WEBVTT 00:00:00.499 --> 00:00:02.766 The following content is provided under a Creative 00:00:02.766 --> 00:00:03.620 Commons license. 00:00:03.620 --> 00:00:06.730 Your support will help MIT OpenCourseWare 00:00:06.730 --> 00:00:10.050 continue to offer high quality educational resources for free. 00:00:10.050 --> 00:00:13.450 To make a donation, or to view additional materials 00:00:13.450 --> 00:00:16.150 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.150 --> 00:00:19.330 at ocw.mit.edu. 00:00:19.330 --> 00:00:26.300 PROFESSOR STRANG: Shall we start? 00:00:26.300 --> 00:00:31.390 The main job of today is eigenvalues and eigenvectors. 00:00:31.390 --> 00:00:34.330 The next section in the book and a very big topic 00:00:34.330 --> 00:00:37.170 and things to say about it. 00:00:37.170 --> 00:00:41.170 I do want to begin with a recap of what I didn't quite 00:00:41.170 --> 00:00:45.130 finish last time. 00:00:45.130 --> 00:00:50.360 So what we did was solve this very straightforward equation. 00:00:50.360 --> 00:00:53.330 Straightforward except that it has a point source, 00:00:53.330 --> 00:00:54.910 a delta function. 00:00:54.910 --> 00:00:58.350 And we solved it, both the fixed-fixed case 00:00:58.350 --> 00:01:02.720 when a straight line went up and back down 00:01:02.720 --> 00:01:05.520 and in the free-fixed case when it 00:01:05.520 --> 00:01:11.790 was a horizontal line and then down with slope minus one. 00:01:11.790 --> 00:01:15.060 And there are different ways to get to this answer. 00:01:15.060 --> 00:01:18.570 But once you have it, you can look at it 00:01:18.570 --> 00:01:20.230 and say, well is it right? 00:01:20.230 --> 00:01:22.830 Certainly the boundary conditions are correct. 00:01:22.830 --> 00:01:26.060 Zero slope, went through zero, that's good. 00:01:26.060 --> 00:01:28.470 And then the only thing you really 00:01:28.470 --> 00:01:33.230 have to check is does the slope drop by one 00:01:33.230 --> 00:01:36.780 at the point of the impulse? 00:01:36.780 --> 00:01:40.580 Because that's what this is forcing us to do. 00:01:40.580 --> 00:01:43.180 It's saying the slope should drop by one. 00:01:43.180 --> 00:01:46.550 And here the slope is 1-a going up. 00:01:46.550 --> 00:01:51.330 And if I take the derivative, it's -a going down. 00:01:51.330 --> 00:01:54.740 1-a dropped to -a, good. 00:01:54.740 --> 00:01:56.640 Here the slope was zero. 00:01:56.640 --> 00:02:00.100 Here the slope was minus one, good. 00:02:00.100 --> 00:02:01.850 So those are the right answers. 00:02:01.850 --> 00:02:11.120 And this is simple, but really a great example. 00:02:11.120 --> 00:02:14.330 And then, what I wanted to do was 00:02:14.330 --> 00:02:17.500 catch the same thing for the matrices. 00:02:17.500 --> 00:02:24.640 So those matrices, we all know what K is and what T is. 00:02:24.640 --> 00:02:27.380 So I'm solving, I'm really solving 00:02:27.380 --> 00:02:31.890 K K inverse equal identity. 00:02:31.890 --> 00:02:33.520 That's the equation I'm solving. 00:02:33.520 --> 00:02:37.340 So I'm looking for K inverse and trying 00:02:37.340 --> 00:02:40.090 to get the columns of the identity. 00:02:40.090 --> 00:02:42.930 And you realize the columns of the identity 00:02:42.930 --> 00:02:45.660 are just like delta vectors. 00:02:45.660 --> 00:02:48.000 They've got a one in one spot, they're 00:02:48.000 --> 00:02:52.620 a point load just like this thing. 00:02:52.620 --> 00:02:56.610 So can I just say how I remember K inverse? 00:02:56.610 --> 00:02:58.640 I finally, you know-- again there 00:02:58.640 --> 00:03:01.790 are different ways to get to it. 00:03:01.790 --> 00:03:03.900 One way is MATLAB, just do it. 00:03:03.900 --> 00:03:08.620 But I guess maybe the whole point 00:03:08.620 --> 00:03:13.100 is, the whole point of these and the eigenvalues that 00:03:13.100 --> 00:03:16.370 are coming too, is this. 00:03:16.370 --> 00:03:25.230 That we have here the chance to see important special cases 00:03:25.230 --> 00:03:26.580 that work out. 00:03:26.580 --> 00:03:28.710 Normally we don't find the inverse, 00:03:28.710 --> 00:03:30.640 print out the inverse of a matrix. 00:03:30.640 --> 00:03:32.370 It's not nice. 00:03:32.370 --> 00:03:36.870 Normally we just let eig find the eigenvalues. 00:03:36.870 --> 00:03:39.590 Because that's an even worse calculation, 00:03:39.590 --> 00:03:42.130 to find eigenvalues, in general. 00:03:42.130 --> 00:03:47.240 I'm talking here about our matrices of all sizes n by n. 00:03:47.240 --> 00:03:51.680 Nobody finds the eigenvalues by hand of n by n matrices. 00:03:51.680 --> 00:03:55.880 But these have terrific eigenvalues 00:03:55.880 --> 00:03:58.380 and especially eigenvectors. 00:03:58.380 --> 00:04:04.010 So in a way this is a little bit like, typical of math. 00:04:04.010 --> 00:04:07.700 That you ask about general stuff or you 00:04:07.700 --> 00:04:14.690 write the equation with a matrix A. 00:04:14.690 --> 00:04:17.540 So that's the general information. 00:04:17.540 --> 00:04:21.110 And then there's the specific, special guys 00:04:21.110 --> 00:04:22.980 with special functions. 00:04:22.980 --> 00:04:27.170 And here there'll be sines and cosines and exponentials. 00:04:27.170 --> 00:04:29.240 Other places in applied math, there 00:04:29.240 --> 00:04:31.710 are Bessel functions and Legendre functions. 00:04:31.710 --> 00:04:33.210 Special guys. 00:04:33.210 --> 00:04:37.080 So here, these are special. 00:04:37.080 --> 00:04:40.880 And how do I complete K inverse? 00:04:40.880 --> 00:04:44.810 So this four, three, two, one. 00:04:44.810 --> 00:04:46.180 Let me complete T inverse. 00:04:46.180 --> 00:04:48.460 You probably know T inverse already. 00:04:48.460 --> 00:04:52.090 So T, this is, four, three, two, one, 00:04:52.090 --> 00:04:57.090 is when the load is way over at the far left end 00:04:57.090 --> 00:04:59.250 and it's just descending. 00:04:59.250 --> 00:05:05.770 And now I'm going to-- Let me show you how I write it in. 00:05:05.770 --> 00:05:07.890 Pay attention here to the diagonal. 00:05:07.890 --> 00:05:15.810 So this will be three, three, two, one. 00:05:15.810 --> 00:05:24.110 Do you see that's the solution that's sort of like this one? 00:05:24.110 --> 00:05:28.310 That's the second column of the inverse so it's solving, 00:05:28.310 --> 00:05:31.820 I'm solving, T T inverse equals I here. 00:05:31.820 --> 00:05:34.700 It's the-- The second column is the guy 00:05:34.700 --> 00:05:39.030 with a one in the second place. 00:05:39.030 --> 00:05:42.400 So that's where the load is, in position number two. 00:05:42.400 --> 00:05:46.000 So I'm level, three, three, up to that load. 00:05:46.000 --> 00:05:51.620 And then I'm dropping after the load. 00:05:51.620 --> 00:05:56.720 What's the third column of T inverse? 00:05:56.720 --> 00:05:59.230 I started with that first column and I 00:05:59.230 --> 00:06:01.700 knew that the answer would be symmetric 00:06:01.700 --> 00:06:03.710 because T is symmetric, so that allowed 00:06:03.710 --> 00:06:05.890 me to write the first row. 00:06:05.890 --> 00:06:08.760 And now we can fill in the rest. 00:06:08.760 --> 00:06:12.450 So what do you think, if the point load is-- Now, 00:06:12.450 --> 00:06:16.250 I'm looking at the third column, third column of the identity, 00:06:16.250 --> 00:06:19.350 the load has moved down to position number three. 00:06:19.350 --> 00:06:22.080 So what do I have there and there? 00:06:22.080 --> 00:06:23.760 Two and two. 00:06:23.760 --> 00:06:26.241 And what do I have last? 00:06:26.241 --> 00:06:26.740 One. 00:06:26.740 --> 00:06:28.290 It's dropping to zero. 00:06:28.290 --> 00:06:31.730 You could put zero in green here if you wanted. 00:06:31.730 --> 00:06:38.810 Zero is the unseen last boundary, 00:06:38.810 --> 00:06:42.370 you know, row at this end. 00:06:42.370 --> 00:06:45.990 And finally, what's happening here? 00:06:45.990 --> 00:06:50.080 What do I get from that? 00:06:50.080 --> 00:06:54.420 All one, one, one to the diagonal. 00:06:54.420 --> 00:06:57.890 And then sure enough it drops to zero. 00:06:57.890 --> 00:07:01.010 So this would be a case where the load is there. 00:07:01.010 --> 00:07:06.010 It would be one, one, one, one and then boom. 00:07:06.010 --> 00:07:07.112 No, it wouldn't be. 00:07:07.112 --> 00:07:08.070 It'd be more like this. 00:07:08.070 --> 00:07:14.780 One, one, one, one and then down to-- 00:07:14.780 --> 00:07:15.280 Okay. 00:07:15.280 --> 00:07:18.380 That's a pretty clean inverse. 00:07:18.380 --> 00:07:22.240 That's a very beautiful matrix. 00:07:22.240 --> 00:07:24.110 Don't you admire that matrix? 00:07:24.110 --> 00:07:27.100 I mean, if they were all like that, gee, 00:07:27.100 --> 00:07:29.630 this would be a great world. 00:07:29.630 --> 00:07:40.760 But of course it's not sparse. 00:07:40.760 --> 00:07:43.510 That's why we don't often use the inverse. 00:07:43.510 --> 00:07:46.090 Because we had a sparse matrix T that 00:07:46.090 --> 00:07:48.380 was really fast to compute with. 00:07:48.380 --> 00:07:50.830 And here, if you tell me the inverse, 00:07:50.830 --> 00:07:52.630 you've actually slowed me down. 00:07:52.630 --> 00:07:58.720 Because you've given me now a dense matrix, no zeroes even 00:07:58.720 --> 00:08:04.830 and multiplying T inverse times the right side 00:08:04.830 --> 00:08:08.800 would be slower than just doing elimination. 00:08:08.800 --> 00:08:11.400 Now this is the kind of more interesting one. 00:08:11.400 --> 00:08:16.000 Because this is the one that has to go up to the diagonal 00:08:16.000 --> 00:08:18.490 and then down. 00:08:18.490 --> 00:08:22.680 So let me-- can I fill in what I think-- way this one goes? 00:08:22.680 --> 00:08:26.370 I'm going upwards to the diagonal 00:08:26.370 --> 00:08:28.170 and then I'm coming down to zero. 00:08:28.170 --> 00:08:31.600 Remember that I'm coming down to zero on this K. 00:08:31.600 --> 00:08:37.590 So Zero, zero, zero, zero is kind of the row number. 00:08:37.590 --> 00:08:41.540 If that's row number zero, here's one, two, three, four, 00:08:41.540 --> 00:08:42.700 the real thing. 00:08:42.700 --> 00:08:48.400 And then row five is getting back to zero again. 00:08:48.400 --> 00:08:52.660 So what do you think, finish the rest of that column. 00:08:52.660 --> 00:08:56.550 So you're telling me now the response to the load 00:08:56.550 --> 00:08:57.850 in position two. 00:08:57.850 --> 00:08:59.910 So it's going to look like this. 00:08:59.910 --> 00:09:03.010 In fact, it's going to look very like this. 00:09:03.010 --> 00:09:06.010 There's the three and then this is in position two. 00:09:06.010 --> 00:09:08.750 And then I'm going to have something here and something 00:09:08.750 --> 00:09:12.510 here and it'll drop to zero. 00:09:12.510 --> 00:09:14.050 What do I get? 00:09:14.050 --> 00:09:15.630 Four, two. 00:09:15.630 --> 00:09:17.830 Six, four, two, zero. 00:09:17.830 --> 00:09:19.490 It's dropping to zero. 00:09:19.490 --> 00:09:21.580 I'm going to finish this in but then I'm 00:09:21.580 --> 00:09:25.660 going to look back and see have I really got it right. 00:09:25.660 --> 00:09:28.880 How does this go now? 00:09:28.880 --> 00:09:31.480 Two, let's see. 00:09:31.480 --> 00:09:36.800 Now it's going up from zero to two to four to six. 00:09:36.800 --> 00:09:38.330 That's on the diagonal. 00:09:38.330 --> 00:09:39.620 Now it starts down. 00:09:39.620 --> 00:09:43.300 It's got to get to zero, so that'll be a three. 00:09:43.300 --> 00:09:48.320 Here is a one going up to two to three to four. 00:09:48.320 --> 00:09:49.360 Is that right? 00:09:49.360 --> 00:09:52.120 And then dropped fast to zero. 00:09:52.120 --> 00:09:55.600 Is that correct? 00:09:55.600 --> 00:09:57.410 Think so, yep. 00:09:57.410 --> 00:10:01.260 Except, wait a minute now. 00:10:01.260 --> 00:10:03.740 We've got the right overall picture. 00:10:03.740 --> 00:10:06.170 Climbing up, dropping down. 00:10:06.170 --> 00:10:07.970 Climbing up, dropping down. 00:10:07.970 --> 00:10:09.550 Climbing up, dropping down. 00:10:09.550 --> 00:10:10.420 All good. 00:10:10.420 --> 00:10:17.190 But we haven't yet got, we haven't checked yet 00:10:17.190 --> 00:10:23.130 that the change in the slope is supposed to be one. 00:10:23.130 --> 00:10:24.680 And it's not. 00:10:24.680 --> 00:10:29.130 Here the slope is like, three, It's going up by threes 00:10:29.130 --> 00:10:33.840 and then it's going down by twos. 00:10:33.840 --> 00:10:38.250 So we've gone from going up at a slope of three 00:10:38.250 --> 00:10:41.010 to down to a slope of two. 00:10:41.010 --> 00:10:44.140 Up three, down just like this. 00:10:44.140 --> 00:10:47.520 But that would be a change in slope of five. 00:10:47.520 --> 00:10:51.490 Therefore there's a 1/5. 00:10:51.490 --> 00:10:54.330 So this is going up with a slope of four and down 00:10:54.330 --> 00:10:55.760 with a slope of one. 00:10:55.760 --> 00:11:00.050 Four dropping to one when I divide by the five, that's 00:11:00.050 --> 00:11:01.070 what I like. 00:11:01.070 --> 00:11:04.060 Here is up by twos, down by threes, again 00:11:04.060 --> 00:11:07.090 it's a change of five so I need the five. 00:11:07.090 --> 00:11:09.720 Up by ones, down by four. 00:11:09.720 --> 00:11:12.740 Sudden, that's a fast drop of four. 00:11:12.740 --> 00:11:16.190 Again, the slope changed by five, dividing by five, 00:11:16.190 --> 00:11:17.610 that's got it. 00:11:17.610 --> 00:11:18.690 So that's my picture. 00:11:18.690 --> 00:11:22.420 You could now create K inverse for any size. 00:11:22.420 --> 00:11:30.160 And more than that, sort of see into K inverse 00:11:30.160 --> 00:11:32.250 what those numbers are. 00:11:32.250 --> 00:11:35.350 Because if I wrote the five by five or six 00:11:35.350 --> 00:11:39.470 by six, doing it a column at a time, 00:11:39.470 --> 00:11:42.290 it would look like a bunch of numbers. 00:11:42.290 --> 00:11:44.050 But you see it now. 00:11:44.050 --> 00:11:46.880 Do you see the pattern? 00:11:46.880 --> 00:11:51.560 Right. 00:11:51.560 --> 00:11:54.260 This is one way to get to those inverses, 00:11:54.260 --> 00:11:57.540 and homework problems are offering other ways. 00:11:57.540 --> 00:12:04.240 T, in particular, is quite easy to invert. 00:12:04.240 --> 00:12:07.400 Do I have any other comment on inverses 00:12:07.400 --> 00:12:12.510 before the lecture on eigenvalues really starts? 00:12:12.510 --> 00:12:18.200 Maybe I do have one comment, one important comment. 00:12:18.200 --> 00:12:20.910 It's this, and I won't develop it in full, 00:12:20.910 --> 00:12:24.530 but let's just say it. 00:12:24.530 --> 00:12:28.730 What if the load is not a delta function? 00:12:28.730 --> 00:12:31.860 What if I have other loads? 00:12:31.860 --> 00:12:35.980 Like the uniform load of all ones or any other load? 00:12:35.980 --> 00:12:45.660 What if the discrete load here is not a delta vector? 00:12:45.660 --> 00:12:50.350 I now know the responses to each column of the identity, right? 00:12:50.350 --> 00:12:54.140 If I put a load in position one, there's the response. 00:12:54.140 --> 00:12:58.480 If I put a load in position two, there is the response. 00:12:58.480 --> 00:13:03.340 Now, what if I have other loads? 00:13:03.340 --> 00:13:05.140 Let me take a typical load. 00:13:05.140 --> 00:13:10.190 What if the load was, well, the one we looked at before. 00:13:10.190 --> 00:13:13.490 If the load was [1, 1, 1, 1]. 00:13:13.490 --> 00:13:20.760 So that I had, the bar was hanging by its own weight, 00:13:20.760 --> 00:13:24.930 let's say. 00:13:24.930 --> 00:13:29.090 In other words, could I solve all problems 00:13:29.090 --> 00:13:31.480 by knowing these answers? 00:13:31.480 --> 00:13:33.510 That's what I'm trying to get to. 00:13:33.510 --> 00:13:36.920 If I know these special delta loads, 00:13:36.920 --> 00:13:41.340 then can I get the solution for every load? 00:13:41.340 --> 00:13:42.250 Yes, no? 00:13:42.250 --> 00:13:43.490 What do you think? 00:13:43.490 --> 00:13:45.730 Yes, right. 00:13:45.730 --> 00:13:47.890 Now with this matrix it's kind of 00:13:47.890 --> 00:13:51.040 easy to see because if you know the inverse matrix, well 00:13:51.040 --> 00:13:53.020 you're obviously in business. 00:13:53.020 --> 00:13:59.390 If I had another load, say another load f for load, 00:13:59.390 --> 00:14:03.450 I would just multiply by K inverse, no problem. 00:14:03.450 --> 00:14:05.490 But I want to look a little deeper. 00:14:05.490 --> 00:14:11.800 Because if I had other loads here than a delta function, 00:14:11.800 --> 00:14:14.710 obviously if I had two delta functions 00:14:14.710 --> 00:14:17.700 I could just combine the two solutions. 00:14:17.700 --> 00:14:20.470 That's linearity that we're using all the time. 00:14:20.470 --> 00:14:23.570 If I had ten delta functions I could combine them. 00:14:23.570 --> 00:14:29.420 But then suppose I had instead of a bunch of spikes, 00:14:29.420 --> 00:14:31.330 instead of a bunch of point loads, 00:14:31.330 --> 00:14:33.710 I had a distributed load. 00:14:33.710 --> 00:14:38.110 Like all ones, how could I do it? 00:14:38.110 --> 00:14:39.690 Main point is I could. 00:14:39.690 --> 00:14:40.410 Right? 00:14:40.410 --> 00:14:44.220 If I know these answers, I know all answers. 00:14:44.220 --> 00:14:47.780 If I know the response to a load at each point, 00:14:47.780 --> 00:14:50.850 then-- come back to the discrete one. 00:14:50.850 --> 00:14:57.140 What would be the answer if the load was [1, 1, 1, 1]? 00:14:57.140 --> 00:15:06.370 Suppose I now try to solve the equation Ku=ones(4,1), 00:15:06.370 --> 00:15:08.509 so all ones. 00:15:08.509 --> 00:15:09.550 What would be the answer? 00:15:09.550 --> 00:15:12.960 How would I get it? 00:15:12.960 --> 00:15:15.510 I would just add the columns. 00:15:15.510 --> 00:15:20.990 Now why would I do that? 00:15:20.990 --> 00:15:21.590 Right. 00:15:21.590 --> 00:15:24.680 Because this, the right-hand side, 00:15:24.680 --> 00:15:29.630 the input is the sum of the four columns, the four 00:15:29.630 --> 00:15:31.330 special inputs. 00:15:31.330 --> 00:15:36.000 So the output is the sum of the four outputs, right. 00:15:36.000 --> 00:15:39.140 In other words, as you saw, we must know everything. 00:15:39.140 --> 00:15:41.950 And that's the way we really know it. 00:15:41.950 --> 00:15:42.520 By linearity. 00:15:42.520 --> 00:15:46.580 If the input is a combination of these, 00:15:46.580 --> 00:15:49.950 the output is the same combination of those. 00:15:49.950 --> 00:15:50.770 Right. 00:15:50.770 --> 00:16:00.240 So, for example, in this T case, if input was, if I did Tu=ones, 00:16:00.240 --> 00:16:06.760 I would just add those and the output would be [10 9, 7, 4]. 00:16:06.760 --> 00:16:10.380 That would be the output from [1, 1, 1, 1]. 00:16:10.380 --> 00:16:20.470 And now, oh boy. 00:16:20.470 --> 00:16:24.720 Actually, let me just introduce a guy's name 00:16:24.720 --> 00:16:31.550 for these solutions and not today show you. 00:16:31.550 --> 00:16:33.970 You have the idea, of course. 00:16:33.970 --> 00:16:37.930 Here we added because everything was discrete. 00:16:37.930 --> 00:16:40.550 So you know what we're going to do over here. 00:16:40.550 --> 00:16:44.290 We'll take integrals, right? 00:16:44.290 --> 00:16:51.430 A general load will be an integral over point loads. 00:16:51.430 --> 00:16:53.090 That's the idea. 00:16:53.090 --> 00:16:54.220 A fundamental idea. 00:16:54.220 --> 00:17:00.960 That some other load, f(x), is an integral of these guys. 00:17:00.960 --> 00:17:05.320 So the solution will be the same integral of these guys. 00:17:05.320 --> 00:17:08.700 Let me not go there except to tell you the name, 00:17:08.700 --> 00:17:11.020 because it's a very famous name. 00:17:11.020 --> 00:17:14.830 This solution u with the delta function 00:17:14.830 --> 00:17:17.480 is called the Green's function. 00:17:17.480 --> 00:17:21.660 So I've now introduced the idea, this is the Green's function. 00:17:21.660 --> 00:17:30.320 This guy is the Green's function for the fixed-fixed problem. 00:17:30.320 --> 00:17:33.010 And this guy is the Green's function 00:17:33.010 --> 00:17:36.280 for the free-fixed problem. 00:17:36.280 --> 00:17:38.820 And the whole point is, maybe this 00:17:38.820 --> 00:17:44.700 is the one point I want you to sort of see always by analogy. 00:17:44.700 --> 00:17:50.450 The Green's function is just like the inverse. 00:17:50.450 --> 00:17:52.080 What is the Green's function? 00:17:52.080 --> 00:17:59.070 The Green's function is the response at x, the u at x, 00:17:59.070 --> 00:18:03.010 when the input, when the impulse is at a. 00:18:03.010 --> 00:18:04.800 So it sort of depends on two things. 00:18:04.800 --> 00:18:09.940 It depends on the position a of the input and it tells you 00:18:09.940 --> 00:18:14.990 the response at position x. 00:18:14.990 --> 00:18:19.260 And often we would use the letter G for Green. 00:18:19.260 --> 00:18:23.680 So it depends on x and a. 00:18:23.680 --> 00:18:29.650 And maybe I'm happy if you just sort of see in some way 00:18:29.650 --> 00:18:33.200 what we did there is just like what we did here. 00:18:33.200 --> 00:18:35.220 And therefore the Green's function 00:18:35.220 --> 00:18:40.820 must be just a differential, continuous version 00:18:40.820 --> 00:18:46.330 of an inverse matrix. 00:18:46.330 --> 00:18:52.760 Let's move on to eigenvalues with that point 00:18:52.760 --> 00:18:59.230 sort of made, but not driven home by many, many examples. 00:18:59.230 --> 00:19:15.820 Question, I'll take a question, shoot. 00:19:15.820 --> 00:19:21.220 Why did I increase zero, three, six and then decrease six? 00:19:21.220 --> 00:19:29.350 Well intuitively it's because this is copying this. 00:19:29.350 --> 00:19:32.660 What's wonderful is that it's a perfect copy. 00:19:32.660 --> 00:19:37.460 I mean, intuitively the solution to our difference equation 00:19:37.460 --> 00:19:40.710 should be like the solution to our differential equation. 00:19:40.710 --> 00:19:44.530 That's why if we have some computational, 00:19:44.530 --> 00:19:47.070 some differential equation that we can't solve, 00:19:47.070 --> 00:19:49.930 which would be much more typical than this one, 00:19:49.930 --> 00:19:53.220 that we couldn't solve it exactly by pencil and paper, 00:19:53.220 --> 00:19:58.930 we would replace derivatives by differences and go over here 00:19:58.930 --> 00:20:02.890 and we would hope that they were like pretty close. 00:20:02.890 --> 00:20:07.820 Here they're right, they're the same. 00:20:07.820 --> 00:20:08.880 Oh the other columns? 00:20:08.880 --> 00:20:09.630 Absolutely. 00:20:09.630 --> 00:20:11.610 These guys? 00:20:11.610 --> 00:20:14.880 Zero, two, four, six going up. 00:20:14.880 --> 00:20:18.180 Six, three, zero coming back. 00:20:18.180 --> 00:20:25.520 So that's a discrete thing of one like that. 00:20:25.520 --> 00:20:27.670 And then the next guy and the last guy 00:20:27.670 --> 00:20:30.310 would be going up one, two, three, four 00:20:30.310 --> 00:20:34.360 and then sudden drop. 00:20:34.360 --> 00:20:35.940 Thanks for all questions. 00:20:35.940 --> 00:20:39.590 I mean, this sort of, by adding these guys in, 00:20:39.590 --> 00:20:41.750 the first one actually went up that way. 00:20:41.750 --> 00:20:45.500 You see the Green's functions. 00:20:45.500 --> 00:20:47.750 But of course this has a Green's function 00:20:47.750 --> 00:20:53.300 for every a. x and a are running all the way from zero to one. 00:20:53.300 --> 00:20:58.550 Here they're just discrete positions. 00:20:58.550 --> 00:21:02.660 Thanks. 00:21:02.660 --> 00:21:06.140 So playing with these delta functions 00:21:06.140 --> 00:21:09.120 and coming up with this solution, 00:21:09.120 --> 00:21:12.910 well, as I say, different ways to do it. 00:21:12.910 --> 00:21:16.710 I worked through one way in class last time. 00:21:16.710 --> 00:21:18.470 It takes practice. 00:21:18.470 --> 00:21:21.610 So that's what the homework's really for. 00:21:21.610 --> 00:21:24.820 You can see me come up with this thing, 00:21:24.820 --> 00:21:28.810 then you can, with leisure, you can follow the steps, 00:21:28.810 --> 00:21:32.740 but you've gotta do it yourself to see. 00:21:32.740 --> 00:21:36.460 Eigenvalues and, of course, eigenvectors. 00:21:36.460 --> 00:21:46.320 We have to give them a fair shot. 00:21:46.320 --> 00:21:49.950 Square matrix. 00:21:49.950 --> 00:21:53.470 So I'm talking about general, what 00:21:53.470 --> 00:21:57.100 eigenvectors and eigenvalues are and why do we want them. 00:21:57.100 --> 00:22:01.400 I'm always trying to say what's the purpose, you know, 00:22:01.400 --> 00:22:07.320 not doing this just for abstract linear algebra. 00:22:07.320 --> 00:22:10.430 We do this, we look for these things 00:22:10.430 --> 00:22:13.510 because they tremendously simplify a problem 00:22:13.510 --> 00:22:16.260 if we can find it. 00:22:16.260 --> 00:22:19.970 So what's an eigenvector? 00:22:19.970 --> 00:22:23.510 The eigenvalue is this number, lambda, 00:22:23.510 --> 00:22:26.840 and the eigenvector is this vector y. 00:22:26.840 --> 00:22:33.120 And now, how do I think about those? 00:22:33.120 --> 00:22:37.850 Suppose I take a vector and I multiply by A. 00:22:37.850 --> 00:22:42.250 So the vector is headed off in some direction. 00:22:42.250 --> 00:22:45.240 Here's a vector v. If I multiply, 00:22:45.240 --> 00:22:47.070 and I'm given this matrix, so I'm 00:22:47.070 --> 00:22:51.030 given the matrix, whatever my matrix is. 00:22:51.030 --> 00:22:54.280 Could be one of those matrices, any other matrix. 00:22:54.280 --> 00:23:00.130 If I multiply that by v, I get some result, Av. 00:23:00.130 --> 00:23:01.410 What do I do? 00:23:01.410 --> 00:23:06.490 I look at that and I say that v was not an eigenvector. 00:23:06.490 --> 00:23:10.290 Eigenvectors are the special vectors which 00:23:10.290 --> 00:23:12.970 come out in the same direction. 00:23:12.970 --> 00:23:18.750 Av comes out parallel to v. So this was not an eigenvector. 00:23:18.750 --> 00:23:21.360 Very few vectors are eigenvectors, 00:23:21.360 --> 00:23:22.890 they're very special. 00:23:22.890 --> 00:23:25.960 Most vectors, that'll be a typical picture. 00:23:25.960 --> 00:23:33.180 But there's a few of them where I've a vector y 00:23:33.180 --> 00:23:36.840 and I multiply by A. And then what's the point? 00:23:36.840 --> 00:23:42.530 Ay is in the same direction. 00:23:42.530 --> 00:23:45.090 It's on that same line as y. 00:23:45.090 --> 00:23:48.890 It could be, it might be twice as far out. 00:23:48.890 --> 00:23:51.880 That would be Ay=2y. 00:23:51.880 --> 00:23:53.710 It might go backwards. 00:23:53.710 --> 00:23:56.070 This would be a possibility, Ay=-y. 00:23:58.830 --> 00:24:02.200 It could be just halfway. 00:24:02.200 --> 00:24:05.020 It could be, not move at all. 00:24:05.020 --> 00:24:06.400 That's even a possibility. 00:24:06.400 --> 00:24:07.960 Ay=0y. 00:24:07.960 --> 00:24:10.910 Count that. 00:24:10.910 --> 00:24:16.870 Those y's are eigenvectors and the eigenvalue is just, 00:24:16.870 --> 00:24:19.910 from this point of view, the eigenvalue has come in second 00:24:19.910 --> 00:24:25.390 because it's-- So y was a special vector that kept its 00:24:25.390 --> 00:24:26.290 direction. 00:24:26.290 --> 00:24:32.430 And then lambda is just the number, the two, the zero, 00:24:32.430 --> 00:24:39.050 the minus one, the 1/2 that tells you stretching, 00:24:39.050 --> 00:24:41.260 shrinking, reversing, whatever. 00:24:41.260 --> 00:24:42.760 That's the number. 00:24:42.760 --> 00:24:45.830 But y is the vector. 00:24:45.830 --> 00:24:53.230 And notice that if I knew y and I 00:24:53.230 --> 00:24:56.440 knew it was an eigenvector, then of course if I multiply by A, 00:24:56.440 --> 00:24:59.360 I'll learn the eigenvalue. 00:24:59.360 --> 00:25:01.370 And if I knew an eigenvalue, you'll 00:25:01.370 --> 00:25:03.790 see how I could find the eigenvector. 00:25:03.790 --> 00:25:06.070 Problem is you have to find them both. 00:25:06.070 --> 00:25:07.810 And they multiply each other. 00:25:07.810 --> 00:25:11.050 So we're not talking about linear equations anymore. 00:25:11.050 --> 00:25:13.540 Because one unknown is multiplying another. 00:25:13.540 --> 00:25:19.780 But we'll find a way to look to discover eigenvectors 00:25:19.780 --> 00:25:23.790 and eigenvalues. 00:25:23.790 --> 00:25:27.750 I said I would try to make clear what's the purpose. 00:25:27.750 --> 00:25:36.460 The purpose is that in this direction on this y line, line 00:25:36.460 --> 00:25:43.080 of multiples of y, A is just acting like a number. 00:25:43.080 --> 00:25:48.050 A is some big n by n, 1,000 by 1,000 matrix. 00:25:48.050 --> 00:25:50.050 So a million numbers. 00:25:50.050 --> 00:25:58.110 But on this line, if we find it, if we find an eigenline, 00:25:58.110 --> 00:26:03.010 you could say, an eigendirection in that direction, 00:26:03.010 --> 00:26:06.070 all the complications of A are gone. 00:26:06.070 --> 00:26:08.200 It's just acting like a number. 00:26:08.200 --> 00:26:14.410 So in particular we could solve 1,000 differential equations 00:26:14.410 --> 00:26:23.680 with 1,000 unknown u's with this 1,000 by 1,000 matrix. 00:26:23.680 --> 00:26:26.620 We can find a solution and this is 00:26:26.620 --> 00:26:29.550 where the eigenvector and eigenvalue 00:26:29.550 --> 00:26:34.250 are going to pay off. 00:26:34.250 --> 00:26:35.250 You recognize this. 00:26:35.250 --> 00:26:38.100 Matrix A is of size 1,000. 00:26:38.100 --> 00:26:41.940 And u is a vector of 1,000 unknowns. 00:26:41.940 --> 00:26:44.330 So that's a system of 1,000 equations. 00:26:44.330 --> 00:26:50.870 But if we have found an eigenvector and its eigenvalue 00:26:50.870 --> 00:26:56.600 then the equation will, if it starts in that direction 00:26:56.600 --> 00:26:59.770 it'll stay in that direction and the matrix will just 00:26:59.770 --> 00:27:01.200 be acting like a number. 00:27:01.200 --> 00:27:03.660 And we know how to solve u'=lambda*u. 00:27:06.850 --> 00:27:10.550 That one by one scalar problem we know how to solve. 00:27:10.550 --> 00:27:13.770 The solution to that is e to the lambda*t. 00:27:17.430 --> 00:27:21.270 And of course it could have a constant in it. 00:27:21.270 --> 00:27:25.950 Don't forget that these equations are linear. 00:27:25.950 --> 00:27:29.450 If I multiply it, if I take 2e^(lambda*t), 00:27:29.450 --> 00:27:32.180 I have a two here and a two here and it's just as good. 00:27:32.180 --> 00:27:37.660 So I better allow that as well. 00:27:37.660 --> 00:27:41.600 A constant times e^(lambda*t) times y. 00:27:41.600 --> 00:27:43.240 Notice this is a vector. 00:27:43.240 --> 00:27:47.490 It's a number times a number, the growth. 00:27:47.490 --> 00:27:50.520 So the lambda is now, for the differential equation, 00:27:50.520 --> 00:27:54.570 the lambda, this number lambda is crucial. 00:27:54.570 --> 00:27:58.810 It's telling us whether the solution grows, whether it 00:27:58.810 --> 00:28:01.330 decays, whether it oscillates. 00:28:01.330 --> 00:28:04.680 And we're just looking at this one normal mode, 00:28:04.680 --> 00:28:09.560 you could say normal mode, for eigenvector y. 00:28:09.560 --> 00:28:18.010 We certainly have not found all possible solutions. 00:28:18.010 --> 00:28:25.070 If we have an eigenvector, we found that one. 00:28:25.070 --> 00:28:30.340 And there's other uses and then, let me think. 00:28:30.340 --> 00:28:31.510 Other uses, what? 00:28:31.510 --> 00:28:33.950 So let me write again the fundamental equation, 00:28:33.950 --> 00:28:34.450 Ay=lambda*y. 00:28:37.330 --> 00:28:41.150 So that was a differential equation. 00:28:41.150 --> 00:28:43.200 Going forward in time. 00:28:43.200 --> 00:28:49.090 Now if we go forward in steps we might multiply by A 00:28:49.090 --> 00:28:55.460 at every step. 00:28:55.460 --> 00:28:59.140 Tell me an eigenvector of A squared. 00:28:59.140 --> 00:29:01.960 I'm looking for a vector that doesn't change direction 00:29:01.960 --> 00:29:06.810 when I multiply twice by A. You're going to tell me 00:29:06.810 --> 00:29:10.550 it's y. y will work. 00:29:10.550 --> 00:29:15.320 If I multiply once by A I get lambda times y. 00:29:15.320 --> 00:29:20.620 When I multiply again by A I get lambda squared times y. 00:29:20.620 --> 00:29:29.980 You see eigenvalues are great for powers of a matrix, 00:29:29.980 --> 00:29:33.840 for differential equations. 00:29:33.840 --> 00:29:37.440 The nth power will just take the eigenvalue to the nth. 00:29:37.440 --> 00:29:42.800 The nth power of A will just have lambda to the nth there. 00:29:42.800 --> 00:29:47.350 You know, the pivots of a matrix are all 00:29:47.350 --> 00:29:49.360 messed up when I square it. 00:29:49.360 --> 00:29:52.480 I can't see what's happening with the pivots. 00:29:52.480 --> 00:29:56.870 The eigenvalues are a different way to look at a matrix. 00:29:56.870 --> 00:30:01.770 The pivots are critical numbers for steady-state problems. 00:30:01.770 --> 00:30:04.930 The eigenvalues are the critical numbers 00:30:04.930 --> 00:30:09.250 for moving problems, dynamic problems, 00:30:09.250 --> 00:30:13.520 things are oscillating or growing or decaying. 00:30:13.520 --> 00:30:19.370 And by the way, let's just recognize since this is 00:30:19.370 --> 00:30:24.990 the only thing that's changing in time, 00:30:24.990 --> 00:30:31.400 what would be the-- I'll just go down here, e^(lambda*t). 00:30:31.400 --> 00:30:32.850 Let's just look and see. 00:30:32.850 --> 00:30:35.630 When would I have decay? 00:30:35.630 --> 00:30:38.160 Which you might want to call stability. 00:30:38.160 --> 00:30:40.800 A stable problem. 00:30:40.800 --> 00:30:42.840 What would be the condition on lambda 00:30:42.840 --> 00:30:46.980 to get-- for this to decay. 00:30:46.980 --> 00:30:49.000 Lambda less than zero. 00:30:49.000 --> 00:30:52.600 Now there's one little bit of bad news. 00:30:52.600 --> 00:30:55.180 Lambda could be complex. 00:30:55.180 --> 00:30:58.000 Lambda could be 3+4i. 00:31:00.850 --> 00:31:03.860 It can be a complex number, these eigenvalues, 00:31:03.860 --> 00:31:09.200 even if A is real. 00:31:09.200 --> 00:31:11.370 You'll say, how did that happen, let me see? 00:31:11.370 --> 00:31:13.200 I didn't think. 00:31:13.200 --> 00:31:14.980 Well, let me finish this thought. 00:31:14.980 --> 00:31:18.040 Suppose lambda was 3+4i. 00:31:22.090 --> 00:31:26.610 So I'm thinking about what would e to the lambda*t do in that 00:31:26.610 --> 00:31:27.900 case? 00:31:27.900 --> 00:31:30.820 So this is small example. 00:31:30.820 --> 00:31:32.510 If I had lambda is (3+4i), t. 00:31:35.860 --> 00:31:40.520 What does that do as time grows? 00:31:40.520 --> 00:31:42.410 It's going to grow and oscillate. 00:31:42.410 --> 00:31:45.160 And what decides the growth? 00:31:45.160 --> 00:31:46.680 The real part. 00:31:46.680 --> 00:31:49.130 So it's really the decay or growth 00:31:49.130 --> 00:31:51.850 is decided by the real part. 00:31:51.850 --> 00:31:55.810 The three, e to the 3t, that would be a growth. 00:31:55.810 --> 00:31:58.430 Let me put growth. 00:31:58.430 --> 00:32:01.190 And that would be, of course, unstable. 00:32:01.190 --> 00:32:05.000 And that's a problem when I have a real part 00:32:05.000 --> 00:32:07.910 of lambda bigger than zero. 00:32:07.910 --> 00:32:12.340 And then if lambda has a zero real part, 00:32:12.340 --> 00:32:17.640 so it's pure oscillation, let me just take a case like that. 00:32:17.640 --> 00:32:18.250 So e^(4it). 00:32:20.990 --> 00:32:24.580 So that would be, oscillating, right? 00:32:24.580 --> 00:32:31.570 It's cos(4t) + i*sin(4t), it's just oscillating. 00:32:31.570 --> 00:32:39.430 So in this discussion we've seen growth and decay. 00:32:39.430 --> 00:32:41.490 Tell me the parallels because I'm always 00:32:41.490 --> 00:32:43.150 shooting for the parallels. 00:32:43.150 --> 00:32:45.930 What about the growth of A? 00:32:45.930 --> 00:32:51.300 What matrices, how can I recognize 00:32:51.300 --> 00:32:56.500 a matrix whose powers grow? 00:32:56.500 --> 00:33:02.950 How can I recognize a matrix whose powers go to zero? 00:33:02.950 --> 00:33:05.630 I'm asking you for powers here, over there 00:33:05.630 --> 00:33:08.940 for exponentials somehow. 00:33:08.940 --> 00:33:15.400 So here would be A to higher and higher powers 00:33:15.400 --> 00:33:18.670 goes to zero, the zero matrix. 00:33:18.670 --> 00:33:20.740 In other words, when I multiply, multiply, 00:33:20.740 --> 00:33:24.440 multiply by that matrix I get smaller and smaller and smaller 00:33:24.440 --> 00:33:26.760 matrices, zero in the limit. 00:33:26.760 --> 00:33:33.730 What do you think's the test on the lambda now? 00:33:33.730 --> 00:33:37.500 So what are the eigenvalues of A to the k? 00:33:37.500 --> 00:33:38.000 Let's see. 00:33:38.000 --> 00:33:40.980 If A had eigenvalues lambda, A squared 00:33:40.980 --> 00:33:43.000 will have eigenvalues lambda squared, 00:33:43.000 --> 00:33:45.510 A cubed will have eigenvalues lambda cubed, 00:33:45.510 --> 00:33:48.165 A to the thousandth will have eigenvalues lambda 00:33:48.165 --> 00:33:49.780 to the thousandth. 00:33:49.780 --> 00:33:54.820 And what's the test for that to be getting small? 00:33:54.820 --> 00:33:58.640 Lambda less than one. 00:33:58.640 --> 00:34:03.880 So the test for stability will be-- In the discrete case, 00:34:03.880 --> 00:34:07.530 it won't be the real part of lambda, 00:34:07.530 --> 00:34:10.880 it'll be the size of lambda less than one. 00:34:10.880 --> 00:34:16.430 And growth would be the size of lambda greater than one. 00:34:16.430 --> 00:34:19.400 And again, there'd be this borderline case 00:34:19.400 --> 00:34:24.310 when the eigenvalue has magnitude exactly one. 00:34:24.310 --> 00:34:30.000 So you're seeing here and also here the idea 00:34:30.000 --> 00:34:34.850 that we may have to deal with complex numbers here. 00:34:34.850 --> 00:34:37.190 We don't have to deal with the whole world 00:34:37.190 --> 00:34:40.620 of complex functions and everything 00:34:40.620 --> 00:34:45.710 but it's possible for complex numbers to come in. 00:34:45.710 --> 00:34:49.720 Well while I'm saying that, why don't I give an example 00:34:49.720 --> 00:34:58.550 where it would come in. 00:34:58.550 --> 00:35:03.130 This is going to be a real matrix 00:35:03.130 --> 00:35:07.360 with complex eigenvalues. 00:35:07.360 --> 00:35:11.030 Complex lambdas. 00:35:11.030 --> 00:35:19.770 It'll be an example. 00:35:19.770 --> 00:35:22.530 So I guess I'm looking for a matrix 00:35:22.530 --> 00:35:29.320 where y and Ay never come out in the same direction. 00:35:29.320 --> 00:35:34.240 For real y's I know, okay, here's a good matrix. 00:35:34.240 --> 00:35:41.690 Take the matrix that rotates every vector by 90 degrees. 00:35:41.690 --> 00:35:43.570 Or by theta. 00:35:43.570 --> 00:35:48.450 But let's say here's a matrix that rotates every vector 00:35:48.450 --> 00:35:55.500 by 90 degrees. 00:35:55.500 --> 00:35:57.350 I'm going to raise this board and hide it 00:35:57.350 --> 00:35:58.970 behind there in a minute. 00:35:58.970 --> 00:36:05.290 I just wanted to-- just to open up this thought that we will 00:36:05.290 --> 00:36:09.960 have to face complex numbers. 00:36:09.960 --> 00:36:15.610 If you know how to multiply two complex numbers and add them, 00:36:15.610 --> 00:36:16.800 you're okay. 00:36:16.800 --> 00:36:20.480 This isn't going to turn into a big deal. 00:36:20.480 --> 00:36:25.020 But let's just realize that-- Suppose that matrix, 00:36:25.020 --> 00:36:30.040 if I put in a vector y and I multiply by that matrix, 00:36:30.040 --> 00:36:33.580 it'll turn it through 90 degrees. 00:36:33.580 --> 00:36:35.420 So y couldn't be an eigenvector. 00:36:35.420 --> 00:36:37.070 That's the point I'm trying to make. 00:36:37.070 --> 00:36:41.860 No real vector could be the eigenvector 00:36:41.860 --> 00:36:46.720 of a rotation matrix because every vector gets turned. 00:36:46.720 --> 00:36:53.230 So that's an example where you'd have to go to complex vectors. 00:36:53.230 --> 00:36:58.960 and I think if I tried the vector [1, i], 00:36:58.960 --> 00:37:02.730 so I'm letting the square root of minus one into here, 00:37:02.730 --> 00:37:05.710 then I think it would come out. 00:37:05.710 --> 00:37:09.040 If I do that multiplication I get minus i. 00:37:09.040 --> 00:37:10.670 And I get one. 00:37:10.670 --> 00:37:15.710 And I think that this is, what is it? 00:37:15.710 --> 00:37:17.930 This is probably minus i times that. 00:37:17.930 --> 00:37:32.230 So this is minus i times the input. 00:37:32.230 --> 00:37:34.120 No big deal. 00:37:34.120 --> 00:37:36.940 That was like, you can forget that. 00:37:36.940 --> 00:37:43.310 It's just complex numbers can come in. 00:37:43.310 --> 00:37:52.650 Now let me come back to the main point about eigenvectors. 00:37:52.650 --> 00:37:57.710 Things can be complex. 00:37:57.710 --> 00:38:02.400 So the main point is how do we use them? 00:38:02.400 --> 00:38:08.690 And how many are there? 00:38:08.690 --> 00:38:10.610 Here's the key. 00:38:10.610 --> 00:38:13.520 A typical, good matrix, which includes 00:38:13.520 --> 00:38:15.810 every symmetric matrix, so it includes 00:38:15.810 --> 00:38:20.870 all of our examples and more, if it's of size 1,000, 00:38:20.870 --> 00:38:24.810 it will have 1,000 different eigenvectors. 00:38:24.810 --> 00:38:29.770 And let me just say for our symmetric matrices 00:38:29.770 --> 00:38:33.250 those eigenvectors will all be real. 00:38:33.250 --> 00:38:37.170 They're great, the eigenvectors of symmetric matrices. 00:38:37.170 --> 00:38:40.980 Oh, let me find them for one particular symmetric matrix. 00:38:40.980 --> 00:38:47.040 Say this guy. 00:38:47.040 --> 00:38:49.950 So that's a matrix, two by two. 00:38:49.950 --> 00:38:53.850 How many eigenvectors am I now looking for? 00:38:53.850 --> 00:38:55.960 Two. 00:38:55.960 --> 00:38:59.980 You could say, how do I find them? 00:38:59.980 --> 00:39:07.740 Maybe with a two by two, I can even just wing it. 00:39:07.740 --> 00:39:13.350 We can come up with a vector that is an eigenvector. 00:39:13.350 --> 00:39:15.840 Actually that's what we're going to do 00:39:15.840 --> 00:39:20.070 here is we're going to guess the eigenvectors 00:39:20.070 --> 00:39:22.330 and then we're going to show that they really 00:39:22.330 --> 00:39:24.990 are eigenvectors and then we'll know the eigenvalues 00:39:24.990 --> 00:39:27.010 and it's fantastic. 00:39:27.010 --> 00:39:31.140 So like let's start here with the two by two case. 00:39:31.140 --> 00:39:33.050 Anybody spot an eigenvector? 00:39:33.050 --> 00:39:35.360 Is [1, 0] an eigenvector? 00:39:35.360 --> 00:39:36.310 Try [1, 0]. 00:39:36.310 --> 00:39:39.530 What comes out of [1, 0]? 00:39:39.530 --> 00:39:43.530 Well that picks the first column, right? 00:39:43.530 --> 00:39:45.860 That's how I see, multiplying by [1, 0], 00:39:45.860 --> 00:39:48.460 that says take one of the first column. 00:39:48.460 --> 00:39:52.450 And is it an eigenvector? 00:39:52.450 --> 00:39:53.960 Yes, no? 00:39:53.960 --> 00:39:55.760 No. 00:39:55.760 --> 00:39:59.300 This vector is not in the same direction as that one. 00:39:59.300 --> 00:40:00.870 No good. 00:40:00.870 --> 00:40:09.360 Now can you tell me one that is? 00:40:09.360 --> 00:40:12.200 You're going to guess it. [1, 1]. 00:40:12.200 --> 00:40:15.000 Try [1, 1]. 00:40:15.000 --> 00:40:21.080 Do the multiplication and what do you get? 00:40:21.080 --> 00:40:23.780 Right? 00:40:23.780 --> 00:40:29.000 If I input this vector y, what do I get out? 00:40:29.000 --> 00:40:33.390 Actually I get y itself. 00:40:33.390 --> 00:40:36.510 Right? 00:40:36.510 --> 00:40:38.620 The point is it didn't change direction, 00:40:38.620 --> 00:40:40.780 and it didn't even change length. 00:40:40.780 --> 00:40:42.840 So what's the eigenvalue for that? 00:40:42.840 --> 00:40:47.840 So I've got one eigenvalue now, one eigenvector. [1, 1]. 00:40:47.840 --> 00:40:51.120 And I've got the eigenvalue. 00:40:51.120 --> 00:40:53.980 So here are the vectors, the y's. 00:40:53.980 --> 00:40:55.690 And here are the lambdas. 00:40:55.690 --> 00:41:01.440 And I've got one of them and it's one, right? 00:41:01.440 --> 00:41:03.390 Would you like to guess the other one? 00:41:03.390 --> 00:41:06.560 I'm only looking for two because it's a two by two matrix. 00:41:06.560 --> 00:41:10.020 So let me erase here, hope that you'll 00:41:10.020 --> 00:41:17.350 come up with another one. [1, -1] is certainly worth a try. 00:41:17.350 --> 00:41:19.220 Let's test it. 00:41:19.220 --> 00:41:21.430 If it's an eigenvector, then it should come out 00:41:21.430 --> 00:41:22.430 in the same direction. 00:41:22.430 --> 00:41:26.290 What do I get when I do that? 00:41:26.290 --> 00:41:28.600 So I do that multiplication. 00:41:28.600 --> 00:41:32.740 Three and I get three and minus three, 00:41:32.740 --> 00:41:35.790 so have we got an eigenvector? 00:41:35.790 --> 00:41:37.390 Yep. 00:41:37.390 --> 00:41:42.390 And what's, so if this was y, what is this vector? 00:41:42.390 --> 00:41:44.220 3y. 00:41:44.220 --> 00:41:47.660 So there's the other eigenvector, is [1, -1], 00:41:47.660 --> 00:41:56.150 and the other eigenvalue is three. 00:41:56.150 --> 00:42:01.370 So we did it by spotting it here. 00:42:01.370 --> 00:42:03.420 MATLAB can't do it that way. 00:42:03.420 --> 00:42:06.310 It's got to figure it out. 00:42:06.310 --> 00:42:12.120 But we're ahead of MATLAB this time. 00:42:12.120 --> 00:42:15.650 So what do I notice? 00:42:15.650 --> 00:42:17.550 What do I notice about this matrix? 00:42:17.550 --> 00:42:20.690 It was symmetric. 00:42:20.690 --> 00:42:25.150 And what do I notice about the eigenvectors? 00:42:25.150 --> 00:42:29.390 If I show you those two vectors, [1, 1] and [1, -1], 00:42:29.390 --> 00:42:32.710 what do you see there? 00:42:32.710 --> 00:42:38.270 They're orthogonal. [1, 1] is orthogonal to [1, -1], 00:42:38.270 --> 00:42:40.910 perpendicular is the same as orthogonal. 00:42:40.910 --> 00:42:49.890 These are orthogonal, perpendicular. 00:42:49.890 --> 00:42:53.090 I can draw them, of course, and see that. [1, 1] 00:42:53.090 --> 00:42:58.260 will go, if this is one, it'll go here. 00:42:58.260 --> 00:43:00.510 So that's [1, 1]. 00:43:00.510 --> 00:43:03.420 And [1, -1] will go there, it'll go down, 00:43:03.420 --> 00:43:06.030 this would be the other one. [1, -1]. 00:43:06.030 --> 00:43:07.820 So there's y_1. 00:43:07.820 --> 00:43:08.780 There's y_2. 00:43:08.780 --> 00:43:11.260 And they are perpendicular. 00:43:11.260 --> 00:43:17.300 But of course I don't draw pictures all the time. 00:43:17.300 --> 00:43:21.920 What's the test for two vectors being orthogonal? 00:43:21.920 --> 00:43:23.440 The dot product. 00:43:23.440 --> 00:43:24.530 The dot product. 00:43:24.530 --> 00:43:31.540 The inner product. y transpose-- y_1 transpose * y_2. 00:43:31.540 --> 00:43:35.510 Do you prefer to write it as y_1 with a dot, y_2? 00:43:38.550 --> 00:43:42.660 This is maybe better because it's matrix notation. 00:43:42.660 --> 00:43:51.420 And the point is orthogonal, the dot product is zero. 00:43:51.420 --> 00:43:53.160 So that's good. 00:43:53.160 --> 00:43:56.240 Very good, in fact. 00:43:56.240 --> 00:43:59.180 So here's a very important fact. 00:43:59.180 --> 00:44:06.730 Symmetric matrices have orthogonal eigenvectors. 00:44:06.730 --> 00:44:09.470 What I'm trying to say is eigenvectors and eigenvalues 00:44:09.470 --> 00:44:13.390 are like a new way to look at a matrix. 00:44:13.390 --> 00:44:16.820 A new way to see into it. 00:44:16.820 --> 00:44:21.270 And when the matrix is symmetric, what we see 00:44:21.270 --> 00:44:25.040 is perpendicular eigenvectors. 00:44:25.040 --> 00:44:28.080 And what comment do you have about the eigenvalues 00:44:28.080 --> 00:44:32.260 of this symmetric matrix? 00:44:32.260 --> 00:44:36.290 Remembering what was on the board 00:44:36.290 --> 00:44:40.330 for this anti-symmetric matrix. 00:44:40.330 --> 00:44:44.290 What was the point about that anti-symmetric matrix? 00:44:44.290 --> 00:44:51.330 Its eigenvalues were imaginary actually, an i there. 00:44:51.330 --> 00:44:53.240 Here it's the opposite. 00:44:53.240 --> 00:44:56.620 What's the property of the eigenvalues 00:44:56.620 --> 00:45:00.940 for a symmetric matrix that you would just guess? 00:45:00.940 --> 00:45:02.080 They're real. 00:45:02.080 --> 00:45:03.670 They're real. 00:45:03.670 --> 00:45:06.280 Symmetric matrices are great because they 00:45:06.280 --> 00:45:18.960 have real eigenvalues and they have perpendicular eigenvectors 00:45:18.960 --> 00:45:22.380 and actually, probably if a matrix has real eigenvalues 00:45:22.380 --> 00:45:27.090 and perpendicular eigenvectors, it's going to be symmetric. 00:45:27.090 --> 00:45:32.230 So symmetry is a great property and it shows up in a great way 00:45:32.230 --> 00:45:38.580 in this real eigenvalue, real lambdas, and orthogonal y's. 00:45:38.580 --> 00:45:48.270 Shows up perfectly in the eigenpicture. 00:45:48.270 --> 00:45:53.660 Here's a handy little check on the eigenvalues 00:45:53.660 --> 00:45:55.580 to see if we got it right. 00:45:55.580 --> 00:45:56.700 Course we did. 00:45:56.700 --> 00:45:59.460 That's one and three we can get. 00:45:59.460 --> 00:46:03.950 But let me just show you two useful checks if you haven't 00:46:03.950 --> 00:46:06.230 seen eigenvalues before. 00:46:06.230 --> 00:46:10.520 If I add the eigenvalues, what do I get? 00:46:10.520 --> 00:46:12.140 Four. 00:46:12.140 --> 00:46:15.060 And I compare that with adding down 00:46:15.060 --> 00:46:17.340 the diagonal of the matrix. 00:46:17.340 --> 00:46:19.350 Two and two, four. 00:46:19.350 --> 00:46:21.800 And that check always works. 00:46:21.800 --> 00:46:25.160 The sum of the eigenvalues matches the sum 00:46:25.160 --> 00:46:26.220 down the diagonal. 00:46:26.220 --> 00:46:30.810 So that's like, if you got all the eigenvalues but one, that 00:46:30.810 --> 00:46:32.150 would tell you the last one. 00:46:32.150 --> 00:46:36.060 Because the sum of the eigenvalues 00:46:36.060 --> 00:46:39.990 matches the sum down the diagonal. 00:46:39.990 --> 00:46:45.130 You have no clue where that comes from but it's true. 00:46:45.130 --> 00:46:48.060 And another useful fact. 00:46:48.060 --> 00:46:52.340 If I multiply the eigenvalues what do I get? 00:46:52.340 --> 00:46:53.620 Three? 00:46:53.620 --> 00:46:58.390 And now, where do you see a three over here? 00:46:58.390 --> 00:47:00.730 The determinant. 00:47:00.730 --> 00:47:03.370 4-1=3. 00:47:03.370 --> 00:47:07.860 Can I just write those two facts with no idea of proof. 00:47:07.860 --> 00:47:12.290 The sum of the lambdas, I could write "sum". 00:47:16.000 --> 00:47:21.680 This is for any matrix, the sum of the lambdas is equal to the, 00:47:21.680 --> 00:47:25.170 it's called the trace, of the matrix. 00:47:25.170 --> 00:47:29.190 The trace of the matrix is the sum down the diagonal. 00:47:29.190 --> 00:47:36.360 And the product of the lambdas, lambda_1 times lambda_2 00:47:36.360 --> 00:47:40.300 is the determinant of the matrix. 00:47:40.300 --> 00:47:42.830 Or if I had ten eigenvalues, I would multiply all ten 00:47:42.830 --> 00:47:47.220 and I'd get the determinant. 00:47:47.220 --> 00:47:51.500 So that's some facts about eigenvalues. 00:47:51.500 --> 00:47:55.960 There's more, of course, in section 1.5 00:47:55.960 --> 00:47:58.260 about how you would find eigenvalues 00:47:58.260 --> 00:48:04.760 and how you would use them. 00:48:04.760 --> 00:48:09.770 That's of course the key point, is how would we use them. 00:48:09.770 --> 00:48:15.770 Let me say something more about that, how to use eigenvalues. 00:48:15.770 --> 00:48:22.480 Suppose I have this system of 1,000 differential equations. 00:48:22.480 --> 00:48:27.600 Linear, constant coefficients, starts from some u(0). 00:48:34.220 --> 00:48:37.380 How do eigenvalues and eigenvectors help? 00:48:37.380 --> 00:48:40.060 Well, first I have to find them, that's the job. 00:48:40.060 --> 00:48:44.200 So suppose I find 1,000 eigenvalues and eigenvectors. 00:48:44.200 --> 00:48:50.680 A times eigenvector number i is eigenvalue number i 00:48:50.680 --> 00:48:52.470 times eigenvector number i. 00:48:52.470 --> 00:49:00.230 So these, y_1 to y_1000, so y_1 to y_1000 are the eigenvectors. 00:49:00.230 --> 00:49:02.920 And each one has its own eigenvalue, 00:49:02.920 --> 00:49:05.830 lambda_1 to lambda_1000. 00:49:05.830 --> 00:49:11.260 And now if I did that work, sort of like, in advance, 00:49:11.260 --> 00:49:14.310 now I come to the differential equation. 00:49:14.310 --> 00:49:21.250 How could I use this? 00:49:21.250 --> 00:49:26.770 This is now going to be the most-- it's 00:49:26.770 --> 00:49:29.460 three steps to use it, three steps 00:49:29.460 --> 00:49:33.600 to use these to get the answer. 00:49:33.600 --> 00:49:37.470 Ready for step one. 00:49:37.470 --> 00:49:44.280 Step one is break u nought into eigenvectors. 00:49:44.280 --> 00:49:48.440 Split, separate, write, express u(0) 00:49:48.440 --> 00:50:02.040 as a combination of eigenvectors. 00:50:02.040 --> 00:50:05.360 Now step two. 00:50:05.360 --> 00:50:08.150 What happens to each eigenvector? 00:50:08.150 --> 00:50:10.550 So this is where the differential equation 00:50:10.550 --> 00:50:11.200 starts from. 00:50:11.200 --> 00:50:13.480 This is the initial condition. 00:50:13.480 --> 00:50:17.490 1,000 components of u at the start and it's 00:50:17.490 --> 00:50:23.390 separated into 1,000 eigenvector pieces. 00:50:23.390 --> 00:50:28.580 Now step two is watch each piece separately. 00:50:28.580 --> 00:50:41.620 So step two will be multiply say c_1 by e^(lambda_1*t), 00:50:41.620 --> 00:50:44.150 by its growth. 00:50:44.150 --> 00:50:47.680 This is following eigenvector number one. 00:50:47.680 --> 00:50:53.250 And in general, I would multiply every one of the c's by e 00:50:53.250 --> 00:50:55.140 to those guys. 00:50:55.140 --> 00:50:59.550 So what would I have now? 00:50:59.550 --> 00:51:01.860 This is one piece of the start. 00:51:01.860 --> 00:51:05.300 And that gives me one piece of the finish. 00:51:05.300 --> 00:51:14.220 So the finish is, the answer is to add up the 1,000 pieces. 00:51:14.220 --> 00:51:18.900 And if you're with me, you see what those 1,000 pieces are. 00:51:18.900 --> 00:51:23.480 Here's a piece, some multiple of the first eigenvector. 00:51:23.480 --> 00:51:26.370 Now if we only were working with that piece, 00:51:26.370 --> 00:51:29.470 we follow it in time by multiplying it 00:51:29.470 --> 00:51:31.640 by e to the lambda_1 * t, and what do we 00:51:31.640 --> 00:51:36.034 have at a later time? 00:51:36.034 --> 00:51:36.950 c_1*e^(lambda_1*t)y_1. 00:51:41.990 --> 00:51:45.520 This piece has grown into that. 00:51:45.520 --> 00:51:48.460 And other pieces have grown into other things. 00:51:48.460 --> 00:51:50.840 And what about the last piece? 00:51:50.840 --> 00:51:57.100 So what is it that I have to add up? 00:51:57.100 --> 00:51:59.690 Tell me what to write here. 00:51:59.690 --> 00:52:05.190 c_1000, however much of eigenvector 1,000 00:52:05.190 --> 00:52:10.700 was in there, and then finally, never 00:52:10.700 --> 00:52:20.090 written left-handed before, e to the who? 00:52:20.090 --> 00:52:23.310 Lambda number 1,000, not 1,000 itself, 00:52:23.310 --> 00:52:31.500 but its eigenvalue, 1,000, t. 00:52:31.500 --> 00:52:36.900 This is just splitting, this is constantly, constantly 00:52:36.900 --> 00:52:41.970 the method, the way to use eigenvalues and eigenvectors. 00:52:41.970 --> 00:52:45.490 Split the problem into the pieces that 00:52:45.490 --> 00:52:48.380 go-- that are eigenvectors. 00:52:48.380 --> 00:52:53.790 Watch each piece, add up the pieces. 00:52:53.790 --> 00:52:56.560 That's why eigenvectors are so important. 00:52:56.560 --> 00:52:59.800 Yeah? 00:52:59.800 --> 00:53:02.600 Yes, right. 00:53:02.600 --> 00:53:08.760 Well, now, very good question. 00:53:08.760 --> 00:53:09.979 Let's see. 00:53:09.979 --> 00:53:11.520 Well, the first thing we have to know 00:53:11.520 --> 00:53:14.760 is that we do find 1,000 eigenvectors. 00:53:14.760 --> 00:53:19.940 And so my answer is going to be for symmetric matrices, 00:53:19.940 --> 00:53:21.850 everything always works. 00:53:21.850 --> 00:53:25.060 For symmetric matrices, if size is 1,000, 00:53:25.060 --> 00:53:28.360 they have 1,000 eigenvectors, and next time we'll 00:53:28.360 --> 00:53:30.330 have a shot at some of these. 00:53:30.330 --> 00:53:33.920 What some of them are for these special matrices. 00:53:33.920 --> 00:53:36.890 So this method always works if I've 00:53:36.890 --> 00:53:42.660 got a full family of independent eigenvectors. 00:53:42.660 --> 00:53:47.780 If it's of size 1,000, I need, you're right, exactly right. 00:53:47.780 --> 00:53:52.690 To see that this was the questionable step. 00:53:52.690 --> 00:53:55.270 If I haven't got 1,000 eigenvectors, 00:53:55.270 --> 00:53:57.270 I'm not going to be able to take that step. 00:53:57.270 --> 00:53:59.890 And it happens. 00:53:59.890 --> 00:54:05.340 I am sad to report that some matrices haven't 00:54:05.340 --> 00:54:07.650 got enough eigenvectors. 00:54:07.650 --> 00:54:11.800 Some matrices, they collapse. 00:54:11.800 --> 00:54:15.300 This always happens in math, somehow. 00:54:15.300 --> 00:54:19.020 Two eigenvectors collapse into one and the matrix 00:54:19.020 --> 00:54:23.950 is defective, like it's a loser. 00:54:23.950 --> 00:54:28.890 So now you have to, of course, the equation 00:54:28.890 --> 00:54:31.720 still has a solution. 00:54:31.720 --> 00:54:34.320 So there has to be something there, 00:54:34.320 --> 00:54:37.890 but the pure eigenvector method is not 00:54:37.890 --> 00:54:40.890 going to make it on those special matrices. 00:54:40.890 --> 00:54:43.530 I could write down one but why should we 00:54:43.530 --> 00:54:46.760 give space to a loser? 00:54:46.760 --> 00:54:51.460 But what happens in that case? 00:54:51.460 --> 00:54:54.310 You might remember from differential equations 00:54:54.310 --> 00:54:58.260 when two of these roots, these are like roots, 00:54:58.260 --> 00:55:00.720 these lambdas are like roots that you 00:55:00.720 --> 00:55:04.910 found in solving a differential equation. 00:55:04.910 --> 00:55:09.290 When two of them come together, that's when the danger is. 00:55:09.290 --> 00:55:11.680 When I have a double eigenvalue, then there's 00:55:11.680 --> 00:55:15.090 a high risk that I've only got one eigenvector. 00:55:15.090 --> 00:55:20.730 And I'll just put in this little thing what the other, 00:55:20.730 --> 00:55:23.910 so the e^(lambda_1*t) is fine. 00:55:23.910 --> 00:55:29.130 But if that y_1 is like, if the lambda_1's in there twice, 00:55:29.130 --> 00:55:30.570 I need something new. 00:55:30.570 --> 00:55:34.760 And the new thing turns out to be t*e^(lambda* t). 00:55:38.220 --> 00:55:40.320 I don't know if anybody remembers. 00:55:40.320 --> 00:55:44.570 This was probably hammered back in differential equations that 00:55:44.570 --> 00:55:49.660 if you had repeated something or other then this, 00:55:49.660 --> 00:55:53.180 you didn't get pure e^(lambda*t)'s, you got also 00:55:53.180 --> 00:55:54.650 a t*e^(lambda*t). 00:55:54.650 --> 00:55:56.660 Anyway that's the answer. 00:55:56.660 --> 00:55:58.870 That if we're short eigenvectors, 00:55:58.870 --> 00:56:02.380 and it can happen, but it won't for our good matrices. 00:56:02.380 --> 00:56:07.580 Okay, so Monday I've got lots to do. 00:56:07.580 --> 00:56:11.403 Special eigenvalues and vectors and then positive definite.