, it matters just how the numbers are organized. R^{2 x 3} means something completely different from what R^{3 x 2} means. And swapping which positions the numbers in the matrix occupy changes what matrix we have, as you might expect.

You can add together matrices, exactly as you can add together vectors. The same rules even apply. You can only add together two matrices of the same size. They have to have the same number of rows and the same number of columns. You add them by adding together the numbers in the corresponding slots. It’s exactly what you would do if you went in without preconceptions.

You can also multiply a matrix by a single number. We called this scalar multiplication back when we were working with vectors. With matrices, we call this scalar multiplication. If it strikes you that we could see vectors as a kind of matrix, yes, we can. Sometimes that’s wise. We can see a vector as a matrix in the set R^{1 x n} or as one in the set R^{n x 1}, depending on just what we mean to do.

It’s trickier to multiply two matrices together. As with vectors multiplying the numbers in corresponding positions together doesn’t give us anything. What we do instead is a time-consuming but not actually hard process. But according to its rules, something in R^{m x n} we can multiply by something in R^{n x k}. “k” is another whole number. The second thing has to have exactly as many rows as the first thing has columns. What we get is a matrix in R^{m x k}.

I grant you maybe didn’t see that coming. Also a potential complication: if you can multiply something in R^{m x n} by something in R^{n x k}, can you multiply the thing in R^{n x k} by the thing in R^{m x n}? … No, not unless k and m are the same number. Even if they are, you can’t count on getting the same product. Matrices are weird things this way. They’re also gateways to weirder things. But it is a productive weirdness, and I’ll explain why in a few paragraphs.

A matrix is a way of organizing terms. Those terms can be anything. Real matrices are surely the most common kind of matrix, at least in mathematical usage. Next in common use would be complex-valued matrices, much like how we get complex-valued vectors. These are written C^{m x n}. A complex-valued matrix is different from a real-valued matrix. The terms inside the matrix can be complex-valued numbers, instead of real-valued numbers. Again, sometimes, these mathematical terms aren’t so tricky.

I’ve heard occasionally of people organizing matrices of other sets. The notation is similar. If you’re building a matrix of “m” rows and “n” columns out of the things you find inside a set we’ll call H, then you write that as H^{m x n}. I’m not saying you should do this, just that if you need to, that’s how to tell people what you’re doing.

Now. We don’t really have a lot of functions that use matrices as domains, and I can think of fewer that use matrices as ranges. There are a couple of valuable ones, ones so valuable they get special names like “eigenvalue” and “eigenvector”. (Don’t worry about what those are.) They take in R^{m x n} or C^{m x n} and return a set of real- or complex-valued numbers, or real- or complex-valued vectors. Not even those, actually. Eigenvectors and eigenfunctions are only meaningful if there are exactly as many rows as columns. That is, for R^{m x m} and C^{m x m}. These are known as “square” matrices, just as you might guess if you were shaken awake and ordered to say what you guessed a “square matrix” might be.

They’re important functions. There are some other important functions, with names like “rank” and “condition number” and the like. But they’re not many. I believe they’re not even thought of as functions, any more than we think of “the length of a vector” as primarily a function. They’re just properties of these matrices, that’s all.

So why are they worth knowing? Besides the joy that comes of knowing something, I mean?

Here’s one answer, and the one that I find most compelling. There is cultural bias in this: I come from an applications-heavy mathematical heritage. We like differential equations, which study how stuff changes in time and in space. It’s very easy to go from differential equations to ordered sets of equations. The first equation may describe how the position of particle 1 changes in time. It might describe how the velocity of the fluid moving past point 1 changes in time. It might describe how the temperature measured by sensor 1 changes as it moves. It doesn’t matter. We get a set of these equations together and we have a majestic set of differential equations.

Now, the dirty little secret of differential equations: we can’t solve them. Most interesting physical phenomena are nonlinear. Linear stuff is easy. Small change 1 has effect A; small change 2 has effect B. If we make small change 1 and small change 2 together, this has effect A plus B. Nonlinear stuff, though … it just doesn’t work. Small change 1 has effect A; small change 2 has effect B. Small change 1 and small change 2 together has effect … A plus B plus some weird A times B thing plus some effect C that nobody saw coming and then C does something with A and B and now maybe we’d best hide.

There are some nonlinear differential equations we can solve. Those are the result of heroic work and brilliant insights. Compared to all the things we would like to solve there’s not many of them. Methods to solve nonlinear differential equations are as precious as ways to slay krakens.

But here’s what we can do. What we usually like to know about in systems are equilibriums. Those are the conditions in which the system stops changing. Those are interesting. We can usually find those points by boring but not conceptually challenging calculations. If we can’t, we can declare x₀ represents the equilibrium. If we still care, we leave calculating its actual values to the interested reader or hungry grad student.

But what’s really interesting is: what happens if we’re near but not exactly at the equilibrium? Sometimes, we stay near it. Think of pushing a swing. However good a push you give, it’s going to settle back to the boring old equilibrium of dangling straight down. Sometimes, we go racing away from it. Think of trying to balance a pencil on its tip; if we did this perfectly it would stay balanced. It never does. We’re never perfect, or there’s some wind or somebody walks by and the perfect balance is foiled. It falls down and doesn’t bounce back up. Sometimes, whether it it stays near or goes away depends on what way it’s away from the equilibrium.

And now we finally get back to matrices. Suppose we are starting out near an equilibrium. We can, usually, approximate the differential equations that describe what will happen. The approximation may only be good if we’re just a tiny bit away from the equilibrium, but that might be all we really want to know. That approximation will be some linear differential equations. (If they’re not, then we’re just wasting our time.) And that system of linear differential equations we can describe using matrices.

If we can write what we are interested in as a set of linear differential equations, then we have won. We can use the many powerful tools of matrix arithmetic — linear algebra, specifically — to tell us everything we want to know about the system. We can say whether a small push away from the equilibrium stays small, or whether it grows, or whether it depends. We can say how fast the small push shrinks, or grows (for a while). We can say how the system will change, approximately.

This is what I love in matrices. It’s not everything there is to them. But it’s enough to make matrices important to me.