WEBVTT 00:00:01.550 --> 00:00:03.920 The following content is provided under a Creative 00:00:03.920 --> 00:00:05.310 Commons license. 00:00:05.310 --> 00:00:07.520 Your support will help MIT OpenCourseWare 00:00:07.520 --> 00:00:11.610 continue to offer high quality educational resources for free. 00:00:11.610 --> 00:00:14.180 To make a donation or to view additional materials 00:00:14.180 --> 00:00:18.140 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:18.140 --> 00:00:19.026 at ocw.mit.edu. 00:00:23.170 --> 00:00:27.920 GILBERT STRANG: OK, so kind of a few things in mind for today. 00:00:27.920 --> 00:00:31.340 One is to answer those two questions on the second line. 00:00:33.860 --> 00:00:38.080 We found those two formulas on the first line last time, 00:00:38.080 --> 00:00:40.850 the derivative of a inverse. 00:00:40.850 --> 00:00:43.460 So the derivative of A squared ought to be easy. 00:00:43.460 --> 00:00:48.020 But if we can't do that, we need to be sure we can. 00:00:48.020 --> 00:00:51.560 And then this was the derivative of an eigenvalue. 00:00:51.560 --> 00:00:54.980 And then it's natural to ask about the derivative 00:00:54.980 --> 00:00:56.870 of the singular value. 00:00:56.870 --> 00:01:00.110 And I had a happy day yesterday in the snow, 00:01:00.110 --> 00:01:03.380 realizing that that has a nice formula too. 00:01:03.380 --> 00:01:06.230 Of course, I'm not the first. 00:01:06.230 --> 00:01:13.260 I'm sure that Wikipedia already knows this formula. 00:01:13.260 --> 00:01:14.970 But it was new to me. 00:01:14.970 --> 00:01:19.460 And I should say Professor Edelman has carried it 00:01:19.460 --> 00:01:21.120 to the second derivative. 00:01:21.120 --> 00:01:27.320 Again, not new, but it's more difficult to find 00:01:27.320 --> 00:01:31.040 second derivatives, and interesting. 00:01:31.040 --> 00:01:34.740 But we'll just stay with first derivatives. 00:01:34.740 --> 00:01:39.050 OK, so that's my first item of sort 00:01:39.050 --> 00:01:41.260 of business from last time. 00:01:41.260 --> 00:01:44.840 And then I'd like to say something about the lab 00:01:44.840 --> 00:01:49.160 homeworks and ask your advice and begin to say something 00:01:49.160 --> 00:01:51.050 about a project. 00:01:51.050 --> 00:01:58.550 And then I will move to these topics in Section 4.4 00:01:58.550 --> 00:02:01.430 that you have already. 00:02:01.430 --> 00:02:06.860 And you might notice I skipped 4.3. 00:02:06.860 --> 00:02:10.610 And the reason is that on Friday, 00:02:10.610 --> 00:02:13.070 actually arriving at MIT tomorrow 00:02:13.070 --> 00:02:19.910 is Professor Townsend, 4.3 is all about his work. 00:02:19.910 --> 00:02:24.320 And he's the best lecturer I know. 00:02:24.320 --> 00:02:29.390 He was here as an instructor and did 18.06 and was 00:02:29.390 --> 00:02:31.540 a big success. 00:02:31.540 --> 00:02:36.110 Actually, he's also just won a prize 00:02:36.110 --> 00:02:44.570 for the SIAG/LA, international prize for young investigators, 00:02:44.570 --> 00:02:48.800 young faculty in applied linear algebra. 00:02:48.800 --> 00:02:53.300 So he goes to Hong Kong to get that prize too. 00:02:53.300 --> 00:02:58.700 Anyway, he will be on the videos and in here in class Friday, 00:02:58.700 --> 00:03:00.860 if all goes well. 00:03:00.860 --> 00:03:06.110 OK, so in order then, the first thing 00:03:06.110 --> 00:03:09.260 is the derivative of A squared. 00:03:09.260 --> 00:03:16.750 And you might think it's 2A dA dt, but it's not. 00:03:16.750 --> 00:03:18.760 And if you realize that it's not, 00:03:18.760 --> 00:03:22.480 then you realize what it is, you will get these things right 00:03:22.480 --> 00:03:23.750 in the future. 00:03:23.750 --> 00:03:32.250 So the answer to the derivative of A squared is not 2A dA dt. 00:03:36.340 --> 00:03:37.690 And why isn't it? 00:03:37.690 --> 00:03:40.720 And what is the right answer? 00:03:40.720 --> 00:03:43.450 So I do that maybe just below here. 00:03:50.590 --> 00:03:53.030 Well, I could ask you to guess the right answer, 00:03:53.030 --> 00:03:55.660 but why don't we do it systematically. 00:03:55.660 --> 00:03:59.800 So how do you find the derivative? 00:03:59.800 --> 00:04:01.180 It's a limit. 00:04:01.180 --> 00:04:03.700 First you have a delta A, right. 00:04:03.700 --> 00:04:05.210 And then you take a limit. 00:04:05.210 --> 00:04:15.700 So I look at A plus delta A squared minus A squared. 00:04:15.700 --> 00:04:18.279 So that's the change in A squared. 00:04:18.279 --> 00:04:21.820 And I divide it by delta t. 00:04:21.820 --> 00:04:24.370 And then delta t goes to 0. 00:04:24.370 --> 00:04:26.920 So that's the derivative I'm looking for, 00:04:26.920 --> 00:04:29.020 the derivative of A squared. 00:04:29.020 --> 00:04:34.210 And now, if I write that out, you'll see why this is wrong, 00:04:34.210 --> 00:04:38.080 but something very close to it, of course-- can't be far away-- 00:04:38.080 --> 00:04:38.980 is right. 00:04:38.980 --> 00:04:41.100 So what happens if I write this out? 00:04:41.100 --> 00:04:44.770 The A squared will cancel the A squared. 00:04:44.770 --> 00:04:45.480 What will I have? 00:04:45.480 --> 00:04:49.540 Will I have 2A delta A? 00:04:49.540 --> 00:04:52.795 Why don't I write 2A delta A next? 00:04:55.900 --> 00:05:01.090 Because when you're squaring a sum of two matrices, 00:05:01.090 --> 00:05:11.180 one term is A delta A, and another term is delta A A. 00:05:11.180 --> 00:05:15.230 And those are different in general. 00:05:15.230 --> 00:05:20.210 And then plus delta A squared. 00:05:20.210 --> 00:05:24.680 And now I divide it all by delta t. 00:05:24.680 --> 00:05:31.550 So you're now seeing my point that now I let delta t go to 0. 00:05:31.550 --> 00:05:34.760 So I'm just doing matrix calculus. 00:05:34.760 --> 00:05:40.490 And it's not altogether simple, but if you follow the rules, 00:05:40.490 --> 00:05:42.200 it comes out right. 00:05:42.200 --> 00:05:48.770 So now what answer do I get as delta t goes to 0? 00:05:48.770 --> 00:05:51.950 I get A dA dt-- 00:05:51.950 --> 00:05:56.240 that's the definition of the-- 00:05:56.240 --> 00:05:58.550 that ratio goes to dA dt. 00:05:58.550 --> 00:06:01.460 That's the whole idea of the derivative of A. 00:06:01.460 --> 00:06:04.730 And now what's the other term? 00:06:04.730 --> 00:06:13.190 It's dA dt A. So it was simply that point 00:06:13.190 --> 00:06:20.510 that I wanted you to pick up on, that the derivative might not 00:06:20.510 --> 00:06:24.770 commute with A. Matrices don't commute in general. 00:06:24.770 --> 00:06:31.415 And so you'll notice that we had a similar expression there. 00:06:35.350 --> 00:06:38.480 We had to pay attention to the order of things there. 00:06:38.480 --> 00:06:39.770 And now we get it right. 00:06:39.770 --> 00:06:51.810 It's not this, but A dA dt plus dA dt A. OK. 00:06:51.810 --> 00:06:52.970 Good. 00:06:52.970 --> 00:06:54.990 Now, can I do the other one? 00:06:54.990 --> 00:07:01.200 Which is a little more serious, but it's a beautiful formula. 00:07:01.200 --> 00:07:04.470 And it's parallel to this guy. 00:07:04.470 --> 00:07:07.050 You might even guess it. 00:07:07.050 --> 00:07:10.050 So I'm looking for the derivative of a singular value. 00:07:10.050 --> 00:07:12.780 The matrix A is changing. 00:07:12.780 --> 00:07:17.830 dA dt tells me how it's changing at the moment, at the instant. 00:07:17.830 --> 00:07:22.400 And I want to know how is sigma changing at that same instant. 00:07:22.400 --> 00:07:26.700 And sort of in parallel with this is a nice-- 00:07:26.700 --> 00:07:27.870 the nice formula-- 00:07:27.870 --> 00:07:36.150 u transpose dA dt v of t. 00:07:36.150 --> 00:07:39.030 Boy, you couldn't ask for a nicer formula than that, right? 00:07:43.310 --> 00:07:46.440 You remember this is the eigenvector. 00:07:46.440 --> 00:07:50.050 And that's the eigenvector of A transpose. 00:07:50.050 --> 00:07:52.650 So this is the singular vector of A. 00:07:52.650 --> 00:07:56.280 And you could say this is a singular vector of A transpose, 00:07:56.280 --> 00:08:04.470 or it's the left singular vector of A. So that's our formula. 00:08:04.470 --> 00:08:07.360 And if we can just recall how to prove it, 00:08:07.360 --> 00:08:10.260 which is going to be parallel to the proof of that one, 00:08:10.260 --> 00:08:14.870 then I'm a happy person and we can get on with life. 00:08:14.870 --> 00:08:20.340 So let's remember this, because it will help us 00:08:20.340 --> 00:08:22.140 to remember the other one, too. 00:08:22.140 --> 00:08:25.620 OK, so where do I start? 00:08:25.620 --> 00:08:28.410 I start with a formula for sigma. 00:08:28.410 --> 00:08:35.340 So I believe that sigma is u transpose times A times 00:08:35.340 --> 00:08:41.780 v. Everybody agree with that? 00:08:41.780 --> 00:08:45.530 Everything's depending on t in this formula. 00:08:45.530 --> 00:08:48.890 As time changes, everything changes. 00:08:48.890 --> 00:08:52.340 But I didn't write in the parentheses, 00:08:52.340 --> 00:08:56.390 t three more times. 00:08:56.390 --> 00:08:59.500 Can we just remember about the SVD. 00:08:59.500 --> 00:09:04.756 The SVD says that A times v equals-- 00:09:04.756 --> 00:09:05.710 AUDIENCE: Sigma u. 00:09:05.710 --> 00:09:06.770 GILBERT STRANG: Sigma u. 00:09:06.770 --> 00:09:08.030 Thanks. 00:09:08.030 --> 00:09:09.490 Av is sigma u. 00:09:09.490 --> 00:09:10.410 That's the SVD. 00:09:13.290 --> 00:09:18.380 So when I put in for Av, I put in sigma u. 00:09:18.380 --> 00:09:19.760 Sigma is just a number. 00:09:19.760 --> 00:09:21.700 So I bring it outside. 00:09:21.700 --> 00:09:24.110 And I'm left with u transpose u. 00:09:24.110 --> 00:09:26.970 And what's u transpose u? 00:09:26.970 --> 00:09:28.660 1. 00:09:28.660 --> 00:09:30.160 So I've used these two facts. 00:09:32.980 --> 00:09:35.890 Or I could have gone the other way 00:09:35.890 --> 00:09:39.580 and said that this is the transpose of-- 00:09:39.580 --> 00:09:43.060 this is A transpose u transpose. 00:09:43.060 --> 00:09:49.060 I could look at it that way times v. 00:09:49.060 --> 00:09:50.860 And if I look at it that way, I'm 00:09:50.860 --> 00:09:53.530 interested in what is A transpose u. 00:09:53.530 --> 00:09:57.370 And what is A transpose u? 00:09:57.370 --> 00:10:04.900 It's sigma v. And it's transpose, so sigma v 00:10:04.900 --> 00:10:07.090 transpose v. 00:10:07.090 --> 00:10:09.643 And what is sigma v transpose v? 00:10:09.643 --> 00:10:10.310 AUDIENCE: Sigma. 00:10:10.310 --> 00:10:12.400 GILBERT STRANG: It's sigma again, of course. 00:10:12.400 --> 00:10:14.070 Got sigma both ways. 00:10:14.070 --> 00:10:15.520 OK. 00:10:15.520 --> 00:10:18.910 Now, I'm ready to take the derivative. 00:10:18.910 --> 00:10:23.680 That's the formula I have for sigma, 00:10:23.680 --> 00:10:25.540 completely parallel to the formula 00:10:25.540 --> 00:10:27.970 that we started out with for lambda. 00:10:27.970 --> 00:10:31.900 The eigenvalue was y transpose Ax. 00:10:31.900 --> 00:10:34.510 And now we've got u transpose Av. 00:10:34.510 --> 00:10:38.170 And, by the way, when would those two formulas 00:10:38.170 --> 00:10:40.410 be one and the same? 00:10:40.410 --> 00:10:45.060 When does the SVD just tell us nothing new 00:10:45.060 --> 00:10:50.520 beyond the eigenvalue stuff for what matrices are the singular 00:10:50.520 --> 00:10:53.430 values, the same as the eigenvalues, and singular 00:10:53.430 --> 00:10:57.870 vectors the same as this as the eigenvectors for-- 00:10:57.870 --> 00:10:58.755 For? 00:10:58.755 --> 00:11:00.240 AUDIENCE: Symmetric. 00:11:00.240 --> 00:11:02.340 GILBERT STRANG: Symmetric, good. 00:11:02.340 --> 00:11:08.490 Symmetric, square, and-- the two words 00:11:08.490 --> 00:11:11.820 that I'm always looking for in this course. 00:11:11.820 --> 00:11:13.560 If you want an A in this course, just 00:11:13.560 --> 00:11:17.970 write down positive definite in the answer to any question, 00:11:17.970 --> 00:11:21.510 because sigmas are by definition positive. 00:11:21.510 --> 00:11:24.570 And if they're going to agree totally with the lambdas, 00:11:24.570 --> 00:11:26.460 then the lambdas have to be positive. 00:11:26.460 --> 00:11:30.237 Or could be 0, so positive semidefinite definite 00:11:30.237 --> 00:11:31.320 would be the right answer. 00:11:31.320 --> 00:11:33.060 Anyway, this is our start. 00:11:36.170 --> 00:11:38.460 And what do we do with that formula? 00:11:38.460 --> 00:11:42.340 So this was all the same, because v transpose v was 1. 00:11:45.870 --> 00:11:48.000 Here I had v transpose v. And that's 1. 00:11:48.000 --> 00:11:49.260 So it gave me sigma. 00:11:49.260 --> 00:11:50.240 Yeah, good. 00:11:50.240 --> 00:11:52.190 Everybody's with us. 00:11:52.190 --> 00:11:53.580 OK, what do I do? 00:11:53.580 --> 00:11:55.110 Take the derivative. 00:11:55.110 --> 00:11:58.160 Takes the derivative of that equation in the box. 00:11:58.160 --> 00:12:00.570 It's exactly what I did last time 00:12:00.570 --> 00:12:03.480 with the corresponding equation for lambda. 00:12:03.480 --> 00:12:04.620 Same thing. 00:12:04.620 --> 00:12:07.140 And I'm going to get again-- 00:12:07.140 --> 00:12:11.130 it's a product rule, because I have three things multiplied 00:12:11.130 --> 00:12:12.760 on the right-hand side. 00:12:12.760 --> 00:12:15.700 So I've got three terms from the product rule. 00:12:15.700 --> 00:12:21.780 So d sigma dt, coming from the box, 00:12:21.780 --> 00:12:35.430 is du transpose dt Av plus u transpose dA dt v 00:12:35.430 --> 00:12:41.370 plus the third guy, which will be u transpose A dv dt. 00:12:44.530 --> 00:12:46.090 Did I get the three terms there? 00:12:46.090 --> 00:12:48.200 Yep. 00:12:48.200 --> 00:12:49.980 And which term do I want? 00:12:49.980 --> 00:12:54.300 Which term do I believe is going to survive and be the answer? 00:12:57.090 --> 00:12:59.620 Well, this is what I'm after. 00:12:59.620 --> 00:13:01.750 So it's the middle term. 00:13:01.750 --> 00:13:03.220 The middle term is just right. 00:13:06.320 --> 00:13:09.770 And the other two terms had better be zero. 00:13:09.770 --> 00:13:12.200 So that will be the proof. 00:13:12.200 --> 00:13:14.540 The other two terms will be zero. 00:13:14.540 --> 00:13:17.840 So can we just take one of those two terms 00:13:17.840 --> 00:13:22.100 and show that it's zero like this one? 00:13:22.100 --> 00:13:24.140 OK, what have I got here? 00:13:24.140 --> 00:13:27.030 I want to know that that term is 0. 00:13:27.030 --> 00:13:28.060 So what have I got. 00:13:28.060 --> 00:13:37.370 I've got du transpose dt times Av. 00:13:37.370 --> 00:13:43.200 And everybody says, OK, in place of Av, write in sigma u. 00:13:43.200 --> 00:13:47.760 And sigma's a number, so I don't mind putting it there. 00:13:47.760 --> 00:13:53.310 So I've got sigma, a number of times the derivative of u times 00:13:53.310 --> 00:13:55.140 u itself, the dot product-- 00:13:55.140 --> 00:13:58.410 the derivative of u with dot product with u. 00:13:58.410 --> 00:14:00.990 And that equals? 00:14:00.990 --> 00:14:05.280 0, I hope, because of this. 00:14:08.100 --> 00:14:09.120 Because of that. 00:14:12.240 --> 00:14:14.790 This comes from the derivative of that. 00:14:17.580 --> 00:14:23.210 But you see, now we've got dot products, ordinary dot 00:14:23.210 --> 00:14:26.800 products, and a number on the right-hand side. 00:14:26.800 --> 00:14:29.750 We're in dimension 1, you could say. 00:14:29.750 --> 00:14:34.300 So this tells me immediately that the derivative 00:14:34.300 --> 00:14:44.870 of u with u plus u transpose times the derivative of u 00:14:44.870 --> 00:14:51.680 is the derivative of 1, which is 0. 00:14:51.680 --> 00:14:56.120 All I'm saying is that these are the same. 00:14:56.120 --> 00:15:01.535 You know, vectors, x transpose y is the same as y transpose 00:15:01.535 --> 00:15:04.460 x when I'm talking about real numbers. 00:15:04.460 --> 00:15:08.030 If I was doing complex things, which I could do, 00:15:08.030 --> 00:15:16.070 then I'd have to pay attention and take complex conjugates 00:15:16.070 --> 00:15:16.920 at the right moment. 00:15:16.920 --> 00:15:19.250 But let's not bother. 00:15:19.250 --> 00:15:23.600 So you see, this is just two of these. 00:15:23.600 --> 00:15:26.960 And it gives me 0. 00:15:26.960 --> 00:15:28.070 So that term's gone. 00:15:30.690 --> 00:15:34.470 And similarly, totally similarly, this term is gone. 00:15:34.470 --> 00:15:40.820 This is A transpose u, all transpose. 00:15:40.820 --> 00:15:45.410 I'm just doing the same thing times dv dt. 00:15:45.410 --> 00:15:48.300 And what is A transpose u? 00:15:48.300 --> 00:15:55.580 It's sigma v. So this is sigma v transpose dv dt. 00:15:55.580 --> 00:15:59.090 And again 0, because of this. 00:16:02.630 --> 00:16:07.630 So in a way this was a slightly easier thing-- 00:16:07.630 --> 00:16:12.830 the last time was completely parallel computation. 00:16:12.830 --> 00:16:17.410 But the first and third terms had to cancel each other with 00:16:17.410 --> 00:16:19.840 the x's and y's. 00:16:19.840 --> 00:16:29.690 Now, they disappear separately, leaving the right answer. 00:16:29.690 --> 00:16:32.860 You might think, how did we get into derivatives 00:16:32.860 --> 00:16:35.470 of singular values? 00:16:35.470 --> 00:16:40.210 Well, I think if we're going to understand the SVD, 00:16:40.210 --> 00:16:45.040 then the first derivative of the sigma is-- 00:16:45.040 --> 00:16:47.320 well, except that I've survived all these years 00:16:47.320 --> 00:16:48.230 without knowing it. 00:16:48.230 --> 00:16:50.440 So you could say it's not-- 00:16:53.340 --> 00:16:58.330 you can live without it, but it's a pretty nice formula. 00:16:58.330 --> 00:17:05.780 OK, that completes that Section 3.1. 00:17:05.780 --> 00:17:09.770 And more to say about 3.2, which was the interlacing 00:17:09.770 --> 00:17:11.960 part that I introduced. 00:17:11.960 --> 00:17:14.720 OK, so where am I? 00:17:14.720 --> 00:17:26.220 I guess I'm thinking about the neat topics about interlacing 00:17:26.220 --> 00:17:28.060 of eigenvalues. 00:17:28.060 --> 00:17:33.810 So may I pick up on that theme, interlacing of eigenvalues 00:17:33.810 --> 00:17:39.730 and say what's in the notes and what's the general idea? 00:17:39.730 --> 00:17:40.230 OK. 00:17:43.290 --> 00:17:48.480 So we're leaving the derivatives and moving 00:17:48.480 --> 00:17:54.570 to finite changes in the eigenvalues and singular 00:17:54.570 --> 00:17:58.020 values, and we are recognizing that we 00:17:58.020 --> 00:18:02.730 can't get exact formulas for the change, 00:18:02.730 --> 00:18:06.200 but we can get bounds for change. 00:18:06.200 --> 00:18:07.990 And they are pretty cool. 00:18:07.990 --> 00:18:12.060 So let me remind you what that is, what they are. 00:18:12.060 --> 00:18:15.260 So I have a matrix-- 00:18:15.260 --> 00:18:18.450 let's see, a symmetric matrix S that 00:18:18.450 --> 00:18:22.080 has eigenvalues lambda 1, greater equal lambda 2, 00:18:22.080 --> 00:18:25.920 greater equal so on. 00:18:25.920 --> 00:18:28.680 Then I change S by some amount. 00:18:28.680 --> 00:18:35.520 I think in the notes there is a number, theta times 1 matrix. 00:18:35.520 --> 00:18:40.080 That has eigenvalues mu 1, greater equal mu 2, 00:18:40.080 --> 00:18:43.530 greater equal something. 00:18:43.530 --> 00:18:49.610 And these are what I can't give you an exact formula for. 00:18:49.610 --> 00:18:52.850 You just would have to compute them. 00:18:52.850 --> 00:18:57.410 But I can give you bounds for them. 00:18:57.410 --> 00:18:59.470 And the bounds come from the lambdas. 00:19:02.030 --> 00:19:04.100 So this was a positive. 00:19:04.100 --> 00:19:05.700 This is a positive change. 00:19:09.590 --> 00:19:14.140 So the eigenvalues will go up, or stay still, 00:19:14.140 --> 00:19:16.760 but they won't go down. 00:19:16.760 --> 00:19:20.830 So the mu's will be bigger than the lambdas. 00:19:20.830 --> 00:19:27.130 But the neat thing is that mu 2 will not pass up lambda 1. 00:19:27.130 --> 00:19:29.240 So here is the interlacing. 00:19:29.240 --> 00:19:32.110 Mu 1 is greater equal lambda 1. 00:19:32.110 --> 00:19:35.350 That says that the highest eigenvalue, the top eigenvalue 00:19:35.350 --> 00:19:39.690 went up, or didn't move. 00:19:39.690 --> 00:19:44.640 But mu 2 is below lambda 1. 00:19:44.640 --> 00:19:46.540 This is the new-- everybody's with me here? 00:19:46.540 --> 00:19:50.210 This is a new, and this is the old. 00:19:50.210 --> 00:19:56.510 New and old being old is S, new is with the change in S. 00:19:56.510 --> 00:20:01.540 And that mu 2 is greater equal lambda 2. 00:20:01.540 --> 00:20:04.010 So the second eigenvalues went up. 00:20:04.010 --> 00:20:05.158 And then so on. 00:20:10.910 --> 00:20:13.790 That's a great fact. 00:20:13.790 --> 00:20:16.810 And I guess that I sent out a puzzle question. 00:20:16.810 --> 00:20:19.215 Did it arrive in email? 00:20:25.100 --> 00:20:29.820 Did anybody see that puzzle question and think about it? 00:20:29.820 --> 00:20:31.060 It worried me for a while. 00:20:36.480 --> 00:20:41.820 Suppose this is the second eigenvalue value-- 00:20:41.820 --> 00:20:44.310 eigenvector. 00:20:44.310 --> 00:20:50.520 So I'm adding on, I'm hyping up the second eigenvector, 00:20:50.520 --> 00:20:52.830 hyping up the matrix in the direction 00:20:52.830 --> 00:20:54.370 of the second eigenvector. 00:20:57.890 --> 00:21:02.250 So the second eigenvalue was lambda 2. 00:21:02.250 --> 00:21:05.280 And its mu 2, the new second eigenvalue, 00:21:05.280 --> 00:21:06.860 is going to be bigger by theta. 00:21:11.190 --> 00:21:15.120 But then I lost a little sleep in thinking, OK, 00:21:15.120 --> 00:21:20.130 if the second eigenvalue is mu 2 plus theta-- 00:21:20.130 --> 00:21:22.980 sorry, if the second eigenvalue mu 2-- 00:21:22.980 --> 00:21:24.300 so let me write it here. 00:21:24.300 --> 00:21:35.460 If mu 2, the second eigenvalue, is the old lambda 2 plus theta 00:21:35.460 --> 00:21:45.390 then bad news, because theta can be as big as I want. 00:21:45.390 --> 00:21:48.180 It can be 20, 200, 2,000. 00:21:48.180 --> 00:21:57.300 And if I'm just adding theta to lambda 2 to get the second-- 00:21:57.300 --> 00:22:01.440 because it's a second eigenvector that's 00:22:01.440 --> 00:22:10.140 getting pumped up, then after a while, mu 2 will pass lambda 1. 00:22:10.140 --> 00:22:11.520 This will be totally true. 00:22:11.520 --> 00:22:13.200 I have no worries about this. 00:22:13.200 --> 00:22:14.610 The old lambda 1-- 00:22:14.610 --> 00:22:18.150 actually, the old-- 00:22:18.150 --> 00:22:21.000 I'll even have equality here, because 00:22:21.000 --> 00:22:27.600 for this particular change, it's not affecting lambda 1. 00:22:27.600 --> 00:22:30.430 So I think mu 1 would be lambda 1 00:22:30.430 --> 00:22:34.080 in my hypothetical possibility. 00:22:34.080 --> 00:22:35.550 What I'm trying to get you to do is 00:22:35.550 --> 00:22:39.210 to think through what this means, because it's quite 00:22:39.210 --> 00:22:43.170 easy to write that line there. 00:22:43.170 --> 00:22:46.950 But then when you think about it, you get some questions. 00:22:46.950 --> 00:22:50.810 And it looks as if it might fail, 00:22:50.810 --> 00:22:57.110 because if theta is really big, that mu 2 would pass up 00:22:57.110 --> 00:22:57.860 lambda 1. 00:22:57.860 --> 00:23:00.500 And the thing would fail. 00:23:00.500 --> 00:23:02.570 And there has to be a catch. 00:23:02.570 --> 00:23:05.960 There has to be a catch. 00:23:05.960 --> 00:23:11.540 So does anybody-- you saw that in the email. 00:23:11.540 --> 00:23:16.400 And I'll now explain what how I understood 00:23:16.400 --> 00:23:24.650 that everything can work and I'm not reaching a contradiction. 00:23:24.650 --> 00:23:27.110 And here's my thinking. 00:23:27.110 --> 00:23:32.810 So it's perfectly true that the eigenvalue that goes with u2-- 00:23:32.810 --> 00:23:36.320 or maybe I should be calling them x2, because usually I 00:23:36.320 --> 00:23:38.750 call the eigenvectors x2-- 00:23:38.750 --> 00:23:42.950 it's perfectly true that mu 2, that that one goes up. 00:23:46.020 --> 00:23:51.900 But what happens when it reaches lambda 1? 00:23:51.900 --> 00:23:54.435 Actually, lambda 1, the first eigenvalue, 00:23:54.435 --> 00:23:57.930 is staying put, because it's not getting any push from this. 00:23:57.930 --> 00:24:01.700 But the second eigenvalue is getting a push of size theta. 00:24:01.700 --> 00:24:07.290 So what happens when lambda 2 plus theta, which is mu 2-- 00:24:07.290 --> 00:24:09.480 mu 2 is lambda 2 plus theta-- 00:24:09.480 --> 00:24:12.720 what happens when it comes up to lambda 1 00:24:12.720 --> 00:24:15.120 and I start worrying that it passes lambda 1? 00:24:18.410 --> 00:24:21.740 Do you see what's happening there? 00:24:21.740 --> 00:24:25.250 What happens when mu 2 passes-- 00:24:25.250 --> 00:24:26.720 when mu 2, which is-- 00:24:26.720 --> 00:24:28.220 I'm just going to copy here-- 00:24:28.220 --> 00:24:31.875 it's the old lambda 2 plus the theta, the number. 00:24:31.875 --> 00:24:34.250 What happens when theta gets bigger and bigger and bigger 00:24:34.250 --> 00:24:37.670 and this hits this thing and then goes beyond? 00:24:37.670 --> 00:24:40.850 Just to see the logic here. 00:24:40.850 --> 00:24:46.760 What happens is that this lambda 2 plus theta, which was mu 2, 00:24:46.760 --> 00:24:49.070 mu 2 until they got here. 00:24:49.070 --> 00:24:55.570 But what is lambda 2 plus theta after it passes lambda 1? 00:24:55.570 --> 00:24:56.800 It's lambda 1 now. 00:24:59.340 --> 00:25:02.190 It passed up, so it's the top eigenvalue 00:25:02.190 --> 00:25:07.390 of the altered matrix. 00:25:07.390 --> 00:25:10.380 And therefore, it's just fine. 00:25:10.380 --> 00:25:11.130 It's out here. 00:25:11.130 --> 00:25:13.740 No problem. 00:25:13.740 --> 00:25:15.810 Maybe I'll just say it again. 00:25:15.810 --> 00:25:20.010 When theta is big enough that mu 2 reaches 00:25:20.010 --> 00:25:23.520 lambda 1, if I increase theta beyond that, 00:25:23.520 --> 00:25:30.060 then this becomes not mu 2 any more, but mu 1. 00:25:30.060 --> 00:25:35.130 And then totally everybody's happy. 00:25:35.130 --> 00:25:40.260 I won't say more on that, because that's just like a way 00:25:40.260 --> 00:25:44.760 that I found to make me think, what do these things mean? 00:25:44.760 --> 00:25:48.070 OK, enough said on that small point. 00:25:48.070 --> 00:25:51.730 But then the main point is, why is this true? 00:25:51.730 --> 00:25:59.240 This interlacing, which is really a nice, beautiful fact. 00:25:59.240 --> 00:26:05.500 And you could imagine that we have 00:26:05.500 --> 00:26:09.220 more different perturbations than just rank 1s. 00:26:13.300 --> 00:26:19.750 So let me tell you the inequality, so named 00:26:19.750 --> 00:26:23.650 after the discoverer, Weyl's inequality. 00:26:27.790 --> 00:26:39.400 So his inequality is for the eigenvalues of S plus T. 00:26:39.400 --> 00:26:41.980 So T is the change. 00:26:41.980 --> 00:26:43.170 S is where I start. 00:26:43.170 --> 00:26:45.340 It has eigenvalues lambda. 00:26:45.340 --> 00:26:48.520 But now, I'm looking at the eigenvalues of S plus T. 00:26:48.520 --> 00:26:50.860 So I'm making a change. 00:26:50.860 --> 00:26:53.350 Over here, in my little puzzle question, 00:26:53.350 --> 00:26:56.710 that was T. It was a rank 1 change. 00:26:56.710 --> 00:26:59.860 Now I will allow other ranks. 00:26:59.860 --> 00:27:03.430 So I want to estimate lambdas of S plus t 00:27:03.430 --> 00:27:10.880 in terms of lambdas of S and lambdas of T. 00:27:10.880 --> 00:27:13.710 And I want some inequality sign there. 00:27:17.000 --> 00:27:21.680 And it's supposed to be true for any symmetric matrices, 00:27:21.680 --> 00:27:26.800 symmetric S and T. 00:27:26.800 --> 00:27:32.360 And then a totally identical Weyl inequality-- 00:27:32.360 --> 00:27:33.980 actually, Weyl was one of the people 00:27:33.980 --> 00:27:36.380 who discovered singular values. 00:27:36.380 --> 00:27:39.350 And when he did it, he asked about his inequality. 00:27:39.350 --> 00:27:42.370 And he found that it still worked the way we've 00:27:42.370 --> 00:27:44.180 found this morning earlier. 00:27:47.210 --> 00:27:49.490 I haven't completed that yet, because I 00:27:49.490 --> 00:27:54.790 haven't told you which lambdas I'm talking about. 00:27:54.790 --> 00:27:58.420 So let me do that. 00:27:58.420 --> 00:28:01.050 So now, I'll tell you Weyl's inequality. 00:28:01.050 --> 00:28:03.280 So S and T are symmetric. 00:28:03.280 --> 00:28:05.770 And so the lambdas are real. 00:28:05.770 --> 00:28:07.670 And we want to know-- 00:28:07.670 --> 00:28:10.060 we want to get them in order. 00:28:10.060 --> 00:28:11.740 OK, so here it goes. 00:28:15.170 --> 00:28:21.460 Weyl allowed the i-th eigenvalue of S and the j-th eigenvalue 00:28:21.460 --> 00:28:27.850 of T and figured out that this was bounded by that eigenvalue 00:28:27.850 --> 00:28:32.650 of S plus T. So that's Weyl's great inequality, 00:28:32.650 --> 00:28:42.730 which reduces to the one I wrote here, 00:28:42.730 --> 00:28:44.680 if I make the right choice-- 00:28:44.680 --> 00:28:47.940 yeah, probably, if I take j equal to 1. 00:28:47.940 --> 00:28:51.340 So you see the beauty of this. 00:28:51.340 --> 00:28:56.560 It tells you about any eigenvalues of S, 00:28:56.560 --> 00:28:57.760 eigenvalues of T. 00:28:57.760 --> 00:29:00.670 So I'm using lambdas here. 00:29:00.670 --> 00:29:02.950 Lambda of S are the eigenvalues of S. 00:29:02.950 --> 00:29:07.480 I'm using lambda again for T and lambda again for S plus T. 00:29:07.480 --> 00:29:11.830 So you have to pay attention to which matrix I'm 00:29:11.830 --> 00:29:13.330 taking the eigenvalues out of. 00:29:13.330 --> 00:29:17.270 So let me take j equal to 1. 00:29:17.270 --> 00:29:21.780 And this says that lambda i, because j is 1, 00:29:21.780 --> 00:29:28.210 S plus T is less or equal to lambda i of S plus lambda 1, 00:29:28.210 --> 00:29:40.170 the top eigenvalue of T. This is lambda max of T. 00:29:40.170 --> 00:29:44.850 Do you see that that's totally reasonable, believable? 00:29:44.850 --> 00:29:49.260 That the eigenvalue when I add on T-- let's 00:29:49.260 --> 00:29:52.260 imagine in our minds that T is positive. 00:29:52.260 --> 00:29:56.640 T is like this thing. 00:29:56.640 --> 00:30:02.280 This could be the T, example of a T. It's what I'm adding on. 00:30:02.280 --> 00:30:06.570 Then the eigenvalues go up. 00:30:06.570 --> 00:30:09.870 But they don't pass that. 00:30:09.870 --> 00:30:12.510 So that tells you how much it could go up by. 00:30:12.510 --> 00:30:19.070 So I guess that Weyl is giving us a less than or equal here. 00:30:19.070 --> 00:30:22.050 Less or equal to lambda 1-- 00:30:22.050 --> 00:30:24.450 so I'm taking i to be 1-- 00:30:24.450 --> 00:30:27.320 plus theta. 00:30:27.320 --> 00:30:32.590 Yeah, so that any equality I've written down there-- 00:30:32.590 --> 00:30:37.020 there's some playing around to do to get practice. 00:30:37.020 --> 00:30:44.310 And it's not so essential for us to be like world grandmasters 00:30:44.310 --> 00:30:48.030 at this thing, but you should see it. 00:30:48.030 --> 00:30:52.010 And you should also see j equal to 2. 00:30:52.010 --> 00:30:55.230 Why will j equal to 2 tell us something? 00:30:55.230 --> 00:30:58.120 I hope it will. 00:30:58.120 --> 00:31:00.310 Let's see what it tells us. 00:31:00.310 --> 00:31:04.720 Lambda i plus 1 now-- j is 2-- 00:31:04.720 --> 00:31:12.640 of S plus T. So it's less than or equal to lambda i of S 00:31:12.640 --> 00:31:19.480 plus lambda 2 of T. I think that's interesting. 00:31:19.480 --> 00:31:34.280 And also, I think I also could get lambda i plus i minus 1. 00:31:34.280 --> 00:31:37.570 Let me write it and see if it's correct. 00:31:37.570 --> 00:31:40.755 Plus lambda i minus 1. 00:31:40.755 --> 00:31:43.690 So those was add up to i plus 2. 00:31:43.690 --> 00:31:51.120 Yeah, I guess lambda i plus 1 plus lambda 1 of T. 00:31:51.120 --> 00:31:55.050 That's what I got by taking-- 00:31:55.050 --> 00:31:57.360 yeah, did I do that right? 00:32:00.480 --> 00:32:03.696 I'm taking j equal to 1. 00:32:03.696 --> 00:32:07.480 No, well, I don't think I got it right. 00:32:10.520 --> 00:32:15.080 What do I want to do here to get a bound on lambda i plus 1? 00:32:15.080 --> 00:32:16.610 I want to take j equal to 2. 00:32:16.610 --> 00:32:23.210 I should just be sensible and plug in j equal to 2 00:32:23.210 --> 00:32:24.270 and i equal to 1. 00:32:29.510 --> 00:32:35.480 All I want to say is that Weyl's inequality is the great fact 00:32:35.480 --> 00:32:38.390 out of which all this interlacing falls 00:32:38.390 --> 00:32:42.650 and more and more, because the interlacing is telling me 00:32:42.650 --> 00:32:46.020 about neighbors. 00:32:46.020 --> 00:32:50.790 And actually if I use Weyl for i and j, different i's and j's, I 00:32:50.790 --> 00:32:56.166 even learn about ones that are not neighbors. 00:32:56.166 --> 00:33:00.600 And I could tell you a proof of Weyl's inequality. 00:33:00.600 --> 00:33:02.340 But I'll save that for the notes. 00:33:07.310 --> 00:33:09.100 So I think maybe that's what I want 00:33:09.100 --> 00:33:14.240 to do about interfacing, just to say what the notes have, 00:33:14.240 --> 00:33:17.550 but not repeat it all in class. 00:33:17.550 --> 00:33:24.230 So the notes have actually two ways to prove this interlacing. 00:33:24.230 --> 00:33:27.830 The standard way that every mathematician would use 00:33:27.830 --> 00:33:30.990 would be Weyl's inequality. 00:33:30.990 --> 00:33:36.600 But last year, Professor Rao, visiting, 00:33:36.600 --> 00:33:42.060 found a nice argument that's also in the notes. 00:33:42.060 --> 00:33:43.700 It ends up with a graph. 00:33:43.700 --> 00:33:47.540 And on that graph, you can see that this is true. 00:33:47.540 --> 00:33:55.530 So for what it's worth, two approaches to this interlacing 00:33:55.530 --> 00:33:58.440 and some examples. 00:33:58.440 --> 00:34:01.710 But I really don't want to spend our lives 00:34:01.710 --> 00:34:05.000 on this eigenvalue topic. 00:34:05.000 --> 00:34:08.940 It's a beautiful fact about symmetric matrices 00:34:08.940 --> 00:34:12.510 and the corresponding fact is true for singular values 00:34:12.510 --> 00:34:18.270 of any matrix, but let's think of leaving it there. 00:34:21.150 --> 00:34:28.670 So now, I'm moving on to the new section. 00:34:28.670 --> 00:34:30.710 The new section involves something 00:34:30.710 --> 00:34:31.897 called compressed sensing. 00:34:31.897 --> 00:34:33.605 I don't know if you've heard those words. 00:34:45.949 --> 00:34:52.699 So these are all topics in Section 4.4, which you have. 00:34:52.699 --> 00:34:55.880 I think we sent it out 10 days ago probably. 00:34:58.660 --> 00:35:04.000 OK, so first let me remember what the nuclear norm is 00:35:04.000 --> 00:35:06.320 of a matrix. 00:35:06.320 --> 00:35:19.635 The nuclear norm a matrix is the sum of the singular values, 00:35:19.635 --> 00:35:22.460 the sum of the singular values. 00:35:22.460 --> 00:35:29.170 So it's like the L1 norm for a vector. 00:35:29.170 --> 00:35:32.080 That's a right way to think about it. 00:35:32.080 --> 00:35:34.030 And do you remember what was special? 00:35:34.030 --> 00:35:38.230 We've talked about using the L1 norm. 00:35:38.230 --> 00:35:42.610 It has this special property that the ordinary L2 00:35:42.610 --> 00:35:45.190 norm absolutely does not have. 00:35:45.190 --> 00:35:48.070 What was it special about the L1 norm? 00:35:48.070 --> 00:35:56.080 If I minimize the L1 norm with some constraint, like ab equal 00:35:56.080 --> 00:36:01.320 b, what's special about the solution, the minimum in the L1 00:36:01.320 --> 00:36:02.040 norm? 00:36:02.040 --> 00:36:02.910 AUDIENCE: Sparse. 00:36:02.910 --> 00:36:04.160 GILBERT STRANG: Sparse, right. 00:36:04.160 --> 00:36:06.920 Sparse. 00:36:06.920 --> 00:36:10.700 So this is moving us up to matrices. 00:36:10.700 --> 00:36:13.670 And that's where compressed sensing comes in. 00:36:13.670 --> 00:36:16.230 Matrix completion comes in. 00:36:16.230 --> 00:36:20.580 So matrix completion would just be-- 00:36:20.580 --> 00:36:23.270 I mentioned-- so this is completion. 00:36:26.120 --> 00:36:28.590 And I'll remember the words Netflix, 00:36:28.590 --> 00:36:31.910 which made the problem famous. 00:36:31.910 --> 00:36:44.140 So I have the matrix A, 3, 2, question mark, question mark, 00:36:44.140 --> 00:36:47.310 question mark, 1, 4, 6, question mark-- 00:36:53.390 --> 00:36:55.780 missing data. 00:36:55.780 --> 00:36:59.650 And so I have to put it in something there, 00:36:59.650 --> 00:37:03.000 because if I don't put in anything, then the numbers 00:37:03.000 --> 00:37:07.930 I do know are useless, because no row or no column 00:37:07.930 --> 00:37:10.100 is complete. 00:37:10.100 --> 00:37:12.100 So it just would give up. 00:37:12.100 --> 00:37:14.710 Somebody that sent me the data, 3 and 2 00:37:14.710 --> 00:37:20.050 and didn't tell me a ranking for the third movie, 00:37:20.050 --> 00:37:22.780 I'd have to say, well, I can't use it. 00:37:22.780 --> 00:37:24.010 That's not possible. 00:37:24.010 --> 00:37:28.990 So we need to think about there. 00:37:28.990 --> 00:37:36.790 And the idea is that the numbers that minimized the nuclear norm 00:37:36.790 --> 00:37:40.180 are a good choice, a good choice. 00:37:40.180 --> 00:37:46.540 So that's just a connection here that we will say more about, 00:37:46.540 --> 00:37:49.300 but not-- 00:37:49.300 --> 00:37:52.240 we could have a whole course in compressed sensing 00:37:52.240 --> 00:37:53.470 and nuclear norm. 00:37:53.470 --> 00:37:59.470 Professor Parrilo in course 6 is an expert on this. 00:38:03.450 --> 00:38:06.980 But you see the point that-- 00:38:06.980 --> 00:38:18.900 so you remember v1 came from the 0 norm. 00:38:21.530 --> 00:38:23.585 And what is the 0 norm of the vector? 00:38:26.940 --> 00:38:27.860 Well, it's not a norm. 00:38:27.860 --> 00:38:31.760 So you could say, forget it, no answer. 00:38:31.760 --> 00:38:34.880 But what do we symbolically mean when 00:38:34.880 --> 00:38:38.310 I write the 0 norm of a vector? 00:38:38.310 --> 00:38:41.070 I mean the number of....? 00:38:41.070 --> 00:38:42.540 Non-zeros. 00:38:42.540 --> 00:38:44.520 The number of non-zeros. 00:38:44.520 --> 00:38:55.430 This was the number of non-zeros in the vector, in v. 00:38:55.430 --> 00:39:03.720 But it's not a norm, because if I take 2 times the vector, 00:39:03.720 --> 00:39:07.330 I have the same number of non-zeros, same norm. 00:39:07.330 --> 00:39:09.662 I can't have the norm of 2v equal the norm 00:39:09.662 --> 00:39:15.890 of v. That would blow away all the properties of norms. 00:39:15.890 --> 00:39:17.920 So v0 is not a norm. 00:39:17.920 --> 00:39:21.500 And then we move it to that sort of appropriate nearest norm. 00:39:21.500 --> 00:39:23.260 And we get v1. 00:39:23.260 --> 00:39:27.320 We get the L1 norm, which is the sum of-- 00:39:27.320 --> 00:39:32.474 everybody remembers that this is the sum of the vi. 00:39:32.474 --> 00:39:37.950 And you remember my pictures of diamonds touching 00:39:37.950 --> 00:39:41.400 planes at sharp points. 00:39:41.400 --> 00:39:44.430 Well, that's what is going on here. 00:39:44.430 --> 00:39:48.660 That problem was called basis pursuit. 00:39:48.660 --> 00:39:51.600 And it comes back again in this section. 00:39:54.120 --> 00:40:01.570 So I minimize this norm subject to the conditions. 00:40:01.570 --> 00:40:07.070 Now, I'm just going to take a jump to the matrix case. 00:40:07.070 --> 00:40:09.920 What's my idea here? 00:40:09.920 --> 00:40:14.420 My idea is that for a matrix, the nuclear norm 00:40:14.420 --> 00:40:15.440 comes from what? 00:40:18.830 --> 00:40:21.570 What's the norm that we sort of start with, 00:40:21.570 --> 00:40:24.220 but it's not a norm? 00:40:24.220 --> 00:40:27.280 And when I sort of take the-- 00:40:27.280 --> 00:40:34.000 because the requirements for a norm don't fail-- 00:40:34.000 --> 00:40:38.170 they fail for what I'm about to write there. 00:40:38.170 --> 00:40:41.920 I could put A 0, but I don't want 00:40:41.920 --> 00:40:44.740 the number of non-zero entries. 00:40:44.740 --> 00:40:47.280 That would be a good guess. 00:40:47.280 --> 00:40:50.480 And probably in some sense it makes sense. 00:40:50.480 --> 00:40:53.960 But it's not the answer I'm looking for. 00:40:53.960 --> 00:41:04.030 What do you think is the 0 norm of a matrix that is not a norm, 00:41:04.030 --> 00:41:09.400 but when I pump it up to the best, to the nearest good norm, 00:41:09.400 --> 00:41:11.710 I get the nuclear norm? 00:41:11.710 --> 00:41:14.500 So this is the question, it's what is A0? 00:41:18.698 --> 00:41:22.160 And it's what? 00:41:22.160 --> 00:41:23.060 AUDIENCE: The rank. 00:41:23.060 --> 00:41:24.290 GILBERT STRANG: The rank. 00:41:24.290 --> 00:41:27.890 The rank of matrix is the equivalent. 00:41:32.630 --> 00:41:34.170 So I don't know about the zero. 00:41:34.170 --> 00:41:35.720 Nobody else calls it A0. 00:41:35.720 --> 00:41:37.460 So I better not. 00:41:37.460 --> 00:41:39.180 It's the rank. 00:41:39.180 --> 00:41:40.890 So again, the rank is not a norm, 00:41:40.890 --> 00:41:43.820 because if I double the matrix, I don't double the rank. 00:41:46.600 --> 00:41:48.220 So I have to move to a norm. 00:41:48.220 --> 00:41:50.410 And it turns out to be the nuclear norm. 00:41:50.410 --> 00:41:52.540 And now, I'll just, with one minute, 00:41:52.540 --> 00:41:57.850 say it's the guess of some people who are working hard 00:41:57.850 --> 00:42:02.920 to prove it, that the deep learning 00:42:02.920 --> 00:42:07.120 algorithm of gradient descent finds 00:42:07.120 --> 00:42:12.850 the solution to the minimum problem in the nuclear norm. 00:42:12.850 --> 00:42:16.140 And we don't know if that's true or not yet. 00:42:16.140 --> 00:42:25.540 For related examples, like this thing, it's proved. 00:42:25.540 --> 00:42:31.870 For the exact problem of deep learning, it's a conjecture. 00:42:31.870 --> 00:42:35.200 So that's what in section 4.4. 00:42:35.200 --> 00:42:38.860 But that word lasso, you want to know what that is. 00:42:38.860 --> 00:42:41.140 Compressed sensing, I'll say a word about. 00:42:41.140 --> 00:42:46.540 So that will be Monday after Alex Townsend's lecture Friday. 00:42:46.540 --> 00:42:51.790 So he's coming to speak to computational 00:42:51.790 --> 00:42:55.960 science students all over MIT tomorrow afternoon. 00:42:55.960 --> 00:43:00.100 I'll certainly go to that, but then he 00:43:00.100 --> 00:43:03.850 said he would come in and take this class Friday. 00:43:03.850 --> 00:43:05.410 So I'll see you Friday. 00:43:05.410 --> 00:43:07.455 And he'll be here too.