Subtitles section Play video Print subtitles Okay. This lecture is mostly about the idea of similar matrixes. I'm going to tell you what that word similar means and in what way two matrixes are called similar. But before I do that, I have a little more to say about positive definite matrixes. You can tell this is a subject I think is really important and I told you what positive definite meant -- it means that this -- this expression, this quadratic form, x transpose I x is always positive. But the direct way to test it was with eigenvalues or pivots or determinants. So I -- we know what it means, we know how to test it, but I didn't really say where positive definite matrixes come from. And so one thing I want to say is that they come from least squares in -- and all sorts of physical problems start with a rectangular matrix -- well, you remember in least squares the crucial combination was A transpose A. So I want to show that that's a positive definite matrix. Can -- so I -- I'm going to speak a little more about positive definite matrixes, just recapping -- so let me ask a question. It may be on the homework. Suppose a matrix A is positive definite. I mean by that it's all -- I'm assuming it's symmetric. That's always built into the definition. So we have a symmetric positive definite matrix. What about its inverse? Is the inverse of a symmetric positive definite matrix also symmetric positive definite? So you quickly think, okay, what do I know about the pivots of the inverse matrix? Not much. What do I know about the eigenvalues of the inverse matrix? Everything, right? The eigenvalues of the inverse are one over the eigenvalues of the matrix. So if my matrix starts out positive definite, then right away I know that its inverse is positive definite, because those positive eigenvalues -- then one over the eigenvalue is also positive. What if I know that A -- a matrix A and a matrix B are both positive definite? But let me ask you this. Suppose if A and B are positive definite, what about -- what about A plus B? In some way, you hope that that would be true. It's -- positive definite for a matrix is kind of like positive for a real number. But we don't know the eigenvalues of A plus B. We don't know the pivots of A plus B. So we just, like, have to go down this list of, all right, which approach to positive definite can we get a handle on? And this is a good one. This is a good one. Can we -- how would we decide that -- if A was like this and if B was like this, then we would look at x transpose A plus B x. I'm sure this is in the homework. Now -- so we have x transpose A x bigger than zero, x transpose B x positive for all -- for all x, so now I ask you about this guy. And of course, you just add that and that and we get what we want. If A and B are positive definites, so is A plus B. So that's what I've shown. So is A plus B. Just -- be sort of ready for all the approaches through eigenvalues and through this expression. And now, finally, one more thought about positive definite is this combination that came up in least squares. Can I do that? So now -- now suppose A is rectangular, m by n. I -- so I'm sorry that I've used the same letter A for the positive definite matrixes in the eigenvalue chapter that I used way back in earlier chapters when the matrix was rectangular. Now, that matrix -- a rectangular matrix, no way its positive definite. It's not symmetric. It's not even square in general. But you remember that the key for these rectangular ones was A transpose A. That's square. That's symmetric. Those are things we knew -- we knew back when we met this thing in the least square stuff, in the projection stuff. But now we know something more -- we can ask a more important question, a deeper question -- is it positive definite? And we sort of hope so. Like, we -- we might -- in analogy with numbers, this is like -- sort of like the square of a number, and that's positive. So now I want to ask the matrix question. Is A transpose A positive definite? Okay, now it's -- so again, it's a rectangular A that I'm starting with, but it's the combination A transpose A that's the square, symmetric and hopefully positive definite matrix. So how -- how do I see that it is positive definite, or at least positive semi-definite? You'll see that. Well, I don't know the eigenvalues of this product. I don't want to work with the pivots. The right thing -- the right quantity to look at is this, x transpose Ax -- A -- x transpose times my matrix times x. I'd like to see that this thing -- that that expression is always positive. I'm not doing it with numbers, I'm doing it with symbols. Do you see -- how do I see that that expression comes out positive? I'm taking a rectangular matrix A and an A transpose -- that gives me something square symmetric, but now I want to see that if I multiply -- that if I do this -- I form this quadratic expression that I get this positive thing that goes upwards when I graph it. How do I see that that's positive, or absolutely it isn't negative anyway? We'll have to, like, spend a minute on the question could it be zero, but it can't be negative. Why can this never be negative? The argument is -- like the one key idea in so many steps in linear algebra -- put those parentheses in a good way. Put the parentheses around Ax and what's the first part? What's this x transpose A transpose? That is Ax transpose. So what do we have? We have the length squared of Ax. We have -- that's the column vector Ax that's the row vector Ax, its length squared, certainly greater than or possibly equal to zero. So we have to deal with this little possibility. Could it be equal? Well, when could the length squared be zero? Only if the vector is zero, right? That's the only vector that has length squared zero. So we have -- we would like to -- I would like to get that possibility out of there. So I want to have Ax never -- never be zero, except of course for the zero vector. How do I assure that Ax is never zero? The -- in other words, how do I show that there's no null space of A? The rank should be -- so now remember -- what's the rank when there's no null space? By no null space, you know what I mean. Only the zero vector in the null space. So if I have a -- if I have an 11 by 5 matrix -- so it's got 11 rows, 5 columns, when is there no null space? So the columns should be independent -- what's the rank? n 5 -- rank n. Independent columns, when -- so if I -- then I conclude yes, positive definite. And this was the assumption -- then A transpose A is invertible -- the least squares equations all work fine. And more than that -- the matrix is even positive definite. And I just to say one comment about numerical things, with a positive definite matrix, you never have to do row exchanges. You never run into unsuitably small numbers or zeroes in the pivot position. They're the right -- they're the great matrixes to compute with,