WEBVTT 00:00:00.000 --> 00:00:11.200 -- one and -- 00:00:11.200 --> 00:00:14.470 the lecture on symmetric matrixes. 00:00:14.470 --> 00:00:17.270 So that's the most important class 00:00:17.270 --> 00:00:19.960 of matrixes, symmetric matrixes. 00:00:19.960 --> 00:00:22.710 A equals A transpose. 00:00:22.710 --> 00:00:26.930 So the first points, the main points of the lecture 00:00:26.930 --> 00:00:31.170 I'll tell you right away. 00:00:31.170 --> 00:00:33.740 What's special about the eigenvalues? 00:00:33.740 --> 00:00:35.900 What's special about the eigenvectors? 00:00:35.900 --> 00:00:42.000 This is -- the way we now look at a matrix. 00:00:42.000 --> 00:00:45.180 We want to know about its eigenvalues and eigenvectors 00:00:45.180 --> 00:00:48.580 and if we have a special type of matrix, 00:00:48.580 --> 00:00:51.490 that should tell us something about eigenvalues 00:00:51.490 --> 00:00:53.220 and eigenvectors. 00:00:53.220 --> 00:00:56.800 Like Markov matrixes, they have an eigenvalue equal 00:00:56.800 --> 00:00:57.440 one. 00:00:57.440 --> 00:01:03.270 Now symmetric matrixes, can I just tell you right off what 00:01:03.270 --> 00:01:06.250 the main facts -- the two main facts are? 00:01:06.250 --> 00:01:09.510 The eigenvalues of a symmetric matrix, real -- 00:01:09.510 --> 00:01:12.320 this is a real symmetric matrix, we -- 00:01:12.320 --> 00:01:15.400 talking mostly about real matrixes. 00:01:15.400 --> 00:01:17.850 The eigenvalues are also real. 00:01:21.440 --> 00:01:25.600 So our examples of rotation matrixes, where -- 00:01:25.600 --> 00:01:29.170 where we got E- eigenvalues that were complex, 00:01:29.170 --> 00:01:31.750 that won't happen now. 00:01:31.750 --> 00:01:34.610 For symmetric matrixes, the eigenvalues are real 00:01:34.610 --> 00:01:37.960 and the eigenvectors are also very special. 00:01:37.960 --> 00:01:41.330 The eigenvectors are perpendicular, orthogonal, 00:01:41.330 --> 00:01:42.520 so which do you prefer? 00:01:42.520 --> 00:01:44.300 I'll say perpendicular. 00:01:44.300 --> 00:01:46.830 Perp- well, they're both long words. 00:01:52.560 --> 00:01:55.290 Okay, right. 00:01:55.290 --> 00:02:01.090 So -- I have a -- you should say "why?" 00:02:01.090 --> 00:02:06.650 and I'll at least answer why for case one, maybe case two, 00:02:06.650 --> 00:02:09.070 the checking the Eigen -- that the eigenvectors are 00:02:09.070 --> 00:02:16.330 perpendicular, I'll leave to, the -- to the book. 00:02:16.330 --> 00:02:19.640 But let's just realize what -- 00:02:19.640 --> 00:02:24.520 well, first I have to say, it -- 00:02:24.520 --> 00:02:29.230 it could happen, like for the identity matrix -- 00:02:29.230 --> 00:02:31.170 there's a symmetric matrix. 00:02:31.170 --> 00:02:33.400 Its eigenvalues are certainly all real, 00:02:33.400 --> 00:02:36.460 they're all one for the identity matrix. 00:02:36.460 --> 00:02:38.480 What about the eigenvectors? 00:02:38.480 --> 00:02:44.020 Well, for the identity, every vector is an eigenvector. 00:02:44.020 --> 00:02:46.930 So how can I say they're perpendicular? 00:02:46.930 --> 00:02:49.400 What I really mean is the -- they -- 00:02:49.400 --> 00:02:57.520 this word are should really be written can be chosen 00:02:57.520 --> 00:02:59.230 perpendicular. 00:02:59.230 --> 00:03:02.650 That is, if we have -- it's the usual case. 00:03:02.650 --> 00:03:05.210 If the eigenvalues are all different, 00:03:05.210 --> 00:03:09.280 then each eigenvalue has one line of eigenvectors 00:03:09.280 --> 00:03:13.250 and those lines are perpendicular here. 00:03:13.250 --> 00:03:16.530 But if an eigenvalue's repeated, then there's 00:03:16.530 --> 00:03:20.730 a whole plane of eigenvectors and all I'm saying 00:03:20.730 --> 00:03:25.330 is that in that plain, we can choose perpendicular ones. 00:03:25.330 --> 00:03:28.520 So that's why it's a can be chosen part, is -- 00:03:28.520 --> 00:03:31.160 this is in the case of a repeated eigenvalue where 00:03:31.160 --> 00:03:34.160 there's some real, substantial freedom. 00:03:34.160 --> 00:03:39.110 But the typical case is different eigenvalues, 00:03:39.110 --> 00:03:43.670 all real, one dimensional eigenvector space, 00:03:43.670 --> 00:03:46.030 Eigen spaces, and all perpendicular. 00:03:46.030 --> 00:03:50.470 So, just -- let's just see the conclusion. 00:03:50.470 --> 00:03:56.270 If we accept those as correct, what happens -- 00:03:56.270 --> 00:03:59.332 and I also mean that there's a full set of them. 00:03:59.332 --> 00:04:01.790 so forgive me for doing such a thing, but, I'll look at the 00:04:01.790 --> 00:04:06.480 I -- so that's part of this picture here, that there -- 00:04:06.480 --> 00:04:08.880 there's a complete set of eigenvectors, 00:04:08.880 --> 00:04:10.160 perpendicular ones. 00:04:10.160 --> 00:04:14.260 So, having a complete set of eigenvectors means -- 00:04:14.260 --> 00:04:16.600 so normal -- 00:04:16.600 --> 00:04:20.649 so the usual -- maybe I put the -- usually -- usual -- 00:04:20.649 --> 00:04:26.060 usual case is that the matrix A we can write in terms of its 00:04:26.060 --> 00:04:31.610 eigenvalue matrix and its eigenvector matrix this way, 00:04:31.610 --> 00:04:33.350 right? 00:04:33.350 --> 00:04:36.430 We can do that in the usual case, 00:04:36.430 --> 00:04:42.460 but now what's special when the matrix is symmetric? 00:04:42.460 --> 00:04:45.740 So this is the usual case, and now 00:04:45.740 --> 00:04:48.960 let me go to the symmetric case. 00:04:52.260 --> 00:04:55.990 So in the symmetric case, A, this -- 00:04:55.990 --> 00:04:58.900 this should become somehow a little special. 00:04:58.900 --> 00:05:03.710 Well, the lambdas on the diagonal are still on the 00:05:03.710 --> 00:05:04.500 diagonal. 00:05:04.500 --> 00:05:08.370 They're -- they're real, but that's where they are. 00:05:08.370 --> 00:05:12.160 What about the eigenvector matrix? 00:05:12.160 --> 00:05:15.620 So what can I do now special about the eigenvector matrix 00:05:15.620 --> 00:05:18.850 when -- when the A itself is symmetric, 00:05:18.850 --> 00:05:22.670 that says something good about the eigenvector matrix, 00:05:22.670 --> 00:05:25.520 so what is this -- what does this lead to? 00:05:25.520 --> 00:05:31.120 This -- these perpendicular eigenvectors, I can not only -- 00:05:31.120 --> 00:05:33.590 I can not only guarantee they're perpendicular, 00:05:33.590 --> 00:05:36.680 I could also make them unit vectors, no problem, 00:05:36.680 --> 00:05:38.910 just s- scale their length to 00:05:38.910 --> 00:05:39.640 one. 00:05:39.640 --> 00:05:41.260 So what do I have? 00:05:41.260 --> 00:05:45.990 I have orthonormal eigenvectors. 00:05:49.950 --> 00:05:55.320 And what does that tell me about the eigenvector matrix? 00:05:55.320 --> 00:05:59.617 What -- what letter should I now use in place of S -- 00:05:59.617 --> 00:06:02.200 I've got -- those two equations are identical,1 remember S has 00:06:02.200 --> 00:06:06.080 the eigenvectors in its columns, but now those columns are 00:06:06.080 --> 00:06:11.810 orthonormal, so the right letter to use is Q. 00:06:11.810 --> 00:06:15.000 So that's where -- so we've got the letter all set up, book. 00:06:15.000 --> 00:06:19.730 so this should be Q lambda Q inverse. 00:06:19.730 --> 00:06:25.150 Q standing in our minds always for this matrix -- in this case 00:06:25.150 --> 00:06:27.550 it's square, it's -- 00:06:27.550 --> 00:06:29.915 so these are the Okay. columns of Q, of course. 00:06:35.190 --> 00:06:39.040 And one more thing. 00:06:39.040 --> 00:06:39.950 What's Q inverse? 00:06:43.070 --> 00:06:45.120 For a matrix that has these orthonormal columns, 00:06:45.120 --> 00:06:47.370 So I took the dot product -- ye, somehow, it didn't -- 00:06:47.370 --> 00:06:52.010 I we know that the inverse is the same as the transpose. 00:06:52.010 --> 00:06:56.710 So here is the beautiful -- 00:06:56.710 --> 00:06:59.150 there is the -- the great haven't learned anything. 00:06:59.150 --> 00:07:03.900 description, the factorization of a symmetric matrix. 00:07:03.900 --> 00:07:06.300 And this is, like, one of the famous theorems 00:07:06.300 --> 00:07:10.230 of linear algebra, that if I have a symmetric matrix, 00:07:10.230 --> 00:07:14.160 it can be factored in this form. 00:07:14.160 --> 00:07:17.230 An orthogonal matrix times diagonal times 00:07:17.230 --> 00:07:20.270 the transpose of that orthogonal matrix. 00:07:20.270 --> 00:07:24.160 And, of course, everybody immediately says yes, 00:07:24.160 --> 00:07:28.510 and if this is possible, then that's 00:07:28.510 --> 00:07:29.780 clearly symmetric, right? 00:07:29.780 --> 00:07:33.890 That -- take -- we've looked at products of three guys like 00:07:33.890 --> 00:07:38.550 that and taken their transpose and we got it back again. 00:07:38.550 --> 00:07:42.060 So do you -- do you see the beauty of this -- 00:07:42.060 --> 00:07:45.030 of this factorization, then? 00:07:45.030 --> 00:07:49.800 It -- it completely displays the eigenvalues and eigenvectors 00:07:49.800 --> 00:07:53.760 the symmetry of the -- of the whole thing, because -- 00:07:53.760 --> 00:07:56.990 that product, Q times lambda times Q transpose, 00:07:56.990 --> 00:08:02.200 if I transpose it, it -- this comes in this position and we 00:08:02.200 --> 00:08:04.260 get that matrix back again. 00:08:04.260 --> 00:08:08.450 So that's -- in mathematics, that's called the spectral 00:08:08.450 --> 00:08:12.260 Spectrum is the set of eigenvalues of a matrix. 00:08:12.260 --> 00:08:12.900 theorem. 00:08:12.900 --> 00:08:17.410 Spec- it somehow comes from the idea of the spectrum of light 00:08:17.410 --> 00:08:22.330 as a combination of pure things -- 00:08:22.330 --> 00:08:28.060 where our matrix is broken down into pure eigenvalues 00:08:28.060 --> 00:08:30.100 and eigenvectors -- 00:08:30.100 --> 00:08:35.940 in mechanics it's often called the principle axis theorem. 00:08:35.940 --> 00:08:37.100 It's very useful. 00:08:37.100 --> 00:08:40.080 It means that if you have -- 00:08:40.080 --> 00:08:42.730 we'll see it geometrically. 00:08:42.730 --> 00:08:45.020 It means that if I have some material -- 00:08:45.020 --> 00:08:49.960 if I look at the right axis, it becomes diagonal, it becomes -- 00:08:49.960 --> 00:08:50.460 the -- the 00:08:50.460 --> 00:08:51.530 I- I've done something dumb, because I've got the -- 00:08:51.530 --> 00:08:52.700 I should've taken the dot product of this guy here with 00:08:52.700 --> 00:08:55.180 -- that's directions don't couple together. 00:08:55.180 --> 00:08:55.920 Okay. 00:08:55.920 --> 00:08:59.420 So that's -- that -- that's what to remember from -- 00:08:59.420 --> 00:09:00.770 from this lecture. 00:09:00.770 --> 00:09:06.020 Now, I would like to say why are the eigenvalues real? 00:09:06.020 --> 00:09:07.080 Can I do that? 00:09:07.080 --> 00:09:10.200 So -- so -- because that -- something useful comes out. 00:09:10.200 --> 00:09:16.251 So I'll just come back -- come to that question why real 00:09:16.251 --> 00:09:16.750 eigenvalues? 00:09:22.910 --> 00:09:24.120 Okay. 00:09:24.120 --> 00:09:25.920 So I have to start from the only thing 00:09:25.920 --> 00:09:28.435 we know, Ax equal lambda x. 00:09:33.710 --> 00:09:34.560 Okay. 00:09:34.560 --> 00:09:37.090 But as far as I know at this moment, 00:09:37.090 --> 00:09:40.100 lambda could be complex. 00:09:40.100 --> 00:09:44.570 I'm going to prove it's not -- and x could be complex. 00:09:44.570 --> 00:09:48.370 In fact, for the moment, even A could be -- 00:09:48.370 --> 00:09:51.120 we could even think, well, what happens if A is complex? 00:09:51.120 --> 00:09:53.750 Well, one thing we can always do -- this is -- 00:09:53.750 --> 00:09:56.000 this is like always -- 00:09:56.000 --> 00:09:58.080 always okay -- 00:09:58.080 --> 00:10:03.030 I can -- if I have an equation, I can take the complex 00:10:03.030 --> 00:10:05.130 conjugate of everything. 00:10:05.130 --> 00:10:09.610 That's -- no -- no -- so A conjugate x conjugate equal 00:10:09.610 --> 00:10:16.036 lambda conjugate x conjugate, it just means that everywhere over 00:10:16.036 --> 00:10:18.410 here that there was a -- an equals x bar transpose lambda 00:10:18.410 --> 00:10:21.780 bar x bar. i, then here I changed it to a-i. 00:10:21.780 --> 00:10:24.500 That's -- that -- you know that that step -- 00:10:24.500 --> 00:10:28.100 that conjugate business, that a+ib, 00:10:28.100 --> 00:10:31.650 if I conjugate it it's a-ib. 00:10:31.650 --> 00:10:36.820 That's the meaning of conjugate -- and products behave right, 00:10:36.820 --> 00:10:39.390 I can conjugate every factor. 00:10:39.390 --> 00:10:43.640 So I haven't done anything yet except to say what would be 00:10:43.640 --> 00:10:53.200 true if, x -- in any case, even if x and lambda were complex. 00:10:53.200 --> 00:10:56.130 Of course, our -- we're speaking about real matrixes A, 00:10:56.130 --> 00:10:58.630 so I can take that out. 00:10:58.630 --> 00:11:04.050 Actually, this already tells me something about real matrixes. 00:11:04.050 --> 00:11:07.060 I haven't used any assumption of A -- 00:11:07.060 --> 00:11:08.950 A transpose yet. 00:11:08.950 --> 00:11:12.440 Symmetry is waiting in the wings to be used. 00:11:12.440 --> 00:11:18.580 This tells me that if a real matrix has an eigenvalue lambda 00:11:18.580 --> 00:11:22.040 what I was going to do. and an eigenvector x, it also has -- 00:11:22.040 --> 00:11:25.040 another of its eigenvalues is lambda bar 00:11:25.040 --> 00:11:27.800 with eigenvector x bar. 00:11:27.800 --> 00:11:32.500 Real matrixes, the eigenvalues come in lambda, lambda bar -- 00:11:32.500 --> 00:11:37.500 the complex eigenvalues come in lambda and lambda bar pairs. 00:11:37.500 --> 00:11:39.600 But, of course, I'm aiming to show 00:11:39.600 --> 00:11:44.190 that they're not complex at all, here, by getting symmetry in. 00:11:44.190 --> 00:11:46.100 So how I going to use symmetry? 00:11:46.100 --> 00:11:51.630 I'm going to transpose this equation to x bar 00:11:51.630 --> 00:11:59.040 transpose A transpose equals x bar transpose lambda bar. 00:11:59.040 --> 00:12:03.150 That's just a number, so I don't mind wear I put that number. 00:12:03.150 --> 00:12:05.750 This is -- this is -- 00:12:05.750 --> 00:12:06.480 this is a -- then 00:12:06.480 --> 00:12:08.250 again okay. 00:12:08.250 --> 00:12:10.000 Ax equals lambda x bar transpose x, right? 00:12:10.000 --> 00:12:14.880 But now I'm ready to use symmetry. 00:12:14.880 --> 00:12:18.890 I'm ready -- so this was all just mechanics. 00:12:18.890 --> 00:12:22.040 Now -- now comes the moment to say, okay, 00:12:22.040 --> 00:12:24.040 if the matrix is this from the right with x bar, 00:12:24.040 --> 00:12:25.665 I get x bar transpose Ax bar symmetric, 00:12:25.665 --> 00:12:29.030 then this A transpose is the same as A. 00:12:29.030 --> 00:12:31.910 You see, at that moment I used the assumption. 00:12:31.910 --> 00:12:34.380 Now let me finish the discussion. 00:12:34.380 --> 00:12:37.610 Here -- here's the way I finish. 00:12:37.610 --> 00:12:42.460 I look at this original equation and I take the inner product. 00:12:42.460 --> 00:12:45.170 I multiply both sides by -- 00:12:45.170 --> 00:12:46.680 oh, maybe I'll do it with this one. 00:12:49.260 --> 00:12:51.540 I take -- 00:12:51.540 --> 00:12:55.200 I multiply both sides by x bar transpose. 00:12:55.200 --> 00:12:58.670 x bar transpose Ax bar equals lambda 00:12:58.670 --> 00:13:03.620 bar x bar transpose x bar. 00:13:03.620 --> 00:13:04.470 Okay, fine. 00:13:08.330 --> 00:13:12.170 All right, now what's the other one? 00:13:12.170 --> 00:13:16.880 Oh, for the other one I'll probably use this guy. 00:13:16.880 --> 00:13:20.580 A- I happy about this? 00:13:20.580 --> 00:13:21.080 No. 00:13:21.080 --> 00:13:22.910 For some reason I'm not. 00:13:22.910 --> 00:13:26.830 I'm -- I want to -- 00:13:26.830 --> 00:14:04.850 if I take the inner product of Okay. 00:14:24.690 --> 00:14:26.660 So that -- that was -- that's fine. 00:14:26.660 --> 00:14:30.230 That comes directly from that, multiplying both sides by x bar 00:14:30.230 --> 00:14:33.860 transpose, but now I'd like to get -- 00:14:33.860 --> 00:14:38.320 why do I have x bars over there? 00:14:38.320 --> 00:14:40.860 Oh, yes. 00:14:40.860 --> 00:14:43.161 Forget this. 00:14:43.161 --> 00:14:43.660 Okay. 00:14:43.660 --> 00:14:45.820 On this one -- right. 00:14:45.820 --> 00:14:49.010 On this one, I took it like that, I multiply on the right 00:14:49.010 --> 00:14:50.400 by x. 00:14:50.400 --> 00:14:54.080 That's the idea. 00:14:54.080 --> 00:14:55.540 Okay. 00:14:55.540 --> 00:15:02.150 Now why I happier with this situation now? 00:15:02.150 --> 00:15:04.180 A proof is coming here. 00:15:04.180 --> 00:15:10.140 Because I compare this guy with this one. 00:15:10.140 --> 00:15:12.570 And they have the same left hand side. 00:15:12.570 --> 00:15:14.410 So they have the same right hand side. 00:15:14.410 --> 00:15:16.240 So comparing those two, can -- 00:15:16.240 --> 00:15:19.910 I'll raise the board to do this comparison -- 00:15:19.910 --> 00:15:25.150 this thing, lambda x bar transpose x 00:15:25.150 --> 00:15:32.880 is equal to lambda bar x bar transpose x. 00:15:32.880 --> 00:15:33.380 Okay. 00:15:35.890 --> 00:15:38.190 And the conclusion I'm going to reach -- 00:15:38.190 --> 00:15:43.280 I -- I on the right track here? 00:15:43.280 --> 00:15:44.830 The conclusion I'm going to reach 00:15:44.830 --> 00:15:47.060 is lambda equal lambda bar. 00:15:52.170 --> 00:15:55.330 I would have to track down the other possibility that this -- 00:15:55.330 --> 00:15:58.400 this thing is zero, but let me -- 00:15:58.400 --> 00:16:01.330 oh -- oh, yes, that's important. 00:16:01.330 --> 00:16:03.370 It's not zero. 00:16:03.370 --> 00:16:08.760 So once I know that this isn't zero, I just cancel it 00:16:08.760 --> 00:16:11.220 and I learn that lambda equals lambda bar. 00:16:11.220 --> 00:16:14.360 And so what can you -- do you -- 00:16:14.360 --> 00:16:17.823 have you got the reasoning altogether? 00:16:20.640 --> 00:16:24.130 What does this tell us? 00:16:24.130 --> 00:16:27.750 Lambda's an eigenvalue of this symmetric matrix. 00:16:27.750 --> 00:16:30.300 We've just proved that it equaled lambda bar, 00:16:30.300 --> 00:16:34.630 so we have just proved that lambda is real, 00:16:34.630 --> 00:16:35.710 right? 00:16:35.710 --> 00:16:39.980 If, if a number is equal to its own complex conjugate, 00:16:39.980 --> 00:16:42.290 then there's no imaginary part at all. 00:16:42.290 --> 00:16:43.280 The number is real. 00:16:43.280 --> 00:16:45.435 So lambda is real. 00:16:48.260 --> 00:16:49.380 Good. 00:16:49.380 --> 00:16:51.740 Good. 00:16:51.740 --> 00:16:56.310 Now, what -- but it depended on this little expression, 00:16:56.310 --> 00:16:59.040 on knowing that that wasn't zero, 00:16:59.040 --> 00:17:03.960 so that I could cancel it out -- so can we just take a second 00:17:03.960 --> 00:17:05.400 on that one? 00:17:05.400 --> 00:17:08.650 Because it's an important quantity. 00:17:08.650 --> 00:17:11.160 x bar transpose x. 00:17:11.160 --> 00:17:18.280 Okay, now remember, as far as we know, x is complex. 00:17:18.280 --> 00:17:20.650 So this is -- 00:17:20.650 --> 00:17:25.260 here -- x is complex, x has these components, x1, 00:17:25.260 --> 00:17:28.310 x2 down to xn. 00:17:28.310 --> 00:17:35.890 And x bar transpose, well, it's transposed and it's conjugated, 00:17:35.890 --> 00:17:42.570 so that's x1 conjugated x2 conjugated up to xn conjugated. 00:17:42.570 --> 00:17:43.520 I'm -- I'm -- 00:17:43.520 --> 00:17:46.390 I'm really reminding you of crucial facts 00:17:46.390 --> 00:17:48.440 about complex numbers that are going 00:17:48.440 --> 00:17:51.230 to come into the next lecture as well as this one. 00:17:51.230 --> 00:17:57.700 So w- what can you tell me about that product -- 00:17:57.700 --> 00:18:01.150 I -- I guess what I'm trying to say is, 00:18:01.150 --> 00:18:05.830 if I had a complex vector, this would be the quantity I would 00:18:05.830 --> 00:18:06.330 -- 00:18:06.330 --> 00:18:07.730 I would like. 00:18:07.730 --> 00:18:09.310 This is the quantity I like. 00:18:09.310 --> 00:18:13.390 I would take the vector times its transpose -- now what -- 00:18:13.390 --> 00:18:17.070 what happens usually if I take a vector -- a -- a -- x transpose 00:18:17.070 --> 00:18:18.070 x? 00:18:18.070 --> 00:18:22.330 I mean, that's a quantity we see all the time, x transpose x. 00:18:22.330 --> 00:18:25.200 That's the length of x squared, right? 00:18:25.200 --> 00:18:28.350 That's this positive length squared, it's Pythagoras, 00:18:28.350 --> 00:18:31.640 it's x1 squared plus x2 squared and so on. 00:18:31.640 --> 00:18:36.100 Now our vector's complex, and you see the effect? 00:18:36.100 --> 00:18:39.090 I'm conjugating one of these guys. 00:18:39.090 --> 00:18:41.180 So now when I do this multiplication, 00:18:41.180 --> 00:18:49.370 I have x1 bar times x1 and x2 bar times x2 and so on. 00:18:49.370 --> 00:18:53.640 So this is an -- this is sum a+ib. 00:18:53.640 --> 00:18:57.960 And this is sum a-ib. 00:18:57.960 --> 00:19:02.440 I mean, what's the point here? 00:19:02.440 --> 00:19:05.450 What's the point -- when I multiply a number by its 00:19:05.450 --> 00:19:11.480 conjugate, a complex number by its conjugate, what do I get? 00:19:11.480 --> 00:19:16.140 I get a n- the -- the imaginary part is gone. 00:19:16.140 --> 00:19:20.680 When I multiply a+ib by its conjugate, what's -- 00:19:20.680 --> 00:19:23.400 what's the result of that -- of each of those separate little 00:19:23.400 --> 00:19:25.210 multiplications? 00:19:25.210 --> 00:19:29.970 There's an a squared and -- and what -- how many -- what's -- 00:19:29.970 --> 00:19:34.010 b squared comes in with a plus or a minus? 00:19:34.010 --> 00:19:35.290 A plus. 00:19:35.290 --> 00:19:39.260 i times minus i is a plus b squared. 00:19:39.260 --> 00:19:41.240 And what about the imaginary part? 00:19:43.830 --> 00:19:46.770 Gone, right? 00:19:46.770 --> 00:19:49.050 An iab and a minus iab. 00:19:49.050 --> 00:19:52.870 So this -- this is the right thing to do. 00:19:52.870 --> 00:20:00.600 If you want a decent answer, then multiply numbers 00:20:00.600 --> 00:20:02.910 by their conjugates. 00:20:02.910 --> 00:20:08.900 Multiply vectors by the conjugates of x transpose. 00:20:08.900 --> 00:20:13.750 So this quantity is positive, this quantity is positive -- 00:20:13.750 --> 00:20:17.460 the whole thing is positive except for the zero vector 00:20:17.460 --> 00:20:23.010 and that allows me to know that this is a positive number, 00:20:23.010 --> 00:20:27.220 which I safely cancel out and I reach the conclusion. 00:20:27.220 --> 00:20:33.290 So actually, in this discussion here, I've done two things. 00:20:33.290 --> 00:20:35.690 If I reached the conclusion that lambda's 00:20:35.690 --> 00:20:38.510 real, which I wanted to do. 00:20:38.510 --> 00:20:41.600 But at the same time, we sort of saw what 00:20:41.600 --> 00:20:43.790 to do if things were complex. 00:20:43.790 --> 00:20:48.490 If a vector is complex, then it's x bar transpose x, 00:20:48.490 --> 00:20:55.350 this is its length squared. 00:20:55.350 --> 00:20:59.510 And as I said, the next lecture Monday, we'll -- 00:20:59.510 --> 00:21:03.220 we'll repeat that this is the right thing and then do 00:21:03.220 --> 00:21:06.280 the right thing for matrixes and all other -- 00:21:06.280 --> 00:21:10.950 all other, complex possibilities. 00:21:10.950 --> 00:21:12.070 Okay. 00:21:12.070 --> 00:21:17.070 But the main point, then, is that the eigenvalues 00:21:17.070 --> 00:21:20.720 of a symmetric matrix, it just -- do you -- do -- 00:21:20.720 --> 00:21:23.620 where did we use symmetry, by the way? 00:21:23.620 --> 00:21:24.860 We used it here, right? 00:21:24.860 --> 00:21:27.870 Let -- can I just -- 00:21:27.870 --> 00:21:31.980 let -- suppose A was a complex. 00:21:31.980 --> 00:21:34.800 Suppose A had been a complex number. 00:21:34.800 --> 00:21:37.230 Could -- could I have made all this work? 00:21:37.230 --> 00:21:40.780 If A was a complex number -- complex matrix, 00:21:40.780 --> 00:21:45.070 then here I should have written A bar. 00:21:45.070 --> 00:21:47.600 I erased the bar because I assumed A was real. 00:21:47.600 --> 00:21:50.560 But now let's suppose for a moment it's not. 00:21:50.560 --> 00:21:55.260 Then when I took this step, what should I have? 00:21:55.260 --> 00:21:56.710 What did I do on that step? 00:21:56.710 --> 00:21:57.820 I transposed. 00:21:57.820 --> 00:22:00.215 So I should have A bar transpose. 00:22:03.560 --> 00:22:05.990 In the symmetric case, that was A, 00:22:05.990 --> 00:22:08.590 and that's what made everything work, right? 00:22:08.590 --> 00:22:12.720 This -- this led immediately to that. 00:22:12.720 --> 00:22:17.840 This one led immediately to this when the matrix was real, 00:22:17.840 --> 00:22:20.300 so that didn't matter, and it was symmetric, 00:22:20.300 --> 00:22:21.790 so that didn't matter. 00:22:21.790 --> 00:22:23.600 Then I got A. 00:22:23.600 --> 00:22:26.760 But -- so now I just get to ask you. 00:22:26.760 --> 00:22:30.900 Suppose the matrix had been complex. 00:22:30.900 --> 00:22:35.620 What's the right equivalent of sym- symmetry? 00:22:38.320 --> 00:22:41.480 So the good matrix -- so here, let me say -- 00:22:41.480 --> 00:22:52.776 good matrixes -- by good I mean real lambdas and perpendicular 00:22:52.776 --> 00:22:53.276 x-s. 00:22:57.310 --> 00:23:02.860 And tell me now, which matrixes are good? 00:23:02.860 --> 00:23:04.800 If they're -- 00:23:04.800 --> 00:23:08.070 If they're real matrixes, the good ones 00:23:08.070 --> 00:23:10.790 are symmetric, because then everything went through. 00:23:10.790 --> 00:23:13.010 The -- so the good -- 00:23:13.010 --> 00:23:15.050 I'm saying now what's good. 00:23:15.050 --> 00:23:17.430 This is -- this is -- these are the good matrixes. 00:23:17.430 --> 00:23:21.030 They have real eigenvalues, perpendicular eigenvectors -- 00:23:21.030 --> 00:23:27.910 good means A equal A transpose if real. 00:23:27.910 --> 00:23:30.560 Then -- then that was what -- our proof worked. 00:23:30.560 --> 00:23:35.690 But if A is complex, all -- our proof will still work provided 00:23:35.690 --> 00:23:38.060 A bar transpose is A. 00:23:38.060 --> 00:23:41.740 Do you see what I'm saying? 00:23:41.740 --> 00:23:47.700 I'm saying if we have complex matrixes and we want to say are 00:23:47.700 --> 00:23:51.330 they -- are they as good as symmetric matrixes, 00:23:51.330 --> 00:23:56.750 then we should not only transpose the thing, 00:23:56.750 --> 00:23:58.760 but conjugate it. 00:23:58.760 --> 00:24:00.800 Those are good matrixes. 00:24:00.800 --> 00:24:03.090 And of course, the most important 00:24:03.090 --> 00:24:06.980 s- the most important case is when they're real, 00:24:06.980 --> 00:24:09.410 this part doesn't matter and I just have 00:24:09.410 --> 00:24:11.330 A equal A transpose symmetric. 00:24:11.330 --> 00:24:12.070 Do you -- I -- 00:24:12.070 --> 00:24:15.260 I'll just repeat that. 00:24:15.260 --> 00:24:20.290 The good matrixes, if complex, are these. 00:24:20.290 --> 00:24:23.590 If real, that doesn't make any difference 00:24:23.590 --> 00:24:25.900 so I'm just saying symmetric. 00:24:25.900 --> 00:24:30.530 And of course, 99% of examples and applications 00:24:30.530 --> 00:24:34.740 to the matrixes are real and we don't have that 00:24:34.740 --> 00:24:38.230 and then symmetric is the key property. 00:24:38.230 --> 00:24:40.950 Okay. 00:24:40.950 --> 00:24:48.170 So that -- that's, these main facts and now let me just -- 00:24:48.170 --> 00:24:53.690 let me just -- so that's this x bar transpose x, okay. 00:24:53.690 --> 00:24:59.210 So I'll just, write it once more in this form. 00:24:59.210 --> 00:25:05.370 So perpendicular orthonormal eigenvectors, real eigenvalues, 00:25:05.370 --> 00:25:08.410 transposes of orthonormal eigenvectors. 00:25:08.410 --> 00:25:13.690 That's the symmetric case, A equal A transpose. 00:25:13.690 --> 00:25:15.320 Okay. 00:25:15.320 --> 00:25:18.030 Good. 00:25:18.030 --> 00:25:23.340 Actually, I'll even take one more step here. 00:25:23.340 --> 00:25:25.860 Suppose -- I -- 00:25:25.860 --> 00:25:29.350 I can break this down to show you really 00:25:29.350 --> 00:25:34.000 what that says about a symmetric matrix. 00:25:34.000 --> 00:25:35.180 I can break that down. 00:25:35.180 --> 00:25:40.110 Let me here -- here go these eigenvectors. 00:25:40.110 --> 00:25:46.540 I -- here go these eigenvalues, lambda one, lambda two and so 00:25:46.540 --> 00:25:47.130 on. 00:25:47.130 --> 00:25:50.355 Here go these eigenvectors transposed. 00:25:54.870 --> 00:25:59.640 And what happens if I actually do out that multiplication? 00:25:59.640 --> 00:26:03.630 Do you see what will happen? 00:26:03.630 --> 00:26:07.700 There's lambda one times q1 transpose. 00:26:07.700 --> 00:26:11.850 So the first row here is just lambda one q1 transpose. 00:26:11.850 --> 00:26:15.650 If I multiply column times row -- 00:26:15.650 --> 00:26:17.850 you remember I could do that? 00:26:17.850 --> 00:26:24.180 When I multiply matrixes, I can multiply columns times rows? 00:26:24.180 --> 00:26:27.130 So when I do that, I get lambda one and then 00:26:27.130 --> 00:26:31.900 the column and then the row and then 00:26:31.900 --> 00:26:34.885 lambda two and then the column and the row. 00:26:41.770 --> 00:26:46.980 Every symmetric matrix breaks up into these pieces. 00:26:46.980 --> 00:26:54.950 So these pieces have real lambdas and they have these 00:26:54.950 --> 00:26:56.833 Eigen -- these orthonormal eigenvectors. 00:27:00.490 --> 00:27:04.560 And, maybe you even could tell me what kind of a matrix 00:27:04.560 --> 00:27:07.370 have I got there? 00:27:07.370 --> 00:27:12.750 Suppose I take a unit vector times its transpose? 00:27:12.750 --> 00:27:17.620 So column times row, I'm getting a matrix. 00:27:17.620 --> 00:27:22.040 That's a matrix with a special name. 00:27:22.040 --> 00:27:24.290 What's it's -- what kind of a matrix is it? 00:27:24.290 --> 00:27:27.860 We've seen those matrixes, now, in chapter four. 00:27:27.860 --> 00:27:32.440 It's -- is A A transpose with a unit vector, 00:27:32.440 --> 00:27:36.210 so I don't have to divide by A transpose A. 00:27:36.210 --> 00:27:40.580 That matrix is a projection matrix. 00:27:40.580 --> 00:27:41.920 That's a projection matrix. 00:27:41.920 --> 00:27:46.550 It's symmetric and if I square it there'll be another -- 00:27:46.550 --> 00:27:50.040 there'll be a q1 transpose q1, which is one. 00:27:50.040 --> 00:27:53.820 So I'll get that matrix back again. 00:27:53.820 --> 00:27:57.070 Every -- so every symmetric matrix -- 00:27:57.070 --> 00:28:06.740 every symmetric matrix is a combination of -- 00:28:06.740 --> 00:28:14.230 of mutually perpendicular -- so perpendicular projection 00:28:14.230 --> 00:28:15.500 matrixes. 00:28:15.500 --> 00:28:17.442 Projection matrixes. 00:28:20.610 --> 00:28:21.530 Okay. 00:28:21.530 --> 00:28:23.940 That's another way that people like 00:28:23.940 --> 00:28:27.770 to think of the spectral theorem, 00:28:27.770 --> 00:28:31.790 that every symmetric matrix can be broken up that way. 00:28:31.790 --> 00:28:35.220 That -- I guess at this moment -- 00:28:35.220 --> 00:28:36.800 first I haven't done an example. 00:28:36.800 --> 00:28:41.700 I could create a symmetric matrix, check that it's -- 00:28:41.700 --> 00:28:44.430 find its eigenvalues, they would come out real, 00:28:44.430 --> 00:28:47.710 find its eigenvectors, they would come out perpendicular 00:28:47.710 --> 00:28:51.230 and you would see it in numbers, but maybe I'll leave it here 00:28:51.230 --> 00:28:54.650 for the moment in letters. 00:28:54.650 --> 00:28:58.520 Oh, I -- maybe I will do it with numbers, for this reason. 00:28:58.520 --> 00:29:02.720 Because there's one more remarkable fact. 00:29:02.720 --> 00:29:05.730 Can I just put this further great fact 00:29:05.730 --> 00:29:07.890 about symmetric matrixes on the board? 00:29:11.720 --> 00:29:13.560 When I have symmetric matrixes, I 00:29:13.560 --> 00:29:18.140 know their eigenvalues are So then I can get interested 00:29:18.140 --> 00:29:22.040 in the question are they positive real. or negative? 00:29:22.040 --> 00:29:23.980 And you remember why that's important. 00:29:23.980 --> 00:29:27.740 For differential equations, that decides between instability 00:29:27.740 --> 00:29:29.920 and stability. 00:29:29.920 --> 00:29:32.620 So I'm -- after I know they're real, 00:29:32.620 --> 00:29:34.970 then the next question is are they positive, 00:29:34.970 --> 00:29:37.480 are they negative? 00:29:37.480 --> 00:29:44.140 And I hate to have to compute those eigenvalues to answer 00:29:44.140 --> 00:29:46.120 that question, right? 00:29:46.120 --> 00:29:49.120 Because computing the eigenvalues of a symmetric 00:29:49.120 --> 00:29:51.230 matrix of order let's say 50 -- 00:29:51.230 --> 00:29:53.930 compute its 50 eigenvalues -- 00:29:53.930 --> 00:29:58.630 is a job. 00:29:58.630 --> 00:30:04.050 I mean, by pencil and paper it's a lifetime's job. 00:30:04.050 --> 00:30:11.480 I mean, which -- and in fact, a few years ago -- well, say, 00:30:11.480 --> 00:30:17.660 20 years ago, or 30, nobody really knew how to do it. 00:30:17.660 --> 00:30:22.140 I mean, so, like, science was stuck on this problem. 00:30:22.140 --> 00:30:24.810 If you have a matrix of order 50 or 100, 00:30:24.810 --> 00:30:27.320 how do you find its eigenvalues? 00:30:27.320 --> 00:30:29.530 Numerically, now, I'm just saying, 00:30:29.530 --> 00:30:34.330 because pencil and paper is -- we're going to run out of time 00:30:34.330 --> 00:30:36.380 or paper or something before we get it. 00:30:38.970 --> 00:30:41.840 Well -- and you might think, okay, 00:30:41.840 --> 00:30:47.850 get Matlab to compute the determinant of lambda minus A, 00:30:47.850 --> 00:30:52.330 A minus lambda I, this polynomial of 50th degree, 00:30:52.330 --> 00:30:53.465 and then find the roots. 00:30:56.230 --> 00:30:59.340 Matlab will do it, but it will complain, 00:30:59.340 --> 00:31:04.920 because it's a very bad way to find the eigenvalues. 00:31:04.920 --> 00:31:08.480 I'm sorry to be saying this, because it's the way I 00:31:08.480 --> 00:31:10.220 taught you to do it, right? 00:31:10.220 --> 00:31:12.000 I taught you to find the eigenvalues 00:31:12.000 --> 00:31:14.830 by doing that determinant and taking 00:31:14.830 --> 00:31:16.790 the roots of that polynomial. 00:31:16.790 --> 00:31:20.030 But now I'm saying, okay, I really meant that for two 00:31:20.030 --> 00:31:21.820 by twos and three by threes but I 00:31:21.820 --> 00:31:24.680 didn't mean you to do it on a 50 by 50 00:31:24.680 --> 00:31:27.980 and you're not too unhappy, probably, 00:31:27.980 --> 00:31:29.620 because you didn't want to do it. 00:31:29.620 --> 00:31:36.650 But -- good, because it would be a very unstable way -- 00:31:36.650 --> 00:31:40.590 the 50 answers that would come out would be highly unreliable. 00:31:40.590 --> 00:31:45.590 So, new ways are -- are much better to find those 50 00:31:45.590 --> 00:31:46.310 eigenvalues. 00:31:46.310 --> 00:31:50.270 That's a -- that's a part of numerical linear algebra. 00:31:50.270 --> 00:31:54.850 But here's the remarkable fact -- 00:31:54.850 --> 00:32:00.170 that Matlab would quite happily find the 50 pivots, right? 00:32:00.170 --> 00:32:03.720 Now the pivots are not the same as the eigenvalues. 00:32:03.720 --> 00:32:06.440 But here's the great thing. 00:32:06.440 --> 00:32:11.070 If I had a real matrix, I could find those 50 pivots 00:32:11.070 --> 00:32:14.340 and I could see maybe 28 of them are positive 00:32:14.340 --> 00:32:15.680 and 22 are negative 00:32:15.680 --> 00:32:16.660 pivots. 00:32:16.660 --> 00:32:20.150 And I can compute those safely and quickly. 00:32:20.150 --> 00:32:23.900 And the great fact is that 28 of the eigenvalues would be 00:32:23.900 --> 00:32:27.410 positive and 22 would be negative -- 00:32:27.410 --> 00:32:31.740 that the sines of the pivots -- so this is, like -- 00:32:31.740 --> 00:32:34.780 I hope you think this -- this is kind of a nice thing, 00:32:34.780 --> 00:32:39.970 that the sines of the pivots -- 00:32:39.970 --> 00:32:42.650 for symmetric, I'm always talking about symmetric 00:32:42.650 --> 00:32:43.700 matrixes -- 00:32:43.700 --> 00:32:45.890 so I'm really, like, trying to convince you 00:32:45.890 --> 00:32:50.100 that symmetric matrixes are better than the rest. 00:32:50.100 --> 00:32:58.700 So the sines of the pivots are same as the sines 00:32:58.700 --> 00:33:02.070 of the eigenvalues. 00:33:02.070 --> 00:33:04.780 The same number. 00:33:04.780 --> 00:33:10.490 The number of pivots greater than zero, 00:33:10.490 --> 00:33:16.120 the number of positive pivots is equal to the number 00:33:16.120 --> 00:33:21.330 of positive eigenvalues. 00:33:21.330 --> 00:33:25.750 So that, actually, is a very useful -- that gives you a g- 00:33:25.750 --> 00:33:31.340 a good start on a decent way to compute eigenvalues, 00:33:31.340 --> 00:33:33.470 because you can narrow them down, 00:33:33.470 --> 00:33:35.410 you can find out how many are positive, 00:33:35.410 --> 00:33:37.490 how many are negative. 00:33:37.490 --> 00:33:42.680 Then you could shift the matrix by seven times the identity. 00:33:42.680 --> 00:33:46.420 That would shift all the eigenvalues by seven. 00:33:46.420 --> 00:33:48.500 Then you could take the pivots of that matrix 00:33:48.500 --> 00:33:52.700 and you would know how many eigenvalues of the original 00:33:52.700 --> 00:33:53.760 were above seven and 00:33:53.760 --> 00:33:54.870 below seven. 00:33:54.870 --> 00:33:59.070 So this -- this neat little theorem, that, 00:33:59.070 --> 00:34:05.490 symmetric matrixes have this connection between the -- 00:34:05.490 --> 00:34:09.150 nobody's mixing up and thinking the pivots are the eigenvalues 00:34:09.150 --> 00:34:10.820 -- 00:34:10.820 --> 00:34:13.050 I mean, the only thing I can think of 00:34:13.050 --> 00:34:15.880 is the product of the pivots equals 00:34:15.880 --> 00:34:19.210 the product of the eigenvalues, why is that? 00:34:19.210 --> 00:34:21.139 So if I asked you for the reason on that, 00:34:21.139 --> 00:34:24.679 why is the product of the pivots for a symmetric matrix 00:34:24.679 --> 00:34:27.639 the same as the product of the eigenvalues? 00:34:27.639 --> 00:34:34.500 Because they both equal the determinant. 00:34:34.500 --> 00:34:35.000 Right. 00:34:35.000 --> 00:34:37.060 The product of the pivots gives the determinant 00:34:37.060 --> 00:34:40.710 if no row exchanges, the product of the eigenvalues 00:34:40.710 --> 00:34:42.340 always gives the determinant. 00:34:42.340 --> 00:34:47.100 So -- so the products -- but that doesn't tell you anything 00:34:47.100 --> 00:34:51.020 about the 50 individual ones, which this does. 00:34:51.020 --> 00:34:51.810 Okay. 00:34:51.810 --> 00:34:57.566 So that's -- those are essential facts about symmetric matrixes. 00:34:57.566 --> 00:34:58.066 Okay. 00:35:00.930 --> 00:35:06.410 Now I -- I said in the -- in the lecture description that I 00:35:06.410 --> 00:35:13.680 would take the last minutes to start on positive definite 00:35:13.680 --> 00:35:15.970 matrixes, because we're right there, 00:35:15.970 --> 00:35:22.506 we're ready to say what's a positive definite matrix? 00:35:31.740 --> 00:35:34.470 It's symmetric, first of all. 00:35:34.470 --> 00:35:37.090 On -- always I will mean symmetric. 00:35:40.090 --> 00:35:43.690 So this is the -- this is the next section of the book. 00:35:43.690 --> 00:35:45.420 It's about this -- 00:35:45.420 --> 00:35:50.030 if symmetric matrixes are good, which was, like, 00:35:50.030 --> 00:35:54.020 the point of my lecture so far, then positive, 00:35:54.020 --> 00:35:57.650 definite matrixes are -- 00:35:57.650 --> 00:36:03.180 a subclass that are excellent, okay. 00:36:03.180 --> 00:36:05.430 Just the greatest. 00:36:05.430 --> 00:36:07.380 so what are they? 00:36:07.380 --> 00:36:10.930 They're matrixes -- they're symmetric matrixes, 00:36:10.930 --> 00:36:13.240 so all their eigenvalues are real. 00:36:13.240 --> 00:36:15.420 You can guess what they are. 00:36:15.420 --> 00:36:20.070 These are symmetric matrixes with all -- 00:36:20.070 --> 00:36:21.190 the eigenvalues are -- 00:36:25.790 --> 00:36:27.270 okay, tell me what to write. 00:36:31.240 --> 00:36:34.040 What -- well, it -- it's hinted, of course, 00:36:34.040 --> 00:36:36.200 by the name for these things. 00:36:36.200 --> 00:36:39.750 All the eigenvalues are positive. 00:36:39.750 --> 00:36:40.250 Okay. 00:36:45.080 --> 00:36:46.900 Tell me about the pivots. 00:36:46.900 --> 00:36:50.200 We can check the eigenvalues or we can check the pivots. 00:36:50.200 --> 00:36:53.270 All the pivots are what? 00:36:58.430 --> 00:36:59.730 And then I'll -- 00:36:59.730 --> 00:37:01.230 then I'll finally give an example. 00:37:01.230 --> 00:37:04.460 I feel awful that I have got to this point in the lecture 00:37:04.460 --> 00:37:05.920 and I haven't given you a single 00:37:05.920 --> 00:37:06.790 example. 00:37:06.790 --> 00:37:08.690 So let me give you one. 00:37:08.690 --> 00:37:13.880 Five three two two. 00:37:13.880 --> 00:37:17.460 That's symmetric, fine. 00:37:17.460 --> 00:37:21.620 It's eigenvalues are real, for sure. 00:37:21.620 --> 00:37:27.850 But more than that, I know the sines of those eigenvalues. 00:37:27.850 --> 00:37:32.440 And also I know the sines of those pivots, 00:37:32.440 --> 00:37:34.430 so what's the deal with the pivots? 00:37:34.430 --> 00:37:39.800 The Ei- if the eigenvalues are all positive and if this little 00:37:39.800 --> 00:37:44.240 fact is true that the pivots and eigenvalues have the same 00:37:44.240 --> 00:37:48.680 sines, then this must be true -- all the pivots are positive. 00:37:51.450 --> 00:37:54.330 And that's the good way to test. 00:37:54.330 --> 00:37:56.090 This is the good test, because I can -- 00:37:56.090 --> 00:37:59.090 what are the pivots for that matrix? 00:37:59.090 --> 00:38:02.610 The pivots for that matrix are five. 00:38:02.610 --> 00:38:08.840 So pivots are five and what's the second pivot? 00:38:08.840 --> 00:38:13.570 Have we, like, noticed the formula for the second pivot 00:38:13.570 --> 00:38:14.560 in a matrix? 00:38:18.332 --> 00:38:19.790 It doesn't necessarily -- you know, 00:38:19.790 --> 00:38:22.200 it may come out a fraction for sure, 00:38:22.200 --> 00:38:24.200 but what is that fraction? 00:38:24.200 --> 00:38:25.070 Can you tell me? 00:38:25.070 --> 00:38:30.070 Well, here, the product of the pivots is the determinant. 00:38:30.070 --> 00:38:31.790 What's the determinant of this matrix? 00:38:34.840 --> 00:38:36.460 Eleven? 00:38:36.460 --> 00:38:41.180 So the second pivot must be eleven over five, 00:38:41.180 --> 00:38:44.600 so that the product is eleven. 00:38:44.600 --> 00:38:47.430 They're both positive. 00:38:47.430 --> 00:38:50.150 Then I know that the eigenvalues of that matrix 00:38:50.150 --> 00:38:51.820 are both positive. 00:38:51.820 --> 00:38:53.140 What are the eigenvalues? 00:38:53.140 --> 00:38:55.550 Well, I've got to take the roots of -- you know, 00:38:55.550 --> 00:38:57.770 do I put in a minus lambda? 00:38:57.770 --> 00:39:03.800 You mentally do this -- lambda squared minus how many lambdas? 00:39:03.800 --> 00:39:04.310 Eight? 00:39:04.310 --> 00:39:04.810 Right. 00:39:04.810 --> 00:39:07.510 Five and three, the trace comes in there, 00:39:07.510 --> 00:39:11.410 plus what number comes here? 00:39:11.410 --> 00:39:14.185 The determinant, the eleven, so I set that to 00:39:14.185 --> 00:39:14.685 zero. 00:39:17.190 --> 00:39:20.190 So the eigenvalues are -- 00:39:20.190 --> 00:39:24.040 let's see, half of that is four, look at that positive number, 00:39:24.040 --> 00:39:28.600 plus or minus the square root of sixteen minus eleven, 00:39:28.600 --> 00:39:29.360 I think five. 00:39:32.480 --> 00:39:35.450 The eigenvalues -- well, two by two they're not so terrible, 00:39:35.450 --> 00:39:37.750 but they're not so perfect. 00:39:37.750 --> 00:39:40.235 Pivots are really simple. 00:39:44.580 --> 00:39:48.370 And this is a -- this is the family of matrixes that you 00:39:48.370 --> 00:39:51.180 really want in differential equations, 00:39:51.180 --> 00:39:55.720 because you know the sines of the eigenvalues, 00:39:55.720 --> 00:39:58.520 so you know the stability or not. 00:39:58.520 --> 00:39:59.340 Okay. 00:39:59.340 --> 00:40:03.700 There's one other related fact I can pop in here in -- 00:40:03.700 --> 00:40:07.745 in the time available for positive definite matrixes. 00:40:10.380 --> 00:40:14.737 The related fact is to ask you about determinants. 00:40:14.737 --> 00:40:15.820 So what's the determinant? 00:40:24.990 --> 00:40:27.470 What can you tell me if I -- remember, 00:40:27.470 --> 00:40:32.090 positive definite means all eigenvalues are positive, 00:40:32.090 --> 00:40:34.710 all pivots are positive, so what can you tell me about 00:40:34.710 --> 00:40:36.890 the determinant? 00:40:36.890 --> 00:40:40.240 It's positive, too. 00:40:40.240 --> 00:40:45.070 But somehow that -- that's not quite enough. 00:40:45.070 --> 00:40:50.730 Here -- here's a matrix minus one minus three, 00:40:50.730 --> 00:40:54.330 what's the determinant of that guy? 00:40:54.330 --> 00:40:55.880 It's positive, right? 00:40:55.880 --> 00:40:58.010 Is this a positive, definite matrix? 00:40:58.010 --> 00:41:00.000 Are the pivots -- what are the pivots? 00:41:00.000 --> 00:41:02.230 Well, negative. 00:41:02.230 --> 00:41:03.400 What are the eigenvalues? 00:41:03.400 --> 00:41:05.470 Well, they're also the same. 00:41:05.470 --> 00:41:12.240 So somehow I don't just want the determinant of the whole 00:41:12.240 --> 00:41:12.850 matrix. 00:41:12.850 --> 00:41:14.770 Here is eleven, that's great. 00:41:14.770 --> 00:41:16.540 Here the determinant of the whole matrix 00:41:16.540 --> 00:41:20.220 is three, that's positive. 00:41:20.220 --> 00:41:26.450 I also -- I've got to check, like, little sub-determinants, 00:41:26.450 --> 00:41:29.330 say maybe coming down from the left. 00:41:29.330 --> 00:41:32.950 So the one by one and the two by two have to be positive. 00:41:32.950 --> 00:41:36.840 So there -- that's where I get the all. 00:41:36.840 --> 00:41:41.140 All -- can I call them sub-determinants -- 00:41:41.140 --> 00:41:43.400 are -- see, I have to -- 00:41:43.400 --> 00:41:45.670 I need to make the thing plural. 00:41:45.670 --> 00:41:51.230 I need to test n things, not just the big determinant. 00:41:51.230 --> 00:41:55.130 All sub-determinants are positive. 00:41:55.130 --> 00:41:58.110 Then I'm okay. 00:41:58.110 --> 00:42:00.610 Then I'm okay. 00:42:00.610 --> 00:42:03.310 This passes the test. 00:42:03.310 --> 00:42:06.830 Five is positive and eleven is positive. 00:42:06.830 --> 00:42:12.220 This fails the test because that minus one there is negative. 00:42:12.220 --> 00:42:16.050 And then the big determinant is positive three. 00:42:16.050 --> 00:42:18.800 So t- this -- 00:42:18.800 --> 00:42:23.320 these -- this fact -- you see that actually the course, like, 00:42:23.320 --> 00:42:24.110 coming together. 00:42:27.210 --> 00:42:29.000 And that's really my point now. 00:42:29.000 --> 00:42:33.850 In the next -- in this lecture and particularly next Wednesday 00:42:33.850 --> 00:42:38.150 and Friday, the course comes together. 00:42:38.150 --> 00:42:41.810 These pivots that we met in the first week, 00:42:41.810 --> 00:42:46.180 these determinants that we met in the middle of the course, 00:42:46.180 --> 00:42:50.810 these eigenvalues that we met most recently -- 00:42:50.810 --> 00:42:56.010 all matrixes are square here, so coming together for square 00:42:56.010 --> 00:43:00.150 matrixes means these three pieces come together and they 00:43:00.150 --> 00:43:05.210 come together in that beautiful fact, that if -- 00:43:05.210 --> 00:43:07.430 that all the -- that if I have one of these, 00:43:07.430 --> 00:43:09.150 I have the others. 00:43:09.150 --> 00:43:10.140 That if I -- 00:43:10.140 --> 00:43:12.180 but for symmetric matrixes. 00:43:12.180 --> 00:43:17.570 So that -- this will be the positive definite section 00:43:17.570 --> 00:43:22.540 and then the real climax of the course is to make everything 00:43:22.540 --> 00:43:27.150 come together for n by n matrixes, 00:43:27.150 --> 00:43:30.330 not necessarily symmetric -- 00:43:30.330 --> 00:43:33.070 bring everything together there and that 00:43:33.070 --> 00:43:34.850 will be the final thing. 00:43:34.850 --> 00:43:35.490 Okay. 00:43:35.490 --> 00:43:38.970 So have a great weekend and don't 00:43:38.970 --> 00:43:40.710 forget symmetric matrixes. 00:43:40.710 --> 00:43:42.260 Thanks.