WEBVTT 00:00:07.380 --> 00:00:08.230 OK. 00:00:08.230 --> 00:00:16.010 So this is the first lecture on eigenvalues and eigenvectors, 00:00:16.010 --> 00:00:19.250 and that's a big subject that will take up 00:00:19.250 --> 00:00:23.400 most of the rest of the course. 00:00:23.400 --> 00:00:31.500 It's, again, matrices are square and we're looking now 00:00:31.500 --> 00:00:34.820 for some special numbers, the eigenvalues, 00:00:34.820 --> 00:00:38.620 and some special vectors, the eigenvectors. 00:00:38.620 --> 00:00:45.360 And so this lecture is mostly about what are these numbers, 00:00:45.360 --> 00:00:48.850 and then the other lectures are about how do we use them, 00:00:48.850 --> 00:00:50.440 why do we want them. 00:00:50.440 --> 00:00:54.200 OK, so what's an eigenvector? 00:00:54.200 --> 00:00:56.480 Maybe I'll start with eigenvector. 00:00:56.480 --> 00:00:57.720 What's an eigenvector? 00:00:57.720 --> 00:01:02.900 So I have a matrix A. 00:01:02.900 --> 00:01:05.570 OK. 00:01:05.570 --> 00:01:07.430 What does a matrix do? 00:01:07.430 --> 00:01:09.290 It acts on vectors. 00:01:09.290 --> 00:01:12.250 It multiplies vectors x. 00:01:12.250 --> 00:01:21.110 So the way that matrix acts is in goes a vector x and out 00:01:21.110 --> 00:01:22.690 comes a vector Ax. 00:01:22.690 --> 00:01:23.930 It's like a function. 00:01:23.930 --> 00:01:27.870 With a function in calculus, in goes 00:01:27.870 --> 00:01:31.360 a number x, out comes f(x). 00:01:31.360 --> 00:01:34.960 Here in linear algebra we're up in more dimensions. 00:01:34.960 --> 00:01:39.350 In goes a vector x, out comes a vector Ax. 00:01:39.350 --> 00:01:43.060 And the vectors I'm specially interested in 00:01:43.060 --> 00:01:47.630 are the ones the come out in the same direction 00:01:47.630 --> 00:01:50.050 that they went in. 00:01:50.050 --> 00:01:52.050 That won't be typical. 00:01:52.050 --> 00:01:59.270 Most vectors, Ax is in -- points in some different direction. 00:01:59.270 --> 00:02:04.456 But there are certain vectors where Ax comes out parallel 00:02:04.456 --> 00:02:04.955 to x. 00:02:07.460 --> 00:02:09.660 And those are the eigenvectors. 00:02:09.660 --> 00:02:12.400 So Ax parallel to x. 00:02:18.130 --> 00:02:19.465 Those are the eigenvectors. 00:02:23.800 --> 00:02:25.990 And what do I mean by parallel? 00:02:25.990 --> 00:02:29.620 Oh, much easier to just state it in an equation. 00:02:29.620 --> 00:02:36.340 Ax is some multiple -- and everybody calls that multiple 00:02:36.340 --> 00:02:38.900 lambda -- of x. 00:02:38.900 --> 00:02:40.170 That's our big equation. 00:02:43.890 --> 00:02:46.780 We look for special vectors -- and remember most vectors 00:02:46.780 --> 00:02:49.780 won't be eigenvectors -- 00:02:49.780 --> 00:02:53.740 that -- for which Ax is in the same direction as x, 00:02:53.740 --> 00:02:57.380 and by same direction I allow it to be the very opposite 00:02:57.380 --> 00:03:03.830 direction, I allow lambda to be negative or zero. 00:03:03.830 --> 00:03:07.950 Well, I guess we've met the eigenvectors that 00:03:07.950 --> 00:03:10.600 have eigenvalue zero. 00:03:10.600 --> 00:03:13.600 Those are in the same direction, but they're -- 00:03:13.600 --> 00:03:18.020 in a kind of very special way. 00:03:18.020 --> 00:03:22.050 So this -- the eigenvector x. 00:03:22.050 --> 00:03:24.830 Lambda, whatever this multiplying factor 00:03:24.830 --> 00:03:31.750 is, whether it's six or minus six or zero or even 00:03:31.750 --> 00:03:34.800 some imaginary number, that's the eigenvalue. 00:03:34.800 --> 00:03:38.350 So there's the eigenvalue, there's the eigenvector. 00:03:38.350 --> 00:03:42.960 Let's just take a second on eigenvalue zero. 00:03:42.960 --> 00:03:45.630 From the point of view of eigenvalues, that's no special 00:03:45.630 --> 00:03:46.140 deal. 00:03:46.140 --> 00:03:48.910 That's, we have an eigenvector. 00:03:48.910 --> 00:03:51.240 If the eigenvalue happened to be zero, 00:03:51.240 --> 00:03:56.380 that would mean that Ax was zero x, in other words zero. 00:03:56.380 --> 00:04:00.280 So what would x, where would we look for -- what are the x-s? 00:04:00.280 --> 00:04:04.460 What are the eigenvectors with eigenvalue zero? 00:04:04.460 --> 00:04:08.200 They're the guys in the null space, Ax equals zero. 00:04:08.200 --> 00:04:12.540 So if our matrix is singular, let me write this down. 00:04:12.540 --> 00:04:23.210 If, if A is singular, then that -- what does singular mean? 00:04:23.210 --> 00:04:26.015 It means that it takes some vector x into zero. 00:04:29.190 --> 00:04:31.570 Some non-zero vector, that's why -- 00:04:31.570 --> 00:04:33.930 will be the eigenvector into zero. 00:04:33.930 --> 00:04:38.945 Then lambda equals zero is an eigenvalue. 00:04:42.530 --> 00:04:46.860 But we're interested in all eigenvalues now, 00:04:46.860 --> 00:04:50.480 lambda equals zero is not, like, so special anymore. 00:04:50.480 --> 00:04:51.140 OK. 00:04:51.140 --> 00:04:57.530 So the question is, how do we find these x-s and lambdas? 00:04:57.530 --> 00:05:01.830 And notice -- we don't have an equation Ax equal B anymore. 00:05:01.830 --> 00:05:02.830 I can't use elimination. 00:05:05.780 --> 00:05:07.670 I've got two unknowns, and in fact they're 00:05:07.670 --> 00:05:08.860 multiplied together. 00:05:08.860 --> 00:05:12.390 Lambda and x are both unknowns here. 00:05:12.390 --> 00:05:18.680 So, we need to, we need a good idea of how to find them. 00:05:18.680 --> 00:05:23.930 But before I, before I do that, and that's 00:05:23.930 --> 00:05:26.590 where determinant will come in, can I just 00:05:26.590 --> 00:05:29.330 give you some matrices? 00:05:29.330 --> 00:05:31.720 Like here you go. 00:05:31.720 --> 00:05:35.261 Take the matrix, a projection matrix. 00:05:35.261 --> 00:05:35.760 OK. 00:05:35.760 --> 00:05:42.950 So suppose we have a plane and our matrix P is -- 00:05:42.950 --> 00:05:45.450 what I've called A, now I'm going to call it P 00:05:45.450 --> 00:05:47.620 for the moment, because it's -- 00:05:47.620 --> 00:05:51.800 I'm thinking OK, let's look at a then this, 00:05:51.800 --> 00:05:57.270 this other new matrix, I just have an Ax, projection matrix. 00:05:57.270 --> 00:06:01.250 What are the eigenvalues of a projection matrix? 00:06:01.250 --> 00:06:03.240 So that's my question. 00:06:03.240 --> 00:06:07.720 What are the x-s, the eigenvectors, and the lambdas, 00:06:07.720 --> 00:06:13.190 the eigenvalues, thing,4 but the roots of that quadratic for -- 00:06:13.190 --> 00:06:17.170 and now let me say a projection matrix. 00:06:17.170 --> 00:06:19.940 My, my point is that we -- 00:06:19.940 --> 00:06:23.300 before we get into determinants and, and formulas 00:06:23.300 --> 00:06:26.280 and all that stuff, let's take some matrices 00:06:26.280 --> 00:06:28.690 where we know what they do. 00:06:28.690 --> 00:06:34.290 We know that if we take a vector b, what this matrix does 00:06:34.290 --> 00:06:38.480 is it projects it down to Pb. 00:06:38.480 --> 00:06:43.340 So is b an eigenvector in, in that picture? 00:06:43.340 --> 00:06:45.010 Is that vector b an eigenvector? 00:06:48.810 --> 00:06:50.270 No. 00:06:50.270 --> 00:06:53.850 Not so, so b is not an eigenvector 00:06:53.850 --> 00:06:59.090 c- because Pb, its projection, is in a different direction. 00:06:59.090 --> 00:07:04.030 So now tell me what vectors are eigenvectors of P? 00:07:04.030 --> 00:07:09.080 What vectors do get projected in the same direction that they 00:07:09.080 --> 00:07:09.580 start? 00:07:09.580 --> 00:07:12.310 So, so answer, tell me some x-s. 00:07:12.310 --> 00:07:16.590 Do you see what3 so it's if Ax equals lambda x, 00:07:16.590 --> 00:07:21.230 In this picture, where could I start with a vector b or x, 00:07:21.230 --> 00:07:27.070 do its projection, and end up in the same direction? 00:07:27.070 --> 00:07:33.360 Well, that would happen if the vector was right in that plane 00:07:33.360 --> 00:07:34.420 already. 00:07:34.420 --> 00:07:41.510 If the vector x was -- so let the vector x -- 00:07:41.510 --> 00:07:49.810 so any vector, any x in the plane will be an eigenvector. 00:07:49.810 --> 00:07:53.140 And what will happen when I multiply by P, 00:07:53.140 --> 00:07:57.110 when I project a vector x -- 00:07:57.110 --> 00:08:00.190 I called it b here, because this is our familiar picture, 00:08:00.190 --> 00:08:03.650 but now I'm going to say that b was no good for, for the, 00:08:03.650 --> 00:08:05.140 for our purposes. 00:08:05.140 --> 00:08:09.850 I'm interested in a vector x that's actually in the plane, 00:08:09.850 --> 00:08:14.100 and I project it, and what do I get back? 00:08:14.100 --> 00:08:15.370 x, of course. 00:08:15.370 --> 00:08:17.920 Doesn't move. can be complex numbers. 00:08:17.920 --> 00:08:24.190 So any x in the plane is unchanged by P, 00:08:24.190 --> 00:08:26.070 and what's that telling me? 00:08:26.070 --> 00:08:28.490 That's telling me that x is an eigenvector, 00:08:28.490 --> 00:08:32.909 and it's also telling me what's the eigenvalue, which is -- 00:08:32.909 --> 00:08:34.470 just compare it with that. 00:08:34.470 --> 00:08:39.549 The eigenvalue, the multiplier, is just one. 00:08:39.549 --> 00:08:41.510 Good. 00:08:41.510 --> 00:08:45.290 So we have actually a whole plane of eigenvectors. 00:08:45.290 --> 00:08:49.050 Now I ask, are there any other eigenvectors? 00:08:49.050 --> 00:08:51.020 And I expect the answer to be yes, 00:08:51.020 --> 00:08:53.290 because I would like to get three, 00:08:53.290 --> 00:08:55.470 if I'm in three dimensions, I would 00:08:55.470 --> 00:08:59.490 like to hope for three independent eigenvectors, two 00:08:59.490 --> 00:09:02.610 of them in the plane and one not in the plane. 00:09:02.610 --> 00:09:03.300 OK. 00:09:03.300 --> 00:09:06.940 So this guy b that I drew there was not any good. 00:09:06.940 --> 00:09:11.330 What's the right eigenvector that's not in the plane? 00:09:11.330 --> 00:09:16.910 The, the good one is the one that's perpendicular to the 00:09:16.910 --> 00:09:17.490 plane. 00:09:17.490 --> 00:09:22.170 There's an, another good x, because what's the projection? 00:09:22.170 --> 00:09:24.300 So these are eigenvectors. 00:09:24.300 --> 00:09:27.000 Another guy here would be another eigenvector. 00:09:27.000 --> 00:09:29.950 But now here is another one. two. 00:09:29.950 --> 00:09:37.710 Any x that's perpendicular to the plane, 00:09:37.710 --> 00:09:41.390 what's Px for that, for that, vector? 00:09:43.950 --> 00:09:47.070 What's the projection of this guy perpendicular to the plane? 00:09:47.070 --> 00:09:50.570 It is zero, of course. 00:09:50.570 --> 00:09:53.640 So -- there's the null space. 00:09:53.640 --> 00:09:58.800 Px and n- for those guys are zero, or zero x if we like, 00:09:58.800 --> 00:10:02.480 and the eigenvalue is zero. 00:10:02.480 --> 00:10:05.697 So my answer to the question is, what are the eigenvalues for 00:10:05.697 --> 00:10:07.280 In our example, the one we worked out, 00:10:07.280 --> 00:10:08.610 a projection matrix? 00:10:08.610 --> 00:10:09.660 There they are. 00:10:09.660 --> 00:10:11.880 One and zero. 00:10:11.880 --> 00:10:13.060 OK. 00:10:13.060 --> 00:10:17.440 We know projection matrices. 00:10:17.440 --> 00:10:22.660 We can write them down as that A, A transpose, A inverse, 00:10:22.660 --> 00:10:28.400 A transpose thing, but without doing that from the picture 00:10:28.400 --> 00:10:31.420 we could see what are the eigenvectors. 00:10:31.420 --> 00:10:32.280 OK. 00:10:32.280 --> 00:10:34.580 Are there other matrices? 00:10:34.580 --> 00:10:36.790 Let me take a second example. 00:10:36.790 --> 00:10:39.650 How about a permutation matrix? 00:10:39.650 --> 00:10:41.619 What about the matrix, I'll call it A now. 00:10:41.619 --> 00:10:42.410 Zero one, one zero. 00:10:42.410 --> 00:10:49.120 A equals zero one one zero, that had eigenvalue one and 00:10:49.120 --> 00:10:53.740 Can you tell me a vector x -- 00:10:53.740 --> 00:10:55.680 see, we'll have a system soon enough, 00:10:55.680 --> 00:10:58.040 so I, I would like to just do these e- 00:10:58.040 --> 00:11:03.950 these couple of examples, just to see the picture before we, 00:11:03.950 --> 00:11:11.300 before we let it all, go into a system 00:11:11.300 --> 00:11:13.850 where that, matrix isn't anything special. 00:11:13.850 --> 00:11:15.590 Because it is special. 00:11:15.590 --> 00:11:20.260 And what, so what vector could I multiply by and end up 00:11:20.260 --> 00:11:21.430 in the same direction? 00:11:21.430 --> 00:11:25.800 Can you spot an eigenvector for this guy? 00:11:25.800 --> 00:11:32.380 That's a matrix that permutes x1 and x2, right? 00:11:32.380 --> 00:11:37.190 It switches the two components of x. 00:11:37.190 --> 00:11:43.710 How could the vector with its x2 x1, with -- 00:11:43.710 --> 00:11:49.630 permuted turn out to be a multiple of x1 x2, 00:11:49.630 --> 00:11:51.060 the vector we start with? 00:11:51.060 --> 00:11:53.300 Can you tell me an eigenvector here for this guy? 00:11:53.300 --> 00:11:56.450 x equal -- what is -- actually, can you tell me one vector that 00:11:56.450 --> 00:11:58.690 which is lambda x, and I have a three x, 00:11:58.690 --> 00:12:01.160 And of course you -- everybody knows that they're -- what, 00:12:01.160 --> 00:12:01.830 has eigenvalue one? 00:12:01.830 --> 00:12:03.670 So what, what vector would have eigenvalue one, 00:12:03.670 --> 00:12:05.544 just above what we2 found here. so that if I, 00:12:05.544 --> 00:12:09.530 if I permute it it doesn't change? right? 00:12:09.530 --> 00:12:13.450 There, that could be one one, thanks. 00:12:13.450 --> 00:12:14.460 One one. 00:12:14.460 --> 00:12:16.780 OK, take that vector one one. 00:12:16.780 --> 00:12:20.470 That will be an eigenvector, because if I do Ax 00:12:20.470 --> 00:12:23.120 I get one one. 00:12:23.120 --> 00:12:27.230 So that's the eigenvalue is one. 00:12:27.230 --> 00:12:28.710 Great. 00:12:28.710 --> 00:12:30.600 That's one eigenvalue. 00:12:30.600 --> 00:12:33.420 But I have here a two by two matrix, 00:12:33.420 --> 00:12:37.620 and I figure there's going to be a second eigenvalue. 00:12:40.180 --> 00:12:42.740 And eigenvector. 00:12:42.740 --> 00:12:44.410 Now, what about that? 00:12:44.410 --> 00:12:50.280 What's a vector, OK, maybe we can just, like, guess it. 00:12:50.280 --> 00:12:55.280 A vector that the other -- actually, 00:12:55.280 --> 00:12:58.940 this one that I'm thinking of is going to be a vector that has 00:12:58.940 --> 00:13:00.630 eigenvalue minus one. 00:13:03.240 --> 00:13:06.670 That's going to be my other eigenvalue for this matrix. 00:13:06.670 --> 00:13:09.430 It's a -- notice the nice positive or not negative 00:13:09.430 --> 00:13:13.240 matrix, but an eigenvalue is going to come out negative. 00:13:13.240 --> 00:13:15.520 And can you guess, spot the x that will work for 00:13:15.520 --> 00:13:18.910 Times x is supposed to give me zero, right? that? 00:13:18.910 --> 00:13:20.960 So I want a, a vector. 00:13:20.960 --> 00:13:27.140 When I multiply by A, which reverses the two components, 00:13:27.140 --> 00:13:31.120 I want the thing to come out minus the original. 00:13:31.120 --> 00:13:34.480 So what shall I send in in that case? 00:13:34.480 --> 00:13:37.650 If I send in negative one one. 00:13:40.550 --> 00:13:47.150 Then when I apply A, I get I do that multiplication, 00:13:47.150 --> 00:13:52.440 and I get one negative one, so it reversed sign. 00:13:52.440 --> 00:13:55.840 So Ax is -x. 00:13:55.840 --> 00:13:57.710 Lambda is minus one. 00:13:57.710 --> 00:14:04.130 Ax -- so Ax was x there and Ax is minus x here. 00:14:04.130 --> 00:14:08.080 Can I just mention, like, jump ahead, have, 00:14:08.080 --> 00:14:10.550 give a perfectly innocent-looking quadratic 00:14:10.550 --> 00:14:14.990 and point out a special little fact about eigenvalues. 00:14:14.990 --> 00:14:18.940 n by n matrices will have n eigenvalues. 00:14:18.940 --> 00:14:23.390 And I get this matrix4 zero zero zero one, 00:14:23.390 --> 00:14:28.390 And it's not like -- suppose n is three or four or more. 00:14:28.390 --> 00:14:32.120 It's not so easy to find them. 00:14:32.120 --> 00:14:35.570 We'd have a third degree or a fourth degree or an n-th degree 00:14:35.570 --> 00:14:36.330 equation. 00:14:36.330 --> 00:14:37.960 But here's one nice fact. 00:14:37.960 --> 00:14:40.270 There, there's one pleasant fact. we -- 00:14:40.270 --> 00:14:42.570 the eigenvalues came out four and two. 00:14:42.570 --> 00:14:45.540 That the sum of the eigenvalues equals the sum 00:14:45.540 --> 00:14:46.530 down the diagonal. 00:14:46.530 --> 00:14:50.150 That's called the trace, and I put that in the lecture 00:14:50.150 --> 00:14:54.590 Now I add three I to that matrix. content specifically. 00:14:54.590 --> 00:15:03.390 So this is a neat fact, the fact that sthe sum of the lambdas, 00:15:03.390 --> 00:15:08.390 add up the lambdas, equals the sum -- 00:15:08.390 --> 00:15:11.220 what would you like me to, shall I write that down? 00:15:11.220 --> 00:15:17.330 What I'm want to say in words is the sum down the diagonal of A. 00:15:17.330 --> 00:15:18.640 Shall I write a11+a22+...+ ann. 00:15:18.640 --> 00:15:22.570 That's add up the diagonal entries. 00:15:27.470 --> 00:15:31.700 In this example, it's zero. 00:15:31.700 --> 00:15:36.560 In other words, once I found this eigenvalue of one, 00:15:36.560 --> 00:15:39.110 I knew the other one had to be minus one 00:15:39.110 --> 00:15:43.710 in this two by two case, because in the two by two case, which 00:15:43.710 --> 00:15:51.970 is a good one to, to, play with, the trace tells you 00:15:51.970 --> 00:15:54.804 right away what the other eigenvalue is. 00:15:54.804 --> 00:15:56.970 So if I tell you one eigenvalue, you can tell me the 00:15:56.970 --> 00:15:57.780 other one. 00:15:57.780 --> 00:16:02.270 We'll, we'll have that -- we'll, minus one and eigenvectors one 00:16:02.270 --> 00:16:05.850 one and eigenvector minus one we'll see that again. 00:16:05.850 --> 00:16:06.350 OK. 00:16:06.350 --> 00:16:07.980 Now can I -- 00:16:07.980 --> 00:16:11.240 I could give more examples, but maybe it's 00:16:11.240 --> 00:16:16.140 time to face the, the equation, Ax equal lambda x, and figure 00:16:16.140 --> 00:16:19.810 how are we going to find x and lambda. 00:16:19.810 --> 00:16:22.571 And that is lambda one times lambda3 00:16:22.571 --> 00:16:23.070 OK. 00:16:23.070 --> 00:16:25.590 So this, so the question now is how 00:16:25.590 --> 00:16:30.630 to find eigenvalues and eigenvectors. 00:16:30.630 --> 00:16:35.430 How to solve, how to solve Ax equal lambda x from the three 00:16:35.430 --> 00:16:43.670 x, so it's just I mean, when we've got two unknowns 00:16:43.670 --> 00:16:47.500 both in the equation. 00:16:47.500 --> 00:16:48.460 OK. 00:16:48.460 --> 00:16:51.330 Here's the trick. 00:16:51.330 --> 00:16:53.240 Simple idea. 00:16:53.240 --> 00:16:58.980 Bring this onto the same side. 00:16:58.980 --> 00:16:59.950 Rewrite. 00:16:59.950 --> 00:17:05.589 Bring this over as A minus lambda times the identity x 00:17:05.589 --> 00:17:09.770 One. equals zero. 00:17:09.770 --> 00:17:10.740 Right? 00:17:10.740 --> 00:17:14.150 I have Ax minus lambda x, so I brought that over 00:17:14.150 --> 00:17:14.260 and I've got zero left on the, on the right-hand side. 00:17:14.260 --> 00:17:16.740 What's the relation between that problem and -- 00:17:16.740 --> 00:17:19.170 let me write 00:17:19.170 --> 00:17:19.980 OK. 00:17:19.980 --> 00:17:23.200 I don't know lambda and I don't know x, but I do know something 00:17:23.200 --> 00:17:23.700 here. 00:17:26.490 --> 00:17:28.460 What I know is if I, if I'm going 00:17:28.460 --> 00:17:31.870 to be able to solve this thing, for some x that's not the zero 00:17:31.870 --> 00:17:35.760 vector, that's not, that's a useless eigenvector, 00:17:35.760 --> 00:17:37.410 doesn't count. 00:17:37.410 --> 00:17:44.860 What I know now is that this matrix must be what? 00:17:44.860 --> 00:17:48.350 If I'm going to be -- if there is an x -- 00:17:48.350 --> 00:17:51.520 I don't -- right now I don't know what it is. 00:17:51.520 --> 00:17:54.490 I'm going to find lambda first, actually. 00:17:54.490 --> 00:18:00.300 And -- but if there is an x, it tells me that this matrix, 00:18:00.300 --> 00:18:05.160 this special combination, which is like the matrix A with 00:18:05.160 --> 00:18:10.430 lambda -- shifted by lambda, shifted by lambda I, 00:18:10.430 --> 00:18:14.550 that it has to be singular. 00:18:14.550 --> 00:18:17.030 This matrix must be singular, otherwise 00:18:17.030 --> 00:18:21.510 the only x would be the zero x, and zero matrix.OK. 00:18:21.510 --> 00:18:22.700 So this is singular. 00:18:22.700 --> 00:18:30.320 And what do I now know about singular matrices? 00:18:30.320 --> 00:18:31.520 So, so take three away. 00:18:31.520 --> 00:18:35.140 Their determinant is zero. 00:18:35.140 --> 00:18:38.790 So I've -- so from the fact that that has to be singular, 00:18:38.790 --> 00:18:44.750 I know that the determinant of A minus lambda I has to be zero. 00:18:48.590 --> 00:18:52.430 And that, now I've got x out of it. 00:18:52.430 --> 00:18:55.680 I've got an equation for lambda, that the key equation -- 00:18:55.680 --> 00:19:01.020 it's called the characteristic equation or the eigenvalue 00:19:01.020 --> 00:19:01.520 equation. 00:19:04.180 --> 00:19:08.370 And that -- in other words, I'm now in a position to find 00:19:08.370 --> 00:19:10.480 lambda first. 00:19:10.480 --> 00:19:15.940 So -- the idea will be to find lambda first. 00:19:18.890 --> 00:19:21.660 And actually, I won't find one lambda, 00:19:21.660 --> 00:19:24.120 I'll find N different lambdas. 00:19:24.120 --> 00:19:28.440 Well, n lambdas, maybe not n different ones. 00:19:28.440 --> 00:19:31.430 A lambda could be repeated. 00:19:31.430 --> 00:19:38.040 A repeated lambda is the source of all trouble in 18.06. 00:19:38.040 --> 00:19:43.620 So, let's hope for the moment that they're not repeated. 00:19:43.620 --> 00:19:48.310 There, there they were different, right? 00:19:48.310 --> 00:19:51.920 One and minus one in that, in that, for that permutation. 00:19:51.920 --> 00:19:52.800 OK. 00:19:52.800 --> 00:19:59.460 So and after I found this lambda, can I just look ahead? 00:19:59.460 --> 00:20:02.200 How I going to find x? 00:20:02.200 --> 00:20:06.670 After I have found this lambda, the lambda being this -- 00:20:06.670 --> 00:20:10.551 one of the numbers that makes this matrix singular. 00:20:10.551 --> 00:20:11.550 Their product was eight. 00:20:11.550 --> 00:20:15.110 Then of course finding x is just by elimination. 00:20:15.110 --> 00:20:15.610 Right? 00:20:15.610 --> 00:20:18.630 It's just -- now I've got a singular matrix, 00:20:18.630 --> 00:20:20.640 I'm looking for the null space. 00:20:20.640 --> 00:20:24.020 We're experts at finding the null space. 00:20:24.020 --> 00:20:26.570 You know, you do elimination, you identify 00:20:26.570 --> 00:20:31.780 the, the, the pivot columns and so on, you're -- 00:20:31.780 --> 00:20:36.900 and, give values to the free variables. 00:20:36.900 --> 00:20:39.300 Probably there'll only be one free variable. 00:20:39.300 --> 00:20:42.480 We'll give it the value one, like there. 00:20:42.480 --> 00:20:44.550 And we find the other variable. 00:20:44.550 --> 00:20:45.390 OK. 00:20:45.390 --> 00:20:52.610 So let's -- find the x second will be a doable job. 00:20:52.610 --> 00:20:54.540 That's my big equation for x. 00:20:54.540 --> 00:20:58.070 Let's go, let's look at the first job of finding lambda. 00:20:58.070 --> 00:20:59.500 Can I take another example? 00:20:59.500 --> 00:21:00.000 OK. 00:21:00.000 --> 00:21:02.130 And let's, let's work that one out. 00:21:02.130 --> 00:21:02.630 OK. 00:21:02.630 --> 00:21:07.590 So let me take the example, say, let me make it easy. 00:21:07.590 --> 00:21:09.240 it's just sitting there. 00:21:09.240 --> 00:21:12.550 Three three one and one. what do you 00:21:12.550 --> 00:21:14.610 know about the complex numbers? 00:21:14.610 --> 00:21:16.680 So I've made it easy. 00:21:16.680 --> 00:21:19.160 I've made it two by two. 00:21:19.160 --> 00:21:20.810 I've made it symmetric. 00:21:20.810 --> 00:21:24.530 And I even made it constant down the diagonal. 00:21:24.530 --> 00:21:27.830 That a matrix, a perfectly real matrix could 00:21:27.830 --> 00:21:32.380 So that -- so the more, like, special properties I stick 00:21:32.380 --> 00:21:36.270 into the matrix, the more special outcome I get 00:21:36.270 --> 00:21:38.370 for the eigenvalues. 00:21:38.370 --> 00:21:42.050 For example, this symmetric matrix, 00:21:42.050 --> 00:21:48.130 I know that it'll come out with real eigenvalues. one. 00:21:48.130 --> 00:21:52.530 The eigenvalues will turn out to be nice real numbers. 00:21:52.530 --> 00:21:57.760 And up in our previous example, that was a symmetric matrix. 00:21:57.760 --> 00:22:02.130 Actually, while we're at it, that was a symmetric matrix. 00:22:02.130 --> 00:22:05.500 Its eigenvalues were nice real numbers, one and minus one. 00:22:05.500 --> 00:22:07.630 And do you notice anything about its eigenvectors? 00:22:07.630 --> 00:22:08.970 And what do you notice? 00:22:08.970 --> 00:22:11.640 Anything particular about those two vectors, one one and minus 00:22:11.640 --> 00:22:14.840 And now comes that thing that I wanted to be reminded of. 00:22:14.840 --> 00:22:15.380 one one? 00:22:15.380 --> 00:22:18.850 They just happen to be -- no, I can't say they just happen 00:22:18.850 --> 00:22:20.610 to be, because that's the whole point, 00:22:20.610 --> 00:22:23.390 is that they had to be -- what? 00:22:23.390 --> 00:22:25.610 What are they? 00:22:25.610 --> 00:22:27.380 They're perpendicular. 00:22:27.380 --> 00:22:30.180 The vector, when I -- if I see a vector one one and a one -- 00:22:30.180 --> 00:22:33.121 and a minus one one, my mind immediately takes that 00:22:33.121 --> 00:22:33.620 dot product. 00:22:33.620 --> 00:22:36.570 It's zero. what's the determinant of that matrix? 00:22:36.570 --> 00:22:38.140 Those vectors are perpendicular. 00:22:38.140 --> 00:22:40.090 That'll happen here too. 00:22:40.090 --> 00:22:42.670 Well, let's find the eigenvalues. 00:22:42.670 --> 00:22:45.770 Actually, oh, my example's too easy. 00:22:45.770 --> 00:22:48.350 My example is too easy. 00:22:48.350 --> 00:22:53.550 Let me tell you in advance what's going to happen. 00:22:53.550 --> 00:22:56.210 May I? 00:22:56.210 --> 00:22:58.510 Or shall I do the determinant of A minus lambda, 00:22:58.510 --> 00:23:00.720 and then point out at the end? 00:23:00.720 --> 00:23:03.180 Will you remind me at the -- after I've found 00:23:03.180 --> 00:23:10.830 the eigenvalues to say why they were -- why they were easy from 00:23:10.830 --> 00:23:17.050 That -- it had to be eight, because we factored into lambda 00:23:17.050 --> 00:23:20.160 the, from the example we did? 00:23:20.160 --> 00:23:23.270 OK, let's do the job here. 00:23:23.270 --> 00:23:27.410 Let's compute determinant of A minus lambda I. 00:23:27.410 --> 00:23:29.490 So that's a determinant. 00:23:29.490 --> 00:23:32.600 And what's, what is this thing? 00:23:32.600 --> 00:23:36.090 It's the matrix A with lambda removed from the diagonal. 00:23:36.090 --> 00:23:38.570 for this matrix? 00:23:38.570 --> 00:23:40.770 So the diagonal matrix is shifted, 00:23:40.770 --> 00:23:43.800 and then I'm taking the determinant. 00:23:43.800 --> 00:23:44.370 OK. 00:23:44.370 --> 00:23:47.390 So I multiply this out. 00:23:47.390 --> 00:23:49.340 So what is that determinant? 00:23:49.340 --> 00:23:52.520 Do you notice, I didn't take lambda away 00:23:52.520 --> 00:23:55.090 from all the entries. 00:23:55.090 --> 00:23:56.790 It's lambda I, so it's lambda along the 00:23:56.790 --> 00:23:58.040 Lambda plus three x. diagonal. 00:23:58.040 --> 00:24:04.040 So I get three minus lambda squared and then minus one, 00:24:04.040 --> 00:24:06.380 right? 00:24:06.380 --> 00:24:08.000 And I want that to be zero. 00:24:08.000 --> 00:24:09.709 And what is A minus lambda I x? 00:24:09.709 --> 00:24:11.000 Well, I'm going to simplify it. 00:24:11.000 --> 00:24:11.050 And what will I get? 00:24:11.050 --> 00:24:11.190 So if I multiply this out, I get lambda squared minus six 00:24:11.190 --> 00:24:11.290 What's -- how is this matrix related to that matrix? 00:24:11.290 --> 00:24:11.330 lambda plus what? 00:24:11.330 --> 00:24:11.350 Plus eight. 00:24:11.350 --> 00:24:11.400 But it's out there. 00:24:11.400 --> 00:24:11.490 And that I'm going to set to zero. 00:24:11.490 --> 00:24:11.560 And I'm going to solve it. 00:24:11.560 --> 00:24:11.640 So and it's, it's a quadratic equation. 00:24:11.640 --> 00:24:11.750 I can use factorization, I can use the quadratic formula. 00:24:11.750 --> 00:24:12.625 I'll get two lambdas. 00:24:12.625 --> 00:24:30.690 Before I do it, tell me what's that number six that's 00:24:30.690 --> 00:24:41.560 showing up in this equation? 00:24:41.560 --> 00:24:48.080 It's the trace. 00:24:48.080 --> 00:25:03.300 That number six is three plus three. 00:25:03.300 --> 00:25:05.820 And while we're at it, what's the number eight that's 00:25:05.820 --> 00:25:08.590 showing up in this equation? 00:25:08.590 --> 00:25:10.340 It's the determinant. 00:25:10.340 --> 00:25:13.290 That our matrix has determinant eight. 00:25:13.290 --> 00:25:16.820 So in a two by two case, it's really nice. 00:25:16.820 --> 00:25:21.630 It's lambda squared minus the trace times lambda -- 00:25:21.630 --> 00:25:24.430 the trace is the linear coefficient -- 00:25:24.430 --> 00:25:27.880 and plus the determinant, the constant term. 00:25:27.880 --> 00:25:28.540 OK. 00:25:28.540 --> 00:25:32.730 So let's -- can, can we find the roots? 00:25:32.730 --> 00:25:36.310 I guess the easy way is to factor that as something times 00:25:36.310 --> 00:25:37.980 something. 00:25:37.980 --> 00:25:41.280 If we couldn't factor it, then we'd have to use the old 00:25:41.280 --> 00:25:46.590 b^2-4ac formula, but I, I think we can factor that into lambda 00:25:46.590 --> 00:25:50.870 minus what times lambda minus what? 00:25:50.870 --> 00:25:54.850 Can you do that factorization? 00:25:54.850 --> 00:25:57.240 Four and two? 00:25:57.240 --> 00:25:59.172 Lambda minus four times lambda minus two. 00:25:59.172 --> 00:26:00.880 So the, the eigenvalues are four and two. 00:26:00.880 --> 00:26:01.210 So the eigenvalues are -- one eigenvalue, lambda one, 00:26:01.210 --> 00:26:01.300 Now I'm looking for x, the eigenvector. let's say, 00:26:01.300 --> 00:26:01.320 is four. 00:26:01.320 --> 00:26:01.490 Lambda two, the other eigenvalue, is two. 00:26:01.490 --> 00:26:03.570 The eigenvalues are four and two. 00:26:03.570 --> 00:26:08.540 And then I can go for the eigenvectors. 00:26:08.540 --> 00:26:15.380 Suppose I have a matrix A, and Ax equal lambda x. 00:26:15.380 --> 00:26:16.620 equals zero. 00:26:16.620 --> 00:26:20.980 You see I got the eigenvalues first. 00:26:20.980 --> 00:26:25.950 So if they, if this had eigenvalue lambda, 00:26:25.950 --> 00:26:26.860 Four and two. 00:26:26.860 --> 00:26:29.030 Now for the eigenvectors. 00:26:29.030 --> 00:26:31.300 So what are the eigenvectors? 00:26:31.300 --> 00:26:34.880 They're these guys in the null space when 00:26:34.880 --> 00:26:40.090 I take away, when I make the matrix singular by taking 00:26:40.090 --> 00:26:42.740 four I or two I away. 00:26:42.740 --> 00:26:46.470 So we're -- we got to do those separately. 00:26:46.470 --> 00:26:50.100 I'll -- let me find the eigenvector for four first. 00:26:50.100 --> 00:26:57.420 So I'll subtract four, so A minus four I is -- 00:26:57.420 --> 00:27:00.590 so taking four away will put minus ones there. 00:27:04.070 --> 00:27:07.000 And what's the point about that matrix? 00:27:07.000 --> 00:27:10.260 If four is an eigenvalue, then A minus four 00:27:10.260 --> 00:27:13.220 I had better be a what kind of matrix? 00:27:13.220 --> 00:27:14.800 Singular. 00:27:14.800 --> 00:27:17.800 If that matrix isn't singular, the four wasn't correct. 00:27:17.800 --> 00:27:21.280 But we're OK, that matrix is singular. 00:27:21.280 --> 00:27:23.110 And what's the x now? 00:27:23.110 --> 00:27:25.290 The x is in the null space. 00:27:25.290 --> 00:27:28.910 So what's the x1 that goes with, with the lambda one? 00:27:28.910 --> 00:27:32.580 eigenvalue, eigenvector, eigenvalue for this, 00:27:32.580 --> 00:27:38.294 So that A -- so this is -- now I'm doing A x1 is lambda one 00:27:38.294 --> 00:27:38.794 x1. 00:27:41.640 --> 00:27:45.160 So I took A minus lambda one I, that's this matrix, 00:27:45.160 --> 00:27:48.630 and now I'm looking for the x1 in its null space, 00:27:48.630 --> 00:27:49.510 and who is he? 00:27:49.510 --> 00:27:51.760 What's the vector x in the null space? 00:27:51.760 --> 00:27:53.380 Of course it's one one. 00:27:53.380 --> 00:27:56.310 So that's the eigenvector that goes with that eigenvalue. 00:27:56.310 --> 00:27:57.610 So, so now -- 00:27:57.610 --> 00:28:00.540 Let's just spend one more minute on this bad 00:28:00.540 --> 00:28:02.490 Now how about the eigenvector that 00:28:02.490 --> 00:28:04.110 goes with the other eigenvalue? 00:28:04.110 --> 00:28:06.040 Can I do that with, with erasing? 00:28:06.040 --> 00:28:08.940 I take A minus two I. 00:28:08.940 --> 00:28:11.380 So now I take two away from the diagonal, 00:28:11.380 --> 00:28:15.670 and that leaves me with a one and a one. 00:28:15.670 --> 00:28:19.490 So A minus two I has again produced a singular matrix, 00:28:19.490 --> 00:28:21.710 as it had to. 00:28:21.710 --> 00:28:24.990 I'm looking for the null space of that guy. 00:28:24.990 --> 00:28:27.210 What vector is in its null space? 00:28:27.210 --> 00:28:29.320 Well, of course, a whole line of vectors. 00:28:32.040 --> 00:28:35.990 So when I say the eigenvector, I'm not speaking correctly. 00:28:35.990 --> 00:28:38.490 There's a whole line of eigenvectors, and you just -- 00:28:38.490 --> 00:28:40.760 I just want a basis. 00:28:40.760 --> 00:28:43.360 And for a line I just want one vector. 00:28:43.360 --> 00:28:48.130 But -- You could, you're -- there's some freedom 00:28:48.130 --> 00:28:50.860 in choosing that one, but choose a reasonable one. 00:28:50.860 --> 00:28:54.200 What's a vector in the null space of that? 00:28:54.200 --> 00:28:58.800 Well, the natural vector to pick as the eigenvector 00:28:58.800 --> 00:29:01.540 with, with lambda two is minus one one. 00:29:05.130 --> 00:29:07.760 If I did elimination on that vector 00:29:07.760 --> 00:29:10.520 and set that, the free variable to be one, 00:29:10.520 --> 00:29:15.010 I would get minus one and get that eigenvector. 00:29:15.010 --> 00:29:22.250 So you see then that I've got eigenvector, 00:29:22.250 --> 00:29:32.890 Now the other neat fact is that the determinant, 00:29:32.890 --> 00:29:36.906 How are those two matrices related? 00:29:40.530 --> 00:29:46.710 Well, one is just three I more than the other one, right? two. 00:29:46.710 --> 00:29:52.410 I just took that matrix and I -- 00:29:52.410 --> 00:30:29.276 I took this matrix and I added three I. 00:30:29.276 --> 00:30:31.900 So my question is, what happened to the minus four times lambda 00:30:31.900 --> 00:30:34.720 minus two. eigenvalues and what happened to the eigenvectors? 00:30:34.720 --> 00:30:37.530 That's the, that's like the question we keep asking now 00:30:37.530 --> 00:30:39.500 in this chapter. 00:30:39.500 --> 00:30:42.690 If I, if I do something to the matrix, what happens if I -- 00:30:42.690 --> 00:30:44.730 or I know something about the matrix, 00:30:44.730 --> 00:30:48.550 what's the what's the conclusion for its eigenvectors 00:30:48.550 --> 00:30:49.280 and eigenvalues? 00:30:49.280 --> 00:30:54.180 Because -- those eigenvalues and eigenvectors are going to tell 00:30:54.180 --> 00:30:57.280 us important information about the matrix. 00:30:57.280 --> 00:31:00.970 And here what are we seeing? 00:31:00.970 --> 00:31:04.330 What's happening to these eigenvalues, one and minus 00:31:04.330 --> 00:31:06.460 one, when I add three I? 00:31:10.040 --> 00:31:13.160 It just added three to the eigenvalues. 00:31:13.160 --> 00:31:16.640 I got four and two, three more than one and minus 00:31:16.640 --> 00:31:18.220 one. 00:31:18.220 --> 00:31:19.960 What happened to the eigenvectors? 00:31:19.960 --> 00:31:21.490 Nothing at all. 00:31:21.490 --> 00:31:25.170 One one is -- and minus -- and one -- and minus one one are -- 00:31:25.170 --> 00:31:27.050 is still the eigenvectors. 00:31:27.050 --> 00:31:34.200 In other words, simple but useful observation. 00:31:34.200 --> 00:31:40.320 If I add three I to a matrix, its eigenvectors don't change 00:31:40.320 --> 00:31:42.870 and its eigenvalues are three bigger. 00:31:42.870 --> 00:31:44.130 Let's, let's just see why. 00:31:44.130 --> 00:32:02.150 Let me keep all this on the same board. but just so you see -- 00:32:02.150 --> 00:32:08.410 so I'll try to do that. 00:32:08.410 --> 00:32:39.970 this has eigenvalue lambda plus three. 00:32:39.970 --> 00:32:46.590 And x, the eigenvector, is the same x for both matrices. 00:32:46.590 --> 00:32:48.790 OK. 00:32:48.790 --> 00:32:50.195 So that's, great. 00:32:53.440 --> 00:32:54.670 Of course, it's special. 00:32:54.670 --> 00:32:57.570 We got the new matrix by adding three I. 00:32:57.570 --> 00:32:59.500 Suppose I had added another matrix. 00:32:59.500 --> 00:33:02.410 Suppose I know the eigenvalues and eigenvectors of A. 00:33:02.410 --> 00:33:07.250 So I took A minus lambda I x, and what kind of a matrix I 00:33:07.250 --> 00:33:09.510 So this is, this, this little board 00:33:09.510 --> 00:33:11.875 here is going to be not so great. 00:33:17.730 --> 00:33:21.270 Suppose I have a matrix A and it has an eigenvector x with 00:33:21.270 --> 00:33:23.260 an eigenvalue lambda. 00:33:23.260 --> 00:33:28.730 You remember, I solve A minus lambda I x 00:33:28.730 --> 00:33:31.020 And now I add on some other matrix. 00:33:31.020 --> 00:33:34.000 So, so what I'm asking you is, if you know the eigenvalues 00:33:34.000 --> 00:33:38.430 of A and you know the eigenvalues of B, 00:33:38.430 --> 00:33:43.990 let me say suppose B -- so this is if -- let me put an if here. 00:33:43.990 --> 00:33:49.230 If Ax equals lambda x, fine, and B has, eigenvalues, 00:33:49.230 --> 00:33:52.240 has eigenvalues -- 00:33:52.240 --> 00:33:56.800 what shall we call them? 00:33:56.800 --> 00:34:02.270 Alpha, alpha one and alpha -- 00:34:02.270 --> 00:34:05.020 let's say -- 00:34:05.020 --> 00:34:07.270 I'll use alpha for the eigenvalues of B 00:34:07.270 --> 00:34:08.199 for no good reason. 00:34:11.489 --> 00:34:16.389 What a- you see what I'm going to ask is, how about A plus B? 00:34:20.949 --> 00:34:24.070 Let me, let me give you the, let me give you, 00:34:24.070 --> 00:34:27.770 what you might think first. 00:34:27.770 --> 00:34:28.730 OK. 00:34:28.730 --> 00:34:35.460 If Ax equals lambda x and if B has an eigenvalue alpha, 00:34:35.460 --> 00:34:40.630 then I allowed to say -- what's the matter with this argument? 00:34:40.630 --> 00:34:43.300 That gave us the constant term eight. 00:34:43.300 --> 00:34:44.070 It's wrong. 00:34:44.070 --> 00:34:47.130 What I'm going to write up is wrong. 00:34:47.130 --> 00:34:50.179 I'm going to say Bx is alpha x. 00:34:50.179 --> 00:34:55.909 Add those up, and you get A plus B x equals lambda plus alpha x. 00:34:55.909 --> 00:35:00.500 So you would think that if you know the eigenvalues of A 00:35:00.500 --> 00:35:03.490 and you knew the eigenvalues of B, 00:35:03.490 --> 00:35:07.900 then if you added you would know the eigenvalues of A plus B. 00:35:07.900 --> 00:35:11.310 But that's false. 00:35:11.310 --> 00:35:17.840 A plus B -- well, when B was three I, that worked great. 00:35:17.840 --> 00:35:21.290 But this is not so great. 00:35:21.290 --> 00:35:25.720 And what's the matter with that argument there? 00:35:25.720 --> 00:35:32.610 We have no reason to believe that x is also 00:35:32.610 --> 00:35:33.500 an eigenvector of 00:35:33.500 --> 00:35:38.930 B has some eigenvalues, B. but it's 00:35:38.930 --> 00:35:43.600 got some different eigenvectors normally. 00:35:43.600 --> 00:35:45.520 It's a different matrix. 00:35:45.520 --> 00:35:47.250 I don't know anything special. 00:35:47.250 --> 00:35:50.150 If I don't know anything special, then as far as I know, 00:35:50.150 --> 00:35:52.670 it's got some different eigenvector y, 00:35:52.670 --> 00:35:55.790 and when I add I get just rubbish. 00:35:55.790 --> 00:35:57.480 I mean, I get -- 00:35:57.480 --> 00:35:59.970 I can add, but I don't learn anything. 00:35:59.970 --> 00:36:05.380 So not so great is A plus B. 00:36:05.380 --> 00:36:09.460 Or A times B. 00:36:09.460 --> 00:36:12.670 Normally the eigenvalues of A plus B 00:36:12.670 --> 00:36:18.630 or A times B are not eigenvalues of A plus eigenvalues of B. 00:36:18.630 --> 00:36:22.500 Ei- eigenvalues are not, like, linear. 00:36:22.500 --> 00:36:24.610 Or -- and they don't multiply. 00:36:24.610 --> 00:36:27.580 Because, eigenvectors are usually different 00:36:27.580 --> 00:36:31.600 and, and there's just no way to find out 00:36:31.600 --> 00:36:33.520 what A plus B does to affect 00:36:33.520 --> 00:36:34.960 What do I do now? it. 00:36:34.960 --> 00:36:35.470 OK. 00:36:35.470 --> 00:36:41.660 So that's, like, a caution. 00:36:41.660 --> 00:36:44.610 Don't, if B is a multiple of the identity, great, 00:36:44.610 --> 00:36:50.340 but if B is some general matrix, then for A plus B you've got 00:36:50.340 --> 00:36:54.580 to find -- you've got to solve the eigenvalue problem. 00:36:54.580 --> 00:36:59.720 Now I want to do another example that brings out a, 00:36:59.720 --> 00:37:02.200 OK. another point about eigenvalues. 00:37:02.200 --> 00:37:06.280 Let me make this example a rotation matrix. 00:37:06.280 --> 00:37:09.390 possibility of complex numbers. 00:37:09.390 --> 00:37:10.170 OK. 00:37:10.170 --> 00:37:13.280 So here's another example. 00:37:13.280 --> 00:37:16.390 So a rotate -- 00:37:16.390 --> 00:37:21.060 oh, I'd better call it Q. 00:37:21.060 --> 00:37:26.500 I often use Q for, for, rotations 00:37:26.500 --> 00:37:32.600 because those are the, like, very important examples 00:37:32.600 --> 00:37:34.580 of orthogonal matrices. 00:37:34.580 --> 00:37:38.290 Let me make it a ninety degree rotation. 00:37:38.290 --> 00:37:41.709 So -- my matrix is going to be the one that rotates every 00:37:41.709 --> 00:37:43.750 And that's the sum, that's lambda one plus lambda 00:37:43.750 --> 00:37:46.640 vector by ninety degrees. 00:37:46.640 --> 00:37:49.480 So do you remember that matrix? 00:37:49.480 --> 00:37:51.950 It's the cosine of ninety degrees, which 00:37:51.950 --> 00:37:54.790 is zero, the sine of ninety degrees, 00:37:54.790 --> 00:38:03.990 which is one, minus the sine of ninety, the cosine of ninety. 00:38:03.990 --> 00:38:09.360 So that matrix deserves the letter Q. 00:38:09.360 --> 00:38:15.500 It's an orthogonal matrix, very, very orthogonal matrix. 00:38:15.500 --> 00:38:21.420 Now I'm interested in its eigenvalues and eigenvectors. 00:38:21.420 --> 00:38:24.500 Two by two, it can't be that tough. 00:38:24.500 --> 00:38:27.200 We know that the eigenvalues add to zero. 00:38:30.780 --> 00:38:33.310 Actually, we know something already here. 00:38:33.310 --> 00:38:35.750 The eigen- what's the sum of the two eigenvalues? 00:38:35.750 --> 00:38:38.880 Just tell me what I just said. 00:38:38.880 --> 00:38:40.430 Zero, right. 00:38:40.430 --> 00:38:42.310 From that trace business. 00:38:42.310 --> 00:38:46.550 The sum of the eigenvalues is, is going to come out zero. 00:38:46.550 --> 00:38:48.440 And the product of the eigenvalues, 00:38:48.440 --> 00:38:50.190 did I tell you about the determinant being 00:38:50.190 --> 00:38:50.870 the product of the eigenvalues? 00:38:50.870 --> 00:38:50.880 No. 00:38:50.880 --> 00:38:50.950 But that's a good thing to know. 00:38:50.950 --> 00:38:51.030 We pointed out how that eight appeared in 00:38:51.030 --> 00:38:52.863 the, in the quadratic equation. eigenvalues, 00:38:52.863 --> 00:39:07.700 we can postpone that evil day, 00:39:07.700 --> 00:39:23.700 So let me just say this. 00:39:23.700 --> 00:39:49.930 The trace is zero plus zero, obviously. 00:39:49.930 --> 00:39:52.675 And that was the determinant. 00:39:52.675 --> 00:39:53.175 OK. 00:39:56.230 --> 00:39:58.390 What I'm leading up to with this example 00:39:58.390 --> 00:40:03.610 is that something's going to go wrong. 00:40:03.610 --> 00:40:09.080 Something goes wrong for rotation 00:40:09.080 --> 00:40:16.410 because what vector can come out parallel to itself 00:40:16.410 --> 00:40:18.820 after a rotation? 00:40:18.820 --> 00:40:23.790 If this matrix rotates every vector by ninety degrees, 00:40:23.790 --> 00:40:26.880 what could be an eigenvector? 00:40:26.880 --> 00:40:31.190 Do you see we're, we're, we're going to have trouble. 00:40:31.190 --> 00:40:33.780 eigenvectors are -- 00:40:36.060 --> 00:40:36.560 Well. 00:40:36.560 --> 00:40:39.220 Our, our picture of eigenvectors, 00:40:39.220 --> 00:40:42.410 of, of coming out in the same direction that they went in, 00:40:42.410 --> 00:40:45.140 there won't be it. 00:40:45.140 --> 00:40:48.820 And with, and with eigenvalues we're going to have trouble. 00:40:48.820 --> 00:40:50.440 From these equations. 00:40:50.440 --> 00:40:51.710 Let's see. 00:40:51.710 --> 00:40:53.780 Why I expecting trouble? 00:40:53.780 --> 00:40:56.640 The, the first equation says that the eigenvalues 00:40:56.640 --> 00:40:57.320 add to zero. 00:40:59.920 --> 00:41:01.170 So there's a plus and a minus. 00:41:01.170 --> 00:41:02.211 So I take the eigenvalue. 00:41:04.360 --> 00:41:06.490 But then the second equation says 00:41:06.490 --> 00:41:09.450 that the product is plus one. 00:41:09.450 --> 00:41:10.880 We're in trouble. 00:41:10.880 --> 00:41:14.810 But there's a way out. 00:41:14.810 --> 00:41:17.860 So how -- let's do the usual stuff. 00:41:17.860 --> 00:41:21.700 Look at determinant of Q minus lambda I. 00:41:21.700 --> 00:41:27.230 So I'll just follow the rules, take the determinant, 00:41:27.230 --> 00:41:31.940 subtract lambda from the diagonal, where I had zeros, 00:41:31.940 --> 00:41:34.400 the rest is the same. 00:41:34.400 --> 00:41:37.180 Rest of Q is just copied. 00:41:37.180 --> 00:41:38.540 Compute that determinant. 00:41:38.540 --> 00:41:42.720 OK, so what does that determinant equal? 00:41:42.720 --> 00:41:47.740 Lambda squared minus minus one plus what? 00:41:51.930 --> 00:41:54.060 What's up? 00:41:54.060 --> 00:41:56.020 There's my equation. 00:41:56.020 --> 00:41:59.050 My equation for the eigenvalues is lambda 00:41:59.050 --> 00:42:00.950 squared plus one equals zero. 00:42:00.950 --> 00:42:04.620 What are the eigenvalues lambda one and lambda two? 00:42:04.620 --> 00:42:27.340 They're I, whatever that is, and minus it, right. 00:42:27.340 --> 00:42:30.240 Those are the right numbers. 00:42:30.240 --> 00:42:36.270 To be real numbers even though the matrix was perfectly real. 00:42:36.270 --> 00:42:38.190 So this can happen. 00:42:41.060 --> 00:42:44.950 Complex numbers are going to -- have to enter eighteen oh six 00:42:44.950 --> 00:42:51.500 at this moment. 00:42:51.500 --> 00:42:54.600 Boo, right. 00:42:54.600 --> 00:42:56.950 All right. 00:42:56.950 --> 00:43:02.970 If I just choose good matrices that have real 00:43:02.970 --> 00:43:08.500 supposed to have here? 00:43:24.360 --> 00:43:35.590 We do know a little information about the, 00:43:35.590 --> 00:43:38.672 the two complex numbers. 00:43:38.672 --> 00:43:40.380 They're complex conjugates of each other. 00:43:45.850 --> 00:43:51.590 If, if lambda is an eigenvalue, then when I change, 00:43:51.590 --> 00:43:54.360 when I go -- you remember what complex conjugates are? 00:43:54.360 --> 00:43:57.170 You switch the sign of the imaginary part. 00:43:57.170 --> 00:43:59.910 Well, this was only imaginary, had no real part, 00:43:59.910 --> 00:44:02.690 so we just switched its sign. 00:44:02.690 --> 00:44:06.720 So that eigenvalues come in pairs like that, 00:44:06.720 --> 00:44:08.440 but they're complex. 00:44:08.440 --> 00:44:11.040 A complex conjugate pair. 00:44:11.040 --> 00:44:14.530 And that can happen with a perfectly real matrix. 00:44:14.530 --> 00:44:17.060 And as a matter of fact -- 00:44:17.060 --> 00:44:18.930 so that was my, my point earlier, 00:44:18.930 --> 00:44:23.240 that if a matrix was symmetric, it wouldn't happen. 00:44:23.240 --> 00:44:27.090 So if we stick to matrices that are symmetric or, like, close 00:44:27.090 --> 00:44:32.610 to symmetric, then the eigenvalues will stay real. 00:44:32.610 --> 00:44:35.400 But if we move far away from symmetric -- 00:44:35.400 --> 00:44:39.520 and that's as far as you can move, because that matrix is -- 00:44:39.520 --> 00:44:44.470 how is Q transpose related to Q for that matrix? 00:44:44.470 --> 00:44:46.760 That matrix is anti-symmetric. 00:44:46.760 --> 00:44:49.520 Q transpose is minus Q. 00:44:49.520 --> 00:44:52.070 That's the very opposite of symmetry. 00:44:52.070 --> 00:44:54.570 When I flip across the diagonal I get -- 00:44:54.570 --> 00:44:56.350 I reverse all the signs. 00:44:56.350 --> 00:45:00.840 Those are the guys that have pure imaginary eigenvalues. 00:45:00.840 --> 00:45:03.320 So they're the extreme case. 00:45:03.320 --> 00:45:07.380 And in between are, are matrices that 00:45:07.380 --> 00:45:10.990 are not symmetric or anti-symmetric but, 00:45:10.990 --> 00:45:13.710 but they have partly a symmetric part 00:45:13.710 --> 00:45:15.110 and an anti-symmetric part. 00:45:15.110 --> 00:45:16.660 OK. 00:45:16.660 --> 00:45:23.220 So I'm doing a bunch of examples here to show the possibilities. 00:45:23.220 --> 00:45:28.320 The good possibilities being perpendicular eigenvectors, 00:45:28.320 --> 00:45:30.360 real eigenvalues. 00:45:30.360 --> 00:45:33.910 The bad possibilities being complex eigenvalues. 00:45:33.910 --> 00:45:37.360 We could say that's bad. 00:45:37.360 --> 00:45:39.455 There's another even worse. 00:45:42.380 --> 00:45:45.740 I'm getting through the, the bad things here today. 00:45:45.740 --> 00:45:51.750 Then, then the next lecture can, can, 00:45:51.750 --> 00:45:56.750 can be like pure happiness. 00:45:56.750 --> 00:45:57.720 OK. 00:45:57.720 --> 00:46:03.260 Here's one more bad thing that could happen. 00:46:03.260 --> 00:46:05.940 So I, again, I'll do it with an example. 00:46:05.940 --> 00:46:10.740 Suppose my matrix is, suppose I take this three three one 00:46:10.740 --> 00:46:13.140 and I change that guy to zero. 00:46:18.150 --> 00:46:21.280 What are the eigenvalues of that matrix? 00:46:21.280 --> 00:46:22.740 What are the eigenvectors? 00:46:22.740 --> 00:46:25.380 This is always our question. 00:46:25.380 --> 00:46:26.860 Of course, the next section we're 00:46:26.860 --> 00:46:29.580 going to show why are, why do we care. 00:46:29.580 --> 00:46:33.121 But for the moment, this lecture is introducing 00:46:33.121 --> 00:46:33.620 them. 00:46:33.620 --> 00:46:35.990 And let's just find them. 00:46:35.990 --> 00:46:36.640 OK. 00:46:36.640 --> 00:46:40.530 What are the eigenvalues of that matrix? 00:46:40.530 --> 00:46:45.910 Let me tell you -- at a glance we could answer that question. 00:46:45.910 --> 00:46:49.490 Because the matrix is triangular. 00:46:49.490 --> 00:46:53.540 It's really useful to know -- if you've got properties like 00:46:53.540 --> 00:46:55.320 a triangular matrix. 00:46:55.320 --> 00:46:57.980 It's very useful to know you can read the eigenvalues 00:46:57.980 --> 00:46:58.610 off. 00:46:58.610 --> 00:47:01.920 They're right on the diagonal. 00:47:01.920 --> 00:47:05.470 So the eigenvalue is three and also three. 00:47:05.470 --> 00:47:07.290 Three is a repeated eigenvalue. 00:47:07.290 --> 00:47:09.330 But let's see that happen. 00:47:09.330 --> 00:47:10.660 Let me do it right. 00:47:10.660 --> 00:47:15.370 The determinant of A minus lambda I, what I always 00:47:15.370 --> 00:47:17.119 have to do is this determinant. 00:47:17.119 --> 00:47:18.660 I take away lambda from the diagonal. 00:47:21.600 --> 00:47:24.070 I leave the rest. 00:47:24.070 --> 00:47:28.450 I compute the determinant, so I get a three minus lambda 00:47:28.450 --> 00:47:32.090 times a three minus lambda. 00:47:32.090 --> 00:47:35.680 And nothing. 00:47:35.680 --> 00:47:38.860 So that's where the triangular part came in. 00:47:38.860 --> 00:47:41.260 Triangular part, the one thing we know about triangular 00:47:41.260 --> 00:47:44.660 matrices is the determinant is just the product down 00:47:44.660 --> 00:47:45.890 the diagonal. 00:47:45.890 --> 00:47:48.990 And in this case, it's this same, repeated -- 00:47:48.990 --> 00:47:51.775 so lambda one is one -- 00:47:51.775 --> 00:47:53.900 sorry, lambda one is three and lambda two is three. 00:47:53.900 --> 00:47:58.540 That was easy. 00:47:58.540 --> 00:48:05.900 I mean, no -- why should I be pessimistic about a matrix 00:48:05.900 --> 00:48:10.650 whose eigenvalues can be read off right away? 00:48:10.650 --> 00:48:14.820 The problem with this matrix is in the eigenvectors. 00:48:14.820 --> 00:48:16.450 So let's go to the eigenvectors. 00:48:16.450 --> 00:48:18.960 So how do I find the eigenvectors? 00:48:18.960 --> 00:48:22.010 I'm looking for a couple of eigenvectors. 00:48:22.010 --> 00:48:22.800 Singular, right? 00:48:22.800 --> 00:48:31.590 It's supposed to be singular. 00:48:31.590 --> 00:48:40.530 And then it's got some vectors -- which it is. 00:48:40.530 --> 00:48:59.410 So it's got some vector x in the null space. 00:49:12.890 --> 00:49:15.670 And what, what's the, what's -- give me a basis for the null 00:49:15.670 --> 00:49:18.360 space for that guy. 00:49:18.360 --> 00:49:21.860 Tell me, what's a vector x in the null space, so that'll 00:49:21.860 --> 00:49:25.710 be the, the eigenvector that goes with lambda one 00:49:25.710 --> 00:49:27.030 equals three. 00:49:27.030 --> 00:49:31.030 The eigenvector is -- so what's in the null space? 00:49:31.030 --> 00:49:32.180 One zero, is it? 00:49:34.700 --> 00:49:35.200 Great. 00:49:38.730 --> 00:49:40.600 Now, what's the other eigenvector? 00:49:40.600 --> 00:49:47.960 What's, what's the eigenvector that goes with lambda two? 00:49:47.960 --> 00:49:51.620 Well, lambda two is three again. 00:49:51.620 --> 00:49:53.150 So I get the same thing again. 00:49:53.150 --> 00:49:55.500 Give me another vector -- 00:49:55.500 --> 00:49:57.750 I want it to be independent. 00:49:57.750 --> 00:49:59.340 If I'm going to write down an x2, 00:49:59.340 --> 00:50:02.440 I'm never going to let it be dependent on x1. 00:50:02.440 --> 00:50:05.240 I'm looking for independent eigenvectors, 00:50:05.240 --> 00:50:08.720 and what's the conclusion? 00:50:08.720 --> 00:50:11.010 There isn't one. 00:50:11.010 --> 00:50:17.050 This is a degenerate matrix. 00:50:17.050 --> 00:50:22.960 It's only got one line of eigenvectors instead of two. 00:50:22.960 --> 00:50:27.250 It's this possibility of a repeated eigenvalue 00:50:27.250 --> 00:50:34.390 opens this further possibility of a shortage of eigenvectors. 00:50:34.390 --> 00:50:43.310 And so there's no second independent eigenvector x2. 00:50:43.310 --> 00:50:48.010 So it's a matrix, it's a two by two matrix, 00:50:48.010 --> 00:50:51.850 but with only one independent eigenvector. 00:50:51.850 --> 00:50:56.910 So that will be -- the matrices that -- 00:50:56.910 --> 00:51:01.830 where eigenvectors are -- don't give the complete story. 00:51:01.830 --> 00:51:02.330 OK. 00:51:02.330 --> 00:51:05.930 My lecture on Monday will give the complete story 00:51:05.930 --> 00:51:11.290 for all the other matrices. 00:51:11.290 --> 00:51:12.360 Thanks. 00:51:12.360 --> 00:51:16.640 Have a good weekend. 00:51:16.640 --> 00:51:22.480 A real New England weekend.