PROF. EISENBUD: So I'm going to tell you about something called matrix factorizations. But the story starts with something I've studied in middle school, and you probably did, too. Do you remember, we—you thought about factoring? So you could factor things like x² − 4, and—if you remember—that's (x - 2) (2 is the square root of 4, that's the point) times (x + 2). And then maybe a little later you were allowed to replace 4 by the square of another variable. So if you have x² − y², the same formula works because you have a square root of y². It's (x − y)(x + y). The advantage of that is you can plug in any numbers you like for x and y, and this will always be true. You could also ask about square roots of polynomials like that. So if you had x² + 2xy + y², then you could not only factor it, you could find a square root. So it's the product of two things that are equal. It's (x+y)². But there are plenty of things you couldn't factor. Maybe the simplest one is something like x² + 3. You couldn't factor it because you couldn't find a square root of −3. But if you put in complex numbers, then you can factor it. So, if you enlarge the domain of things you accept, there's a factorization. Then suddenly it becomes possible to factor. And that's a kind of old story in mathematics. If you want to solve an equation, like, even something simple like 3x − 1 = 0, you can't solve that in integers, so you invent rational numbers, fractions. And then you suddenly can solve it; x is ⅓. Or, if you said 3x + 1 = 0, then you'd have to know about negative numbers, too. And for a while negative numbers were sort of very strange things in mathematics. Then they got ordinary, and we're happy to use them. So this is a kind of old theme in mathematics. If you don't have enough tricks in your bag, put in a new trick. The square root of −1 will let you solve lots more equations than you could have solved before. And that's the story that's gone on in mathematics since the Greek times, really, and been very important, and always there's a kind of complaint that, well, that isn't real. It isn't the same thing, you know, if you solve and get a negative number, what could that mean in reality? But people got over it, and they discovered that negative numbers were just as useful as positive numbers, though they could represent different kind of things. BRADY: It's almost that necessity is the mother of invention. PROF. EISENBUD: That's right, and nature somehow follows along, or really nature was ahead of us there I think. So nature knew about complex numbers, but didn't bother to tell us for a long time. And then we needed them for something, and we realized that they were useful, and they've, you know, now are the basis of lots of physics and everything. So they're really out there in nature, even though they're called imaginary or complex. So there's another story like this, which comes up in the invention of quantum mechanics. Paul Adrien Dirac was one of the great physicists of the 20th century, and he was involved in the beginnings of quantum mechanics. As he worked, he realized he needed to solve a problem like this which didn't seem to have a solution. And the problem was about a complicated-looking formula, a differential operator: −c² times the second derivative with respect to time, plus the sum of the derivatives with respect to the spatial coordinates, so ∂²/∂x² + ∂²/∂y² + ∂²/∂z². So Dirac wanted not to solve this equation—this is an important differential operator— but he wanted a square root of it. Now that's a sort of weird thing to want, because there is no square root, which is a differential operator written like this. And, if you think about it, you can change coordinates (I won't bother to go through the change— you need complex numbers and things) But this is just like the formula t² + x² + y² + z². So Dirac knew right away that finding a square root of this complicated-looking thing is the same as finding a square root of this simpler-looking thing. BRADY: It's got four variables. PROF. EISENBUD: It's got four terms, but you can't find a square root. You can't even find a factorization of it, in fact; if you think about it for a moment, you'll see that. So Dirac had to jump out of the realm of simple polynomials, and what he did was realize that you could factor this in a certain sense in terms of matrices. And in fact, he's famous for inventing matrix mechanics and quantum mechanics. In quantum mechanics, one way to represent a state is as a big matrix. In fact, an infinite matrix, but finite ones are already interesting enough. BRADY: Professor, so, I take it the tool that he's about to pull from the tool bag to get him out of his jam are matrices, but he didn't invent matrices; they were already known. PROF. EISENBUD: Matrices were a big deal already in the 19th Century, when they were used for other mathematical purposes. And they're used in all kinds of ways today. They're everywhere in mathematics and in physics and in all the applications of mathematics. But this was a brand-new idea that Dirac had, and it's an idea that's had legs, as I'll tell you. So, how did he do it? First I have to remind you—what would it even mean to factor something in matrices? You got to multiply things to factor, right? So, how do we multiply matrices? Let me remind you. I'll do it with 2×2 matrices. So, suppose you have two matrices. They could be matrices of numbers or matrices of polynomials, and let's call the other one (a′, b′, c′, d′). A matrix, of course, is just an array of variables, and we're gonna multiply them only when they're square matrices of the same size. So, you do this by taking the rows of the first matrix and multiplying them by the columns of the second matrix. So, for example, the upper corner of the product matrix is gonna be gotten by taking the first row of the first matrix in the first column So it's aa′ + bc′. And then I'm gonna do the same thing for the second entry on the top: ab′ + bd′, and then on the bottom, I'll use the bottom row of the first matrix: a′c +c′d, and then finally b′c + d′d. So that's a formula. That's a formula like any other. The good thing about it is it has lots of the properties that ordinary multiplication has, but one property it doesn't have is that if you take matrices in the other order, you don't get the same answer. They don't commute. That's because we use the rows of this matrix and the columns of that matrix, so it's obvious that there's no reason why they should commute, and in fact, they don't. Dirac's idea was, first of all, to factor this polynomial in terms of matrices. I'm gonna do a slightly simpler looking example, but those of you who know about complex numbers will know that it's the same example written in a different way: xy − uv. That's another polynomial in four variables, but you can transform this into this with complex numbers. And so, how would you factor that in matrices? Well, here's the answer. I'll just write it down. It's x, y, u, v times y, x here, and −u, and −v here. So this is a formula that some people in the audience will surely recognize, and it's a very simple one. Dirac certainly knew it, and knew how to use it. Here's a polynomial on the left and a matrix on the right. So what could that mean? Well, so, this equality is wrong, but what's true is that this is a diagonal matrix xy − uv, and then 0, and 0, and xy − uv. And I think of this as xy − uv, the polynomial, multiplied by the matrix which looks like 1. You know, if you multiply something by 1 you get the same thing back in ordinary arithmetic. So this is like the polynomial xy − uv, and that's the sense in which this is a matrix factorization. And I can even do this with getting a square root by another simple trick that Dirac also probably knew in advance. BRADY: Which is what he really wanted. PROF. EISENBUD: Which is what he wanted. He wanted the square root of this polynomial. So this requires bigger matrices, no problem. So let's call this matrix A and this matrix B. And then I'll make a bigger matrix by taking 0, A—this is now going to be a 4×4 matrix—B,0, and multiplying it by the same thing, 0, A, B, 0. So, that's the square of the matrix, 0, A, B, 0, squared. And that's equal to xy - uv times a bigger matrix, 1, 1, 1, 1 on the diagonal. Still, it's an identity matrix, so this counts in Dirac's book as a square root. So, here… BRADY: Because you can square root. PROF. EISENBUD: You square a matrix and you get something which is the personification in matrices of this polynomial. Dirac was satisfied with this; it served his purpose. He invented matrix mechanics, and quantum mechanics peacefully went on. And that was sort of the end of the story. BRADY: Can I just ask a question though? It feels a bit like you've changed sports mid-game though. Like, it's like you started playing baseball, and then by the end you were playing football, because you were using matrices, like… But so have you not lost any meaning or usefulness by that change of games? PROF. EISENBUD: Surprisingly not. This is another case where nature was ahead of us, and nature said, well, really you should be working with matrices to begin with, and you just found that out. So, Dirac was happy and as I say quantum mechanics is based on this idea: extending the domain of thought in order to get some better result. BRADY: Where's it gone since? You've been playing with this. PROF. EISENBUD: So actually I wrote a paper on this subject. It's actually become my most quoted paper and I'll tell you why that is too - the title of the paper is homological algebra on complete intersections - doesn't seem to have anything to do with this, but it contains a - the theorem, which is a nice one, which says essentially that you can factor any polynomial that even seemed plausibly factorable in terms of matrices. And if you don't exclude a sort of stupid idea, then that becomes a trivial statement. So, of course you can factor any polynomial or anything in the world. You can say f = 1 times f. That's not an interesting factorization. You didn't learn anything from that, and that's not the kind of factorizations we were doing before. So I want to exclude polynomials and matrices that have ones in them or numbers in them, and only look at things with polynomials of higher degree. Then I can't factor the polynomial "x" - it's of degree 1, and if you multiply polynomials of degree 3 and 4, you get a polynomial of degree 7. There's nothing you can do about it. You can't factor x + y^2 either for the same reason because you can't find things that multiply to x so you have to exclude that. So the theorem I proved is that any polynomial with no constant or linear term - linear means degree 1 - can be factored as a product of matrices with no constant terms - no numbers in them in other words. BRADY: Can you show me a few examples of "yes, I can do that one; no, I can't do that one"? Right, just show me - PROF. EISENBUD: Tell me a polynomial, right? x^3 + 79y^2 + 34xz + ... Just, anything, doesn't matter. So I could - and I can write down an algorithm which would produce the matrix factorization of whatever you want or the machine will do it for me. BRADY: Now can you show me one you can't do? PROF. EISENBUD: So you can't do x + y^2, you can't do x plus anything. But that's basically the only things you can't do. BRADY: It sounds to me like you can't do a polynomial if it has a variable like an x in it that isn't multiplied by another variable, and is a power of 1. PROF. EISENBUD: Right, it isn't multiplied by itself or another variable, right? Right, the variable itself. BRADY: So the reason the x was okay here is because it was multiplied by z, right? PROF. EISENBUD: That's right. BRADY: Okay. PROF. EISENBUD: So actually the theorem applies to power series and lots of other things too, and the restriction of not having a linear term or a constant term has a different interpretation in power series Which is more natural in a way So this is a true statement, but it's not the most natural statement for a mathematician. BRADY: Why had Dirac not seen not? PROF. EISENBUD: Good question, I don't and of course, I can't ask him. I don't know the answer. He's dead a long time already, but proving this depends on a theory of finite-free resolutions, of which I'm an expert and have thought about a lot, and I actually got to it by proving something different in that theory. So I proved that if you take a polynomial ring and factor out a polynomial then every free resolution becomes periodic of period 2, and the period 2 part really comes from a matrix factorization of that polynomial. So it's actually an algorithmic way of producing the matrix factorization, and there's more to it than that, of course, but that's the - that was how I got to it. BRADY: Why is it your most cited paper? PROF. EISENBUD: So, It was a - was already a nice paper. I was very proud of it at the time I did it, and people used it for this and that it has something to do with a subject called Cohen-McAuley modules, and representations of finite groups. So it got out in the literature, it was used. But then in 2004, a physicist discovered that he needed this general theorem for the purpose of defining boundary conditions in string theory, and string theory of course is a big deal and physicists tend to cluster in groups with young things so soon there were hundreds of references to this. So that's why it became my most cited paper I think. BRADY: Have you become a bit of an expert on string theory now? Like, if you research why they like it or have you just kind of said, "alright if it makes you happy, go ahead, I don't care." PROF. EISENBUD: A mixture of the two. Actually, a collaborator of mine, Irena Peeva, and I worked on a generalization of this and we thought well the string theorists love this so much, maybe they'll love our generalization too - that would be nice. And so we talked to a bunch of physicists and we got them to give us days of explanation and at the end I understood a little more than I had before, but I certainly didn't become a physicist. I think that's a hard theory to get into if you're not steeped in field theory to begin with. BRADY: What good does it do you if a paper you wrote gets cited lots and lots of times? PROF. EISENBUD: Makes me pleased, that's all really. I'm in a stage of my career where it doesn't really matter to me in career terms at all, but it's it's nice to know that someone is using them, you know. Mathematics has a pretty small audience base - actually, your videos have vastly extended the audience base. But if I look at the colleagues who would be natural candidates for using this theorem It might be in the hundreds, but it's not in the millions certainly. Maybe it's not in the thousands. And so having more people interested in one's work - that feels good. BRADY: So citations are like your video views. PROF. EISENBUD: That's right. Exactly, exactly. BRADY: How many citations has it had? PROF. EISENBUD: I don't know. I haven't looked. My book on commutative algebra has the most citations and that's in the thousands, but that's a lot for any mathematical work.
Referring to this as the origin of matrix mechanics is kind of odd, at least it's not what physicists mean when they use that term. This is instead how Dirac found the equation describing the relativistic motion of electrons.
I was expecting SVD