WEBVTT 00:00:09.720 --> 00:00:12.130 OK, this is quiz review day. 00:00:12.130 --> 00:00:17.750 The quiz coming up on Wednesday will before this 00:00:17.750 --> 00:00:22.520 lecture the quiz will be this hour one o'clock Wednesday 00:00:22.520 --> 00:00:27.740 in Walker, top floor of Walker, closed book, all normal. 00:00:27.740 --> 00:00:33.760 I wrote down what we've covered in this second part 00:00:33.760 --> 00:00:38.830 of the course, and actually I'm impressed as I write it. 00:00:38.830 --> 00:00:42.250 so that's chapter four on orthogonality 00:00:42.250 --> 00:00:45.790 and you're remembering these -- 00:00:45.790 --> 00:00:50.530 what this is suggesting, these are those columns are 00:00:50.530 --> 00:00:56.810 orthonormal vectors, and then we call that matrix Q and the -- 00:00:56.810 --> 00:01:01.270 what's the key -- how do we state the fact that those v- 00:01:01.270 --> 00:01:04.989 those columns are orthonormal in terms of Q, 00:01:04.989 --> 00:01:11.920 it means that Q transpose Q is the identity. 00:01:11.920 --> 00:01:15.480 So that's the matrix statement of the -- 00:01:15.480 --> 00:01:18.930 of the property that the columns are orthonormal, 00:01:18.930 --> 00:01:21.710 the dot products are either one or zero, 00:01:21.710 --> 00:01:30.160 and then we computed the projections onto lines and onto 00:01:30.160 --> 00:01:35.920 subspaces, and we used that to solve problems Ax=b in -- 00:01:35.920 --> 00:01:38.800 in the least square sense, when there was no solution, 00:01:38.800 --> 00:01:40.640 we found the best solution. 00:01:40.640 --> 00:01:43.530 And then finally this Graham-Schmidt idea, 00:01:43.530 --> 00:01:50.790 which takes independent vectors and lines them up, takes -- 00:01:50.790 --> 00:01:54.000 subtracts off the projections of the part 00:01:54.000 --> 00:01:58.390 you've already done, so that the new part is orthogonal 00:01:58.390 --> 00:02:03.850 and so it takes a basis to an orthonormal basis. 00:02:03.850 --> 00:02:07.440 And you -- those calculations involve square roots a lot 00:02:07.440 --> 00:02:11.300 because you're making things unit vectors, 00:02:11.300 --> 00:02:13.440 but you should know that step. 00:02:13.440 --> 00:02:16.540 OK, for determinants, the three big -- 00:02:16.540 --> 00:02:19.720 the big picture is the properties of the determinant, 00:02:19.720 --> 00:02:24.020 one to three d- properties one, two and three, d- 00:02:24.020 --> 00:02:27.590 that define the determinant, and then four, five, 00:02:27.590 --> 00:02:31.110 six through ten were consequences. 00:02:31.110 --> 00:02:35.330 Then the big formula that has n factorial terms, half of them 00:02:35.330 --> 00:02:38.350 have plus signs and half minus signs, and then 00:02:38.350 --> 00:02:40.420 the cofactor formula. 00:02:40.420 --> 00:02:46.240 So and which led us to a formula for the inverse. 00:02:46.240 --> 00:02:48.780 And finally, just so you know what's 00:02:48.780 --> 00:02:53.550 covered in from chapter three, it's section six point 00:02:53.550 --> 00:02:58.920 one and two, so that's the basic idea of eigenvalues 00:02:58.920 --> 00:03:02.660 and eigenvectors, the equation for the eigenvalues, 00:03:02.660 --> 00:03:07.800 the mechanical step, this is really Ax equal lambda 00:03:07.800 --> 00:03:11.450 x for all n eigenvectors at once, 00:03:11.450 --> 00:03:14.480 if we have n independent eigenvectors, 00:03:14.480 --> 00:03:18.520 and then using that to compute powers of a matrix. 00:03:18.520 --> 00:03:22.430 So you notice the differential equations not on this list, 00:03:22.430 --> 00:03:28.390 because that's six point three, that that's for the third quiz. 00:03:28.390 --> 00:03:30.330 OK. 00:03:30.330 --> 00:03:36.140 Shall I what I usually do for review is to take an old exam 00:03:36.140 --> 00:03:40.720 and just try to pick out questions that are significant 00:03:40.720 --> 00:03:44.070 and write them quickly on the board, shall I -- 00:03:44.070 --> 00:03:46.210 shall I proceed that way again? 00:03:46.210 --> 00:03:51.650 This -- this exam is really old. 00:03:51.650 --> 00:03:54.770 November nineteen -- nineteen eighty-four, 00:03:54.770 --> 00:03:59.850 so that was before the Web existed. 00:03:59.850 --> 00:04:04.760 So not only were the lectures not on the Web, 00:04:04.760 --> 00:04:07.260 nobody even had a Web page, my God. 00:04:07.260 --> 00:04:10.760 OK, so can I nevertheless linear algebra 00:04:10.760 --> 00:04:14.530 was still as great as ever. 00:04:14.530 --> 00:04:22.290 So may I and that wasn't meant to be a joke, OK, all right, 00:04:22.290 --> 00:04:25.575 so let me just take these questions as they come. 00:04:28.420 --> 00:04:28.930 All right. 00:04:28.930 --> 00:04:30.260 OK. 00:04:30.260 --> 00:04:32.460 So the first question's about projections. 00:04:32.460 --> 00:04:35.670 It says we're given the line, the -- 00:04:35.670 --> 00:04:39.930 the vector a is the vector two, one, two, 00:04:39.930 --> 00:04:44.320 and I want to find the projection matrix P that 00:04:44.320 --> 00:04:46.900 projects onto the line through a. 00:04:46.900 --> 00:04:51.510 So my picture is, well I'm in three dimensions, of course, 00:04:51.510 --> 00:04:55.710 so there's a vector, two -- there's the vector a, two, one, 00:04:55.710 --> 00:04:59.210 two, let me draw the whole line through it, 00:04:59.210 --> 00:05:05.690 and I want to project any vector b onto that line, 00:05:05.690 --> 00:05:08.490 and I'm looking for the projection matrix. 00:05:08.490 --> 00:05:12.360 So -- so this -- the projection matrix is the matrix that I 00:05:12.360 --> 00:05:15.320 multiply b by to get here. 00:05:15.320 --> 00:05:19.260 And I guess this first part, this is just part one 00:05:19.260 --> 00:05:23.860 a, I'm really asking you the -- the quick way to answer, 00:05:23.860 --> 00:05:28.600 to find P, is just to remember what the formula is. 00:05:28.600 --> 00:05:31.740 And -- and we're in -- we're projecting onto a line, 00:05:31.740 --> 00:05:37.800 so our formula, our -- our usual formula is AA transpose A 00:05:37.800 --> 00:05:43.800 inverse A transpose, but now A is just a column, 00:05:43.800 --> 00:05:49.920 one-column matrix, so it'll be just a, so I'll just call it 00:05:49.920 --> 00:05:53.850 little a, little a transpose, and this is just a number now, 00:05:53.850 --> 00:05:56.970 one by one, so I can put it in the denominator, 00:05:56.970 --> 00:05:58.950 so there's our -- 00:05:58.950 --> 00:06:01.560 that's really what we want to remember, and -- 00:06:01.560 --> 00:06:05.490 and using that two, one, two, what will I get? 00:06:05.490 --> 00:06:08.280 I'm dividing by -- 00:06:08.280 --> 00:06:12.020 what's the length squared of that vector? 00:06:12.020 --> 00:06:13.970 So what's a transpose a? 00:06:13.970 --> 00:06:18.370 Looks like nine, and what's the matrix, well, I'm -- 00:06:18.370 --> 00:06:21.580 I'm doing two, one, two against two, 00:06:21.580 --> 00:06:28.520 one, two, so it's one-ninth of this matrix four, two, four, 00:06:28.520 --> 00:06:30.790 two, one, two, four, two, four. 00:06:34.250 --> 00:06:37.410 Now the next part asked about eigenvalues. 00:06:37.410 --> 00:06:39.450 So you see we're -- since we're learning, 00:06:39.450 --> 00:06:43.070 we know a lot more now, we can make connections between these 00:06:43.070 --> 00:06:47.880 chapters, so what's the eigenvector, 00:06:47.880 --> 00:06:50.200 what are the eigenvalues of P? 00:06:50.200 --> 00:06:53.140 I could ask what's the rank of P. What's 00:06:53.140 --> 00:06:55.860 the rank of that matrix? 00:06:55.860 --> 00:06:58.120 Uh -- one. 00:06:58.120 --> 00:06:59.410 Rank is one. 00:06:59.410 --> 00:07:01.780 What's the column space? 00:07:01.780 --> 00:07:06.780 If I apply P to all vectors then I fill up the column space, 00:07:06.780 --> 00:07:09.650 it's combinations of the columns, so what's the column 00:07:09.650 --> 00:07:10.880 space? 00:07:10.880 --> 00:07:13.370 Well, it's just this line. 00:07:13.370 --> 00:07:17.310 The column space is the line through two, one, two. 00:07:17.310 --> 00:07:20.770 And now what's the eigenvalue? 00:07:20.770 --> 00:07:24.090 So, or since that matrix has rank one, 00:07:24.090 --> 00:07:26.750 so tell me the eigenvalues of this matrix. 00:07:29.330 --> 00:07:31.330 It's a singular matrix, so it certainly 00:07:31.330 --> 00:07:34.970 has an eigenvalue zero. 00:07:34.970 --> 00:07:39.470 Actually, the rank is only one, so that 00:07:39.470 --> 00:07:42.720 means that there like going to be 00:07:42.720 --> 00:07:48.120 t- a two-dimensional null space, there'll be at least two, 00:07:48.120 --> 00:07:50.990 this lambda equals zero will be repeated twice 00:07:50.990 --> 00:07:55.630 because I can find two independent eigenvectors 00:07:55.630 --> 00:07:58.740 with lambda equals zero, and then of course since it's 00:07:58.740 --> 00:08:02.370 got three eigenvalues, what's the third one? 00:08:02.370 --> 00:08:03.300 It's one. 00:08:03.300 --> 00:08:05.080 How do I know it's one? 00:08:05.080 --> 00:08:09.640 Either from the trace, which is nine over nine, which is one, 00:08:09.640 --> 00:08:13.810 or by remembering what -- 00:08:13.810 --> 00:08:16.350 what is the eigenvector, and actually now 00:08:16.350 --> 00:08:18.350 it's going to ask for the eigenvector, 00:08:18.350 --> 00:08:21.545 so what's the eigenvector for that eigenvalue? 00:08:24.150 --> 00:08:28.310 What's the eigenvector for that eigenvalue? 00:08:28.310 --> 00:08:35.020 It's the -- it's the vector that doesn't move, eigenvalue one, 00:08:35.020 --> 00:08:37.460 so the vector that doesn't move 00:08:37.460 --> 00:08:38.630 is a. 00:08:38.630 --> 00:08:41.390 This a is the -- 00:08:41.390 --> 00:08:47.040 is -- is also the eigenvector with lambda equal one, 00:08:47.040 --> 00:08:51.250 because if I apply the projection matrix to a, I get 00:08:51.250 --> 00:08:52.290 a again. 00:08:52.290 --> 00:08:55.580 Everybody sees that if I apply that matrix to a, can I do it 00:08:55.580 --> 00:08:59.380 in little letters, if I apply that matrix to a, 00:08:59.380 --> 00:09:02.320 then I have a transpose a canceling a transpose a 00:09:02.320 --> 00:09:04.190 and I get a again. 00:09:04.190 --> 00:09:06.720 So sure enough, Pa equals a. 00:09:09.620 --> 00:09:12.220 And the eigenvalue is one. 00:09:12.220 --> 00:09:13.350 OK. 00:09:13.350 --> 00:09:16.140 Good. 00:09:16.140 --> 00:09:22.840 now, actually it asks you further to solve this 00:09:22.840 --> 00:09:27.200 difference equation, so this will be -- this is -- 00:09:27.200 --> 00:09:36.030 this is solve u(k+1)=Puk, starting from u0 equal nine, 00:09:36.030 --> 00:09:36.655 nine, zero. 00:09:39.590 --> 00:09:41.740 And find uk. 00:09:44.380 --> 00:09:49.950 So -- so what's up? 00:09:49.950 --> 00:09:53.140 Shall we find u1 first of all? 00:09:53.140 --> 00:09:55.870 So just to get started. 00:09:55.870 --> 00:09:57.780 So what is u1? 00:09:57.780 --> 00:10:01.570 It's Pu0 of course. 00:10:01.570 --> 00:10:04.480 So if I do the projection of -- 00:10:04.480 --> 00:10:10.020 of this vector onto the line, so this is like my vector b now 00:10:10.020 --> 00:10:12.070 that I'm projecting onto the line, 00:10:12.070 --> 00:10:19.410 I get a times a transpose u0 over a transpose a. 00:10:19.410 --> 00:10:23.080 Well, one way or another I just do this multiplication. 00:10:23.080 --> 00:10:26.100 but maybe this is the easiest way to do it. 00:10:26.100 --> 00:10:29.530 a transpose, can I remember what a is on 00:10:29.530 --> 00:10:30.340 this board? 00:10:30.340 --> 00:10:37.110 Two, one, two, so I'm projecting onto the line through there. 00:10:37.110 --> 00:10:41.450 This is the projection, it's P times the vector u0, 00:10:41.450 --> 00:10:44.590 so what do I have for a transpose u0 looks like 00:10:44.590 --> 00:10:47.000 eighteen, looks like twenty-seven, 00:10:47.000 --> 00:10:50.140 and a transpose a we figured was nine, 00:10:50.140 --> 00:10:57.790 so it's three a, so that this is the -- this is the x hat, 00:10:57.790 --> 00:11:00.870 the -- the multiple of a, in -- in our formulas, 00:11:00.870 --> 00:11:03.162 and of course that's six, three, six. 00:11:06.650 --> 00:11:08.320 So that's u1. 00:11:08.320 --> 00:11:11.500 Computed out directly. 00:11:11.500 --> 00:11:17.530 That's on the line through a and it's the closest point to u0, 00:11:17.530 --> 00:11:19.600 and it's just Pu0. 00:11:19.600 --> 00:11:23.070 You just straightforward multiplication produces that. 00:11:23.070 --> 00:11:23.620 OK. 00:11:23.620 --> 00:11:25.300 Now, what's u2? 00:11:28.420 --> 00:11:31.310 Well, u2 is Pu1, I agree. 00:11:31.310 --> 00:11:32.750 Do I need to compute that again? 00:11:35.540 --> 00:11:41.730 No, because once I'm already on the line through A, uk will be, 00:11:41.730 --> 00:11:45.220 I could do the projection k times, 00:11:45.220 --> 00:11:47.635 but it's enough just to do it once. 00:11:50.380 --> 00:11:53.570 It's the same, it's the same, six, three, six. 00:11:53.570 --> 00:11:59.780 So this is a case where I could and -- 00:11:59.780 --> 00:12:03.620 and actually on the quiz if you see one of these, 00:12:03.620 --> 00:12:09.360 which could very well be there, and it could very well be not 00:12:09.360 --> 00:12:16.120 a projection matrix, then we would use all the eigenvalues 00:12:16.120 --> 00:12:16.940 and eigenvectors. 00:12:16.940 --> 00:12:19.790 Let's think for a moment, how do you do those? 00:12:19.790 --> 00:12:24.400 M - the point of this small part of a question was that when P 00:12:24.400 --> 00:12:28.900 is a projection matrix, so that P squared equals P and P cubed 00:12:28.900 --> 00:12:33.600 equals P, then -- then we don't need to get into the mechanics 00:12:33.600 --> 00:12:38.520 of all knowing all the other eigenvalues and eigenvectors. 00:12:38.520 --> 00:12:40.670 We just can go directly. 00:12:40.670 --> 00:12:48.200 But if P was now some other matrix, can you just -- 00:12:48.200 --> 00:12:51.140 let's just remember from these very recent lectures how you 00:12:51.140 --> 00:12:52.560 would proceed. 00:12:52.560 --> 00:12:57.590 from these very recent lectures we know that uk we would -- 00:12:57.590 --> 00:13:02.780 we would expand u0 as a combination of eigenvectors. 00:13:02.780 --> 00:13:07.020 Let me leave -- yeah, as a combination of eigenvectors, 00:13:07.020 --> 00:13:12.650 c1x1, some multiple of the second eigenvector, 00:13:12.650 --> 00:13:15.470 some multiple of the third eigenvector, 00:13:15.470 --> 00:13:22.320 and then A to the ku0 would be c1, so this -- 00:13:22.320 --> 00:13:27.270 we have to find these numbers here, that's the work actually. 00:13:27.270 --> 00:13:30.280 The work is find the eigenvalues, 00:13:30.280 --> 00:13:33.500 find the eigenvectors and find the c-s because they all 00:13:33.500 --> 00:13:35.320 come into the formula. 00:13:35.320 --> 00:13:50.230 We have -- so -- so to do this, you can see what you have 00:13:50.230 --> 00:13:51.280 to compute. 00:13:51.280 --> 00:13:53.320 You have to compute the eigenvalues, 00:13:53.320 --> 00:13:55.430 you have to compute the eigenvectors, 00:13:55.430 --> 00:13:59.230 and then to match u0 you compute the c-s, 00:13:59.230 --> 00:14:01.210 and then you've got it. 00:14:01.210 --> 00:14:06.610 So it's -- it's just that's a formula that shows what pieces 00:14:06.610 --> 00:14:07.810 we need. 00:14:07.810 --> 00:14:12.550 And what would actually happen in the case of this projection 00:14:12.550 --> 00:14:13.070 matrix? 00:14:13.070 --> 00:14:15.310 If this A is a projection matrix, 00:14:15.310 --> 00:14:19.050 then a couple of eigenvalues are zero. 00:14:19.050 --> 00:14:21.200 That's why we just throw those away. 00:14:21.200 --> 00:14:27.780 The other eigenvalue was a one, so that we got the same thing 00:14:27.780 --> 00:14:30.890 every time, c3x3. 00:14:30.890 --> 00:14:33.050 From the first time, second time, 00:14:33.050 --> 00:14:37.160 third, all iterations pro- left us with this constant, 00:14:37.160 --> 00:14:39.620 left us right here at six, three, six. 00:14:39.620 --> 00:14:41.220 But maybe I take -- 00:14:41.220 --> 00:14:43.840 I'm taking this chance to remind you of what 00:14:43.840 --> 00:14:47.900 to do for other matrices. 00:14:47.900 --> 00:14:48.600 OK. 00:14:48.600 --> 00:14:52.910 So that's part way through. 00:14:52.910 --> 00:14:53.460 OK. 00:14:53.460 --> 00:14:55.980 The next question in nineteen eighty-four 00:14:55.980 --> 00:15:01.030 is fitting a straight line to points. 00:15:01.030 --> 00:15:04.712 And actually a straight line through the origin. 00:15:04.712 --> 00:15:06.170 A straight line through the origin. 00:15:06.170 --> 00:15:10.120 So can I go to question two? 00:15:10.120 --> 00:15:15.360 So this is fitting a straight line to these points, can I -- 00:15:15.360 --> 00:15:26.580 I'll just give you the points at t=1 the y is four, at t=2, 00:15:26.580 --> 00:15:30.710 y is five, at t=3, y is eight. 00:15:34.190 --> 00:15:42.630 So we've got points one, two, three, four, five, and eight. 00:15:42.630 --> 00:15:45.720 And I'm trying to fit a straight line through the origin 00:15:45.720 --> 00:15:48.540 to these three values. 00:15:48.540 --> 00:15:57.170 OK, so my equation that I'm allowing myself 00:15:57.170 --> 00:16:00.950 is just y equal Dt. 00:16:00.950 --> 00:16:03.470 So I have only one unknown. 00:16:03.470 --> 00:16:04.950 One degree of freedom. 00:16:04.950 --> 00:16:07.730 One parameter D. 00:16:07.730 --> 00:16:12.510 So I'm expecting to end up my matrix so my -- my -- 00:16:12.510 --> 00:16:14.510 when I try to -- 00:16:14.510 --> 00:16:17.440 when I try to fit a straight line, that 00:16:17.440 --> 00:16:19.660 goes through the origin, that's because it goes 00:16:19.660 --> 00:16:22.570 through the origin, I've lost the constant c here, 00:16:22.570 --> 00:16:27.330 so I have just this should be a quick calculation. 00:16:27.330 --> 00:16:32.590 and I can write down the three equations that -- that -- 00:16:32.590 --> 00:16:33.200 that would -- 00:16:33.200 --> 00:16:36.990 I'd like to solve if the line went through the points, that's 00:16:36.990 --> 00:16:38.080 a good start. 00:16:38.080 --> 00:16:40.015 Because that displays the matrix. 00:16:43.410 --> 00:16:45.270 So can I continue that problem? 00:16:45.270 --> 00:16:48.760 We would like to solve-- 00:16:48.760 --> 00:16:54.200 so y is Dt, so I'd like to solve D times one times 00:16:54.200 --> 00:17:00.030 D equals four, two times D equals five and three 00:17:00.030 --> 00:17:02.690 times D equals eight. 00:17:02.690 --> 00:17:04.500 That would be perfection. 00:17:04.500 --> 00:17:09.150 If I could find such a D, then the line y 00:17:09.150 --> 00:17:13.349 equal Dt would satisfy all three equations, 00:17:13.349 --> 00:17:16.500 would go through all three points, but it doesn't exist. 00:17:16.500 --> 00:17:19.140 So -- so I have to solve this -- so the -- 00:17:19.140 --> 00:17:23.680 my matrix is now you can see my matrix, it just has one column. 00:17:23.680 --> 00:17:27.589 Multiplying a scalar D. 00:17:27.589 --> 00:17:30.260 And you can see the right-hand side. 00:17:30.260 --> 00:17:31.500 This is my Ax=b. 00:17:34.280 --> 00:17:38.860 I don't need three equals signs now because I've got vectors. 00:17:38.860 --> 00:17:39.620 OK. 00:17:39.620 --> 00:17:43.610 There's Ax=b and you take it from there. 00:17:43.610 --> 00:17:47.570 You the -- the best x will be -- will come from -- 00:17:47.570 --> 00:17:49.510 so what's the -- the key equation? 00:17:49.510 --> 00:17:54.960 So this is the A, this is the Ax hat equal b equation. 00:17:54.960 --> 00:17:57.900 Well, Ax=b. 00:17:57.900 --> 00:18:00.590 And what's the equation for x hat? 00:18:00.590 --> 00:18:08.490 The best D, so to find the best D, the best x, 00:18:08.490 --> 00:18:14.660 the equation is A transpose A, the best D, 00:18:14.660 --> 00:18:20.900 is A transpose times the right-hand side. 00:18:20.900 --> 00:18:24.340 This is all coming from projection on a line, our -- 00:18:24.340 --> 00:18:26.620 our matrix only has one column. 00:18:26.620 --> 00:18:30.070 So A transpose A would be maybe fourteen, 00:18:30.070 --> 00:18:35.410 D hat, and A transpose b I'm getting four, ten, 00:18:35.410 --> 00:18:37.520 and twenty-four. 00:18:37.520 --> 00:18:38.590 Is that right? 00:18:38.590 --> 00:18:40.950 Four, ten and twenty-four. 00:18:40.950 --> 00:18:43.000 So thirty-eight. 00:18:43.000 --> 00:18:47.530 So that tells me the best D hat is D hat 00:18:47.530 --> 00:18:52.150 is thirty-eight over fourteen. 00:18:52.150 --> 00:18:53.310 OK. 00:18:53.310 --> 00:18:54.930 Fine. 00:18:54.930 --> 00:18:57.620 All right. 00:18:57.620 --> 00:18:59.240 so we found the best line. 00:18:59.240 --> 00:19:01.830 And now here's a -- here's the next question. 00:19:01.830 --> 00:19:07.050 What vector did I just project onto what line? 00:19:07.050 --> 00:19:12.210 See in this section on least squares here's the key point, 00:19:12.210 --> 00:19:12.740 I'm -- 00:19:12.740 --> 00:19:14.700 I'm asking you to think of the least squares 00:19:14.700 --> 00:19:17.240 problem in two ways. 00:19:17.240 --> 00:19:18.890 Two different pictures. 00:19:18.890 --> 00:19:20.080 Two different graphs. 00:19:20.080 --> 00:19:22.320 One graph is this. 00:19:22.320 --> 00:19:28.060 This is a graph in the -- in the b -- in the tb plane, ty plane. 00:19:28.060 --> 00:19:30.900 The -- the -- the line itself. 00:19:30.900 --> 00:19:32.940 The other picture I'm asking you to think of 00:19:32.940 --> 00:19:35.050 is like my projection picture. 00:19:35.050 --> 00:19:38.610 What -- what projection -- what -- what vector I -- 00:19:38.610 --> 00:19:42.940 I projecting onto what line or what subspace when I -- when I 00:19:42.940 --> 00:19:44.050 do this? 00:19:44.050 --> 00:19:48.700 So the -- my second picture is a projection picture that -- 00:19:48.700 --> 00:19:51.280 that sees the whole thing with vectors. 00:19:51.280 --> 00:19:53.970 Here's my vector of course that I'm projecting. 00:19:53.970 --> 00:20:06.800 I'm projecting that vector b onto the column space of A. 00:20:06.800 --> 00:20:15.800 Of if you like -- it's just a line onto that's the line 00:20:15.800 --> 00:20:18.620 it's just a line, of course. 00:20:18.620 --> 00:20:21.070 That's what this calculation is doing. 00:20:21.070 --> 00:20:25.595 This is computing the best D, which is -- this is the x hat. 00:20:30.230 --> 00:20:34.090 So -- so seeing it as a projection means I don't see 00:20:34.090 --> 00:20:36.570 the projection in this figure, right? 00:20:36.570 --> 00:20:39.060 In this figure I'm not projecting those points 00:20:39.060 --> 00:20:41.760 onto that line or anything of the sort. 00:20:41.760 --> 00:20:48.430 The projection s-picture for -- for least squares is in the -- 00:20:48.430 --> 00:20:51.900 in the space where b lies, the whole vector b, 00:20:51.900 --> 00:20:54.870 and the columns of A. 00:20:54.870 --> 00:21:02.570 And then the x is the best combination 00:21:02.570 --> 00:21:04.600 that gives the projection. 00:21:04.600 --> 00:21:05.100 OK. 00:21:05.100 --> 00:21:07.980 So that's a chance to tell me that. 00:21:07.980 --> 00:21:08.760 OK. 00:21:08.760 --> 00:21:12.310 I'll go -- OK now finally in orthogonality 00:21:12.310 --> 00:21:14.610 there's the Graham-Schmidt idea. 00:21:14.610 --> 00:21:20.230 So that's problem two D here. 00:21:20.230 --> 00:21:24.840 It asks me if I have two vectors, a1 equal one, two, 00:21:24.840 --> 00:21:32.650 three, and a2 equal one, one, one, find 00:21:32.650 --> 00:21:37.650 two orthogonal vectors in that plane. 00:21:37.650 --> 00:21:43.480 So those two vectors give a plane, they give a plane. 00:21:43.480 --> 00:21:47.860 Which is of course the -- the column space of the -- 00:21:47.860 --> 00:21:50.710 of the matrix. 00:21:50.710 --> 00:21:55.050 And I'm looking for an orthogonal basis 00:21:55.050 --> 00:21:55.710 for that plane. 00:21:55.710 --> 00:21:58.300 So I'm looking for two orthogonal vectors. 00:21:58.300 --> 00:22:02.100 And of course there are lots of -- 00:22:02.100 --> 00:22:05.330 I mean, I've got a plane there. 00:22:05.330 --> 00:22:08.700 If I get one orthogonal pair, I can rotate it. 00:22:08.700 --> 00:22:10.910 There's not just one answer here. 00:22:10.910 --> 00:22:16.140 But Graham-Schmidt says OK, start with the first vector, 00:22:16.140 --> 00:22:19.840 and let that be -- and keep that one. 00:22:19.840 --> 00:22:23.300 And then take the second one orthogonal to this. 00:22:23.300 --> 00:22:27.400 So -- so Graham-Schmidt says start with this one and then 00:22:27.400 --> 00:22:31.220 make a second vector B, can I call that second vector B, 00:22:31.220 --> 00:22:35.970 this is going to be orthogonal to, so perpendicular to a1. 00:22:35.970 --> 00:22:40.650 If I can with my chalk create the key equation. 00:22:40.650 --> 00:22:46.740 This vector B is going to be this one, one, one, 00:22:46.740 --> 00:22:50.720 but that one, one -- one, one, one is not perpendicular to a1, 00:22:50.720 --> 00:22:54.340 so I have to subtract off its projection, 00:22:54.340 --> 00:23:00.820 I have to subtract off the B, the -- the B trans- ye the B -- 00:23:00.820 --> 00:23:07.780 the -- the I should say the a1 transpose b over a1 transpose 00:23:07.780 --> 00:23:11.400 a1, that multiple of a1, I've got to remove. 00:23:14.990 --> 00:23:17.090 So I just have to compute what that is, 00:23:17.090 --> 00:23:21.440 and I get ano- I get a vector B that's orthogonal to a1. 00:23:21.440 --> 00:23:24.440 It's the -- it's -- 00:23:24.440 --> 00:23:29.630 it's the original vector minus its projection. 00:23:29.630 --> 00:23:32.110 Oh, so what is -- 00:23:32.110 --> 00:23:33.490 I mean this to be a2. 00:23:36.130 --> 00:23:39.430 So I'm projecting a2 onto the line through a1. 00:23:39.430 --> 00:23:40.190 Yeah. 00:23:40.190 --> 00:23:42.190 That's the part that I don't want because that's 00:23:42.190 --> 00:23:45.250 in the direction I already have, so I subtract off 00:23:45.250 --> 00:23:48.170 that projection and I get the part I want, 00:23:48.170 --> 00:23:50.240 the orthogonal part. 00:23:50.240 --> 00:23:52.540 So that's the Graham-Schmidt thing 00:23:52.540 --> 00:23:54.640 and we can put numbers in. 00:23:54.640 --> 00:23:55.370 OK. 00:23:55.370 --> 00:24:00.710 one, one, one take away a1 transpose a2 is six, 00:24:00.710 --> 00:24:06.900 a1 transpose a1 is fourteen,multiplying a1. 00:24:06.900 --> 00:24:13.400 And that gives us the new orthogonal vector B. 00:24:13.400 --> 00:24:15.340 Because I only ask for orthogonal right now, 00:24:15.340 --> 00:24:19.200 I don't have to divide by the length which 00:24:19.200 --> 00:24:20.590 will involve a square root. 00:24:20.590 --> 00:24:23.210 OK. 00:24:23.210 --> 00:24:23.940 Third question. 00:24:27.540 --> 00:24:28.677 Third question. 00:24:28.677 --> 00:24:29.510 All right, let me -- 00:24:29.510 --> 00:24:33.270 I'll move this board up. 00:24:33.270 --> 00:24:37.930 third question will probably be about eigenvalues. 00:24:37.930 --> 00:24:38.610 OK. 00:24:38.610 --> 00:24:41.200 Three. 00:24:41.200 --> 00:24:44.020 This is a four-by-four matrix. 00:24:44.020 --> 00:24:47.320 Its eigenvalues are lambda one, lambda two, lambda three, 00:24:47.320 --> 00:24:50.310 lambda four. 00:24:50.310 --> 00:24:52.930 Question one. 00:24:52.930 --> 00:24:55.870 What's the condition on the lambdas so 00:24:55.870 --> 00:24:59.380 that the matrix is invertible? 00:24:59.380 --> 00:24:59.990 OK. 00:24:59.990 --> 00:25:02.510 So under what conditions on the lambdas 00:25:02.510 --> 00:25:05.900 will the matrix be invertible? 00:25:05.900 --> 00:25:08.850 So that's easy. 00:25:08.850 --> 00:25:17.780 Invertible if what's the condition on the lambdas? 00:25:17.780 --> 00:25:19.370 None of them are zero. 00:25:19.370 --> 00:25:23.900 A zero eigenvalue would mean something in the null space 00:25:23.900 --> 00:25:28.850 would mean a solution to Ax=0x, but we're invertible, 00:25:28.850 --> 00:25:32.170 so none of them is zero, the product -- 00:25:32.170 --> 00:25:34.120 however you want to say, no -- 00:25:34.120 --> 00:25:38.510 no zero eigenvalues. 00:25:38.510 --> 00:25:39.170 Good. 00:25:39.170 --> 00:25:43.200 OK, what's the determinant of A inverse? 00:25:43.200 --> 00:25:45.555 The determinant of A inverse? 00:25:49.280 --> 00:25:52.120 So where is that going to come from? 00:25:52.120 --> 00:25:55.470 Well, if we knew the eigenvalues of A inverse, 00:25:55.470 --> 00:25:59.780 we could multiply them together to find the determinant. 00:25:59.780 --> 00:26:02.110 And we do know the eigenvalues of A inverse. 00:26:02.110 --> 00:26:04.370 What are they? 00:26:04.370 --> 00:26:08.580 They're just one over lambda one times 00:26:08.580 --> 00:26:11.470 one over lambda two, that's the second eigenvalue, 00:26:11.470 --> 00:26:13.640 the third eigenvalue and the 00:26:13.640 --> 00:26:14.760 fourth. 00:26:14.760 --> 00:26:18.770 So the product of the four eigenvalues of the inverse 00:26:18.770 --> 00:26:21.450 will give us the determinant of the inverse. 00:26:21.450 --> 00:26:21.950 Fine. 00:26:21.950 --> 00:26:23.110 OK. 00:26:23.110 --> 00:26:39.600 And what's the trace of A plus I? 00:26:39.600 --> 00:26:41.170 So what do we know about trace? 00:26:44.160 --> 00:26:46.510 It's the sum down the diagonal, but we 00:26:46.510 --> 00:26:48.630 don't know what our matrix is. 00:26:48.630 --> 00:26:51.960 The trace is also the sum of the eigenvalues, 00:26:51.960 --> 00:26:55.710 and we do know the eigenvalues of A plus I. 00:26:55.710 --> 00:26:57.410 So we just add them up. 00:26:57.410 --> 00:27:02.520 So what -- what's the first eigenvalue of A plus I? 00:27:02.520 --> 00:27:05.610 When the matrix A has eigenvalues lambda one, two, 00:27:05.610 --> 00:27:08.030 three and four, then the eigenvalues 00:27:08.030 --> 00:27:11.610 if I add the identity, that moves all the eigenvalues 00:27:11.610 --> 00:27:17.450 by one, so I just add up lambda one plus one, 00:27:17.450 --> 00:27:22.520 lambda two plus one, and so on, lambda three plus one, lambda 00:27:22.520 --> 00:27:27.180 four plus one, so it's lambda one plus lambda two plus lambda 00:27:27.180 --> 00:27:32.180 three plus lambda four plus four. 00:27:32.180 --> 00:27:34.850 Right. 00:27:34.850 --> 00:27:38.110 That movement by the identity moved all the eigenvalues 00:27:38.110 --> 00:27:42.290 by one, so it moved the whole trace by four. 00:27:42.290 --> 00:27:45.690 So it was the trace of A plus four more. 00:27:45.690 --> 00:27:46.880 OK. 00:27:46.880 --> 00:27:48.240 Let's see. 00:27:48.240 --> 00:27:51.600 We may be finished this quiz twenty minutes early. 00:27:51.600 --> 00:27:52.420 No. 00:27:52.420 --> 00:27:54.700 There's another question. 00:27:54.700 --> 00:27:57.590 Oh, God, OK. 00:27:57.590 --> 00:27:59.270 How did this class ever do it? 00:27:59.270 --> 00:28:02.710 Well, you'll see. you'll be able to do it. 00:28:02.710 --> 00:28:03.510 OK. 00:28:03.510 --> 00:28:06.950 this has got to be a determinant question. 00:28:06.950 --> 00:28:09.290 All right. 00:28:09.290 --> 00:28:13.810 More determinants and cofactors and big formula question. 00:28:13.810 --> 00:28:14.430 OK. 00:28:14.430 --> 00:28:18.760 Let me do that. 00:28:18.760 --> 00:28:22.820 So it's about a matrix, a -- a whole family of matrices. 00:28:22.820 --> 00:28:26.200 Here's the four-by-four one. 00:28:26.200 --> 00:28:29.990 The four-by-four one is, and -- and all the matrices in this 00:28:29.990 --> 00:28:35.240 family are tridiagonal with -- 00:28:35.240 --> 00:28:38.230 with ones. 00:28:38.230 --> 00:28:40.110 Otherwise zeroes. 00:28:40.110 --> 00:28:43.480 So that's the pattern, and we've seen this matrix. 00:28:43.480 --> 00:28:44.840 OK. 00:28:44.840 --> 00:28:47.850 So the -- it's tridiagonal with ones on the diagonal, 00:28:47.850 --> 00:28:52.790 ones above and ones below, and you see the general formula An, 00:28:52.790 --> 00:28:58.780 so I'll use Dn for the determinant of An. 00:28:58.780 --> 00:28:59.950 OK. 00:28:59.950 --> 00:29:00.990 All right. 00:29:00.990 --> 00:29:02.950 So I'm going to do a -- 00:29:02.950 --> 00:29:14.740 the first question is use cofactors to show that Dn is 00:29:14.740 --> 00:29:20.680 something times D(n-1) plus something times D(n-2). 00:29:20.680 --> 00:29:23.150 And find those somethings. 00:29:23.150 --> 00:29:23.650 OK. 00:29:26.310 --> 00:29:31.020 So this -- the fact that it's tridiagonal with these constant 00:29:31.020 --> 00:29:37.290 diagonals means that there is such a recurrence formula. 00:29:37.290 --> 00:29:39.660 And so the first question is find it. 00:29:39.660 --> 00:29:41.810 Well, what's the recurrence formula? 00:29:41.810 --> 00:29:43.550 OK, how does it go? 00:29:43.550 --> 00:29:47.390 So I'll use cofactors along the first row. 00:29:47.390 --> 00:29:51.570 So I take that number times its cofactor. 00:29:51.570 --> 00:29:56.730 So it's one times its cofactor and what is its cofactor? 00:29:56.730 --> 00:30:00.120 D(n-1), right, exactly, the cofactor is this -- 00:30:00.120 --> 00:30:03.510 is this guy uses up row one and column one, 00:30:03.510 --> 00:30:07.670 so the cofactor is down here, so it's one of those. 00:30:10.230 --> 00:30:12.550 OK, that's the first cofactor term. 00:30:12.550 --> 00:30:17.090 Now the other cofactor term is this guy. 00:30:17.090 --> 00:30:20.740 Which uses up row one and column two 00:30:20.740 --> 00:30:24.870 and what's surprising about that? 00:30:24.870 --> 00:30:30.660 When you use row one and column two that brings in a minus. 00:30:30.660 --> 00:30:32.940 There'll be a minus because the -- 00:30:32.940 --> 00:30:38.490 the cofactor is this determinant times minus one. 00:30:38.490 --> 00:30:44.440 The the one-two cofactor is that determinant with its sign 00:30:44.440 --> 00:30:45.460 changed. 00:30:45.460 --> 00:30:45.960 OK. 00:30:45.960 --> 00:30:47.270 So I have to look at that determinant 00:30:47.270 --> 00:30:48.645 and I have to remember in my head 00:30:48.645 --> 00:30:50.620 a sign is going to get changed. 00:30:50.620 --> 00:30:51.160 OK. 00:30:51.160 --> 00:30:54.970 Now how do I do that determinant? 00:30:54.970 --> 00:30:58.180 How do I make that one clear? 00:30:58.180 --> 00:31:01.810 I -- the -- the neat way to do is -- is here I see I -- 00:31:01.810 --> 00:31:05.570 I'll use cofactors down the first column. 00:31:05.570 --> 00:31:08.880 Because the first column is all zeroes except for that one, 00:31:08.880 --> 00:31:13.220 so this one is now -- and what's its cofactor? 00:31:13.220 --> 00:31:17.490 Within this three-by-three its cofactor will be two-by-two, 00:31:17.490 --> 00:31:19.080 and what is it? 00:31:19.080 --> 00:31:20.970 It's this, right? 00:31:20.970 --> 00:31:25.730 So -- so that part is all gone, so I'm taking that times its 00:31:25.730 --> 00:31:29.550 cofactor, then zero times whatever its cofactor is, 00:31:29.550 --> 00:31:34.040 so it's really just one times and what's this in the general 00:31:34.040 --> 00:31:35.510 n-by-n case? 00:31:35.510 --> 00:31:39.420 It's Dn minus two. 00:31:39.420 --> 00:31:43.660 But now so is this a plus or sign or a minus sign, it's -- 00:31:43.660 --> 00:31:48.360 it's just a one, because there's a one from there and a one from 00:31:48.360 --> 00:31:50.050 there. 00:31:50.050 --> 00:31:52.930 And is it a plus or a minus? 00:31:52.930 --> 00:31:56.010 It's minus I guess because there was a minus the first time 00:31:56.010 --> 00:31:57.970 and then the second time it's a plus, 00:31:57.970 --> 00:32:00.240 so it's overall it's a minus. 00:32:00.240 --> 00:32:04.864 So there's my a and b were one and minus one. 00:32:04.864 --> 00:32:05.530 Those constants. 00:32:05.530 --> 00:32:09.140 Th- that's the -- that's the recurrence. 00:32:09.140 --> 00:32:10.220 OK. 00:32:10.220 --> 00:32:19.180 And oh, then it asks you to then it asks you to solve this thing 00:32:19.180 --> 00:32:22.070 first by writing it as a -- 00:32:22.070 --> 00:32:25.120 as a system. 00:32:25.120 --> 00:32:28.200 So now I'd like to know the solution. 00:32:28.200 --> 00:32:30.600 I -- I better know how it starts, right? 00:32:30.600 --> 00:32:34.020 It starts with D1, what was D1, that's 00:32:34.020 --> 00:32:39.550 just the one-by-one case, so D1 is one, and what is D2? 00:32:39.550 --> 00:32:41.880 Just to get us started and then this would give us 00:32:41.880 --> 00:32:45.520 D3, D4, and forever. 00:32:45.520 --> 00:32:48.390 D2 is this two-by-two that I'm seeing here 00:32:48.390 --> 00:32:51.620 and that determinant is obviously zero. 00:32:51.620 --> 00:32:57.410 So those little ones will start the recurrence and then we take 00:32:57.410 --> 00:32:58.030 off. 00:32:58.030 --> 00:33:02.070 And then the idea is to write this recurrence as -- 00:33:02.070 --> 00:33:11.990 as a Dn, D(n-1) is some matrix times the one before, 00:33:11.990 --> 00:33:16.613 the D(n-1), D(n-2). 00:33:20.250 --> 00:33:22.270 What's the matrix? 00:33:22.270 --> 00:33:26.570 You see, you remember this step of taking a single second order 00:33:26.570 --> 00:33:31.090 equation and by introducing a vector unknown to make it 00:33:31.090 --> 00:33:32.600 into a -- 00:33:32.600 --> 00:33:35.600 to a first order system. 00:33:35.600 --> 00:33:36.200 OK. 00:33:36.200 --> 00:33:41.860 So Dn is one of Dn minus one minus one, I think that -- 00:33:41.860 --> 00:33:43.530 that goes in the first row, right? 00:33:43.530 --> 00:33:45.640 From the equation above? 00:33:45.640 --> 00:33:48.200 And the second one is this is the same as this, 00:33:48.200 --> 00:33:50.010 so one and zero are fine. 00:33:53.500 --> 00:33:55.330 So there's the matrix. 00:33:55.330 --> 00:33:55.920 OK. 00:33:55.920 --> 00:33:58.820 So now how do I proceed? 00:33:58.820 --> 00:34:01.590 We can guess what this examiner's 00:34:01.590 --> 00:34:02.845 got in his little mind. 00:34:07.740 --> 00:34:09.120 well, find the eigenvalues. 00:34:12.760 --> 00:34:19.469 And actually it tells us that the sixth power 00:34:19.469 --> 00:34:23.679 of these eigenvalues turns out to be one. 00:34:23.679 --> 00:34:30.065 Uh, well, can -- can we get the equation for the eigenvalues? 00:34:30.065 --> 00:34:31.940 Let's do it and let's get a formula for them. 00:34:31.940 --> 00:34:32.880 OK. 00:34:32.880 --> 00:34:35.260 So what are the eigenvalues? 00:34:35.260 --> 00:34:39.590 I look at the -- the matrix, this determinant one minus 00:34:39.590 --> 00:34:44.420 lambda and zero minus lambda, and these guys are still there, 00:34:44.420 --> 00:34:49.400 I compute that determinant, I get lambda squared minus lambda 00:34:49.400 --> 00:34:52.310 and then plus one. 00:34:52.310 --> 00:34:54.820 And I set that to zero. 00:34:54.820 --> 00:34:55.820 OK. 00:34:55.820 --> 00:34:59.540 So we're not Fibonacci here. 00:34:59.540 --> 00:35:02.930 We're -- we're not seeing Fibonacci numbers. 00:35:02.930 --> 00:35:07.500 Because the sign -- we had a sign change there. 00:35:07.500 --> 00:35:11.040 And it's not clear right away whether these -- 00:35:11.040 --> 00:35:13.380 whether this -- is it clear? 00:35:13.380 --> 00:35:18.400 Is this matrix stable or unstable? 00:35:18.400 --> 00:35:21.000 When we take -- when we go further and further out? 00:35:21.000 --> 00:35:23.520 Are these Ds increasing? 00:35:23.520 --> 00:35:25.050 Are they going to zero? 00:35:25.050 --> 00:35:27.860 Are they bouncing around periodically? 00:35:27.860 --> 00:35:30.600 the answers have to be here. 00:35:30.600 --> 00:35:34.840 I would like to know how big these lambdas are, right? 00:35:34.840 --> 00:35:37.880 And the point is probably these -- let's -- let's see, 00:35:37.880 --> 00:35:38.930 what's lambda? 00:35:38.930 --> 00:35:42.780 From the quadratic formula lambda is one, 00:35:42.780 --> 00:35:45.390 I switch the sign of that, plus or minus 00:35:45.390 --> 00:35:50.570 the square root of one minus 4ac, I getting a minus three 00:35:50.570 --> 00:35:51.970 there? 00:35:51.970 --> 00:35:52.710 Over two. 00:35:58.860 --> 00:36:00.960 What's up? 00:36:00.960 --> 00:36:03.340 They're complex. 00:36:03.340 --> 00:36:08.240 The -- the eigenvalues are one plus square root of three I 00:36:08.240 --> 00:36:15.600 over two and one minus square root of three I over two. 00:36:15.600 --> 00:36:18.240 What's the magnitude of lambda? 00:36:18.240 --> 00:36:19.765 That's the key point for stability. 00:36:22.560 --> 00:36:26.480 These are two numbers in the complex plane. 00:36:26.480 --> 00:36:30.650 One plus some -- somewhere here, and its complex conjugate 00:36:30.650 --> 00:36:32.820 there. 00:36:32.820 --> 00:36:37.580 I want to know how far from the origin are those numbers. 00:36:37.580 --> 00:36:40.840 What's the magnitude of lambda? 00:36:40.840 --> 00:36:43.230 And do you see what it is? 00:36:43.230 --> 00:36:46.690 Do you recognize this -- a number like that? 00:36:46.690 --> 00:36:50.220 Take the real part squared and the imaginary part squared 00:36:50.220 --> 00:36:51.510 and add. 00:36:51.510 --> 00:36:53.840 What do you get? 00:36:53.840 --> 00:36:57.010 So the real part squared is a quarter. 00:36:57.010 --> 00:36:59.990 The imaginary part squared is three-quarters. 00:36:59.990 --> 00:37:01.200 They add to one. 00:37:01.200 --> 00:37:06.090 That's a number with -- that's on the unit circle. 00:37:06.090 --> 00:37:08.540 That's an e to the i theta. 00:37:08.540 --> 00:37:10.180 That's a cos(theta)+isin(theta). 00:37:10.180 --> 00:37:14.650 And what's theta? 00:37:14.650 --> 00:37:19.330 This -- this is like a complex number that's worth knowing, 00:37:19.330 --> 00:37:23.670 it's not totally obvious but it's nice. 00:37:23.670 --> 00:37:27.480 That's -- I should see that as cos(theta)+isin(theta), 00:37:27.480 --> 00:37:30.770 and the angle that would do that is sixty degrees, 00:37:30.770 --> 00:37:32.580 pi over three. 00:37:32.580 --> 00:37:36.520 So that's a -- let me improve my picture. 00:37:36.520 --> 00:37:39.990 So those -- that's e to the i pi over six -- 00:37:39.990 --> 00:37:40.920 pi over three. 00:37:40.920 --> 00:37:46.840 This is -- this number is e to the i pi over three and e 00:37:46.840 --> 00:37:49.840 to the minus i pi over three. 00:37:49.840 --> 00:37:54.400 We'll be doing more complex numbers briefly 00:37:54.400 --> 00:37:56.545 but a little more in the next two days. 00:37:59.560 --> 00:38:00.940 next two lectures. 00:38:00.940 --> 00:38:05.670 Anyway, the -- so what's the deal with stability, 00:38:05.670 --> 00:38:08.600 what do the Dn-s do? 00:38:08.600 --> 00:38:13.570 Well, look, if -- if I take the sixth power I'm around at one, 00:38:13.570 --> 00:38:16.300 the problem actually told me this. 00:38:16.300 --> 00:38:18.950 The sixth power of those eigenvalues brings me around to 00:38:18.950 --> 00:38:23.350 What does that tell you about the matrix, by the way? one. 00:38:23.350 --> 00:38:26.010 Suppose you know -- this was a great quiz question, 00:38:26.010 --> 00:38:29.510 so I should never have just said it, but popped out. 00:38:29.510 --> 00:38:33.600 Suppose lambda one to the sixth and lambda two to the sixth are 00:38:33.600 --> 00:38:36.930 -- are one, which they are. 00:38:36.930 --> 00:38:40.090 What does that tell me about a m- a matrix? 00:38:40.090 --> 00:38:43.890 About my matrix A here. 00:38:43.890 --> 00:38:47.290 Well, what -- what matrix is connected with lambda one 00:38:47.290 --> 00:38:49.180 to the sixth and lambda two to the sixth? 00:38:49.180 --> 00:38:51.890 It's got to be the matrix A to the sixth. 00:38:51.890 --> 00:38:55.700 So what is A to the sixth for that matrix? 00:38:55.700 --> 00:39:00.170 It's got eigenvalues one and one. 00:39:00.170 --> 00:39:04.720 Because when I take the sixth power, actually, ye, 00:39:04.720 --> 00:39:09.610 if I take the sixth power b- all the sixth power of that is one 00:39:09.610 --> 00:39:12.990 and the sixth power of that is one, the sixth power of this 00:39:12.990 --> 00:39:16.660 is e to the two pi i, that's one, the sixth power of this 00:39:16.660 --> 00:39:19.330 is e to the minus two pi i, that's one. 00:39:19.330 --> 00:39:24.190 So the sixth powers, the -- the sixth power of that matrix has 00:39:24.190 --> 00:39:27.850 eigenvalues one and one, so what is it? 00:39:27.850 --> 00:39:30.400 It's the identity, right. 00:39:30.400 --> 00:39:33.940 So if I operate this -- if I run this thing six times, 00:39:33.940 --> 00:39:35.930 I'm back where I was. 00:39:35.930 --> 00:39:39.340 The sixth power of that matrix is the identity. 00:39:39.340 --> 00:39:40.550 Good. 00:39:40.550 --> 00:39:41.670 OK. 00:39:41.670 --> 00:39:44.890 So it'll loop around, it's -- it doesn't go to zero, 00:39:44.890 --> 00:39:49.080 it doesn't blow up, it just periodically goes around with 00:39:49.080 --> 00:39:50.360 period six. 00:39:50.360 --> 00:39:51.460 OK. 00:39:51.460 --> 00:39:54.440 let's just see if there's a -- 00:39:54.440 --> 00:39:55.095 all right. 00:39:58.520 --> 00:40:00.020 I'll -- let's see. 00:40:00.020 --> 00:40:02.250 Could I also look at a -- 00:40:02.250 --> 00:40:05.155 at a final exam from nineteen ninety-two. 00:40:07.870 --> 00:40:11.570 I think that's yeah, let me do that on this last board. 00:40:14.740 --> 00:40:18.080 It starts -- a lot of the questions in this exam are 00:40:18.080 --> 00:40:20.760 about a family of matrices. 00:40:20.760 --> 00:40:25.300 Let me give you the fourth, the fourth guy in the family is -- 00:40:25.300 --> 00:40:29.880 has a one, so it's zeroes on the diagonal, 00:40:29.880 --> 00:40:34.130 but these are going one, two, three and so on. 00:40:34.130 --> 00:40:37.420 One, two, three, and so on. 00:40:37.420 --> 00:40:42.190 But, for the four-by-four case I'm stopping at four. 00:40:42.190 --> 00:40:47.070 You see the pattern? 00:40:47.070 --> 00:40:49.670 It's a family of matrices which is growing, 00:40:49.670 --> 00:40:52.760 and actually the numbers -- it's symmetric, right, 00:40:52.760 --> 00:40:56.490 it's equal to A4 transpose. 00:40:56.490 --> 00:41:00.880 And we can ask all sorts of questions about its null space, 00:41:00.880 --> 00:41:07.030 its range, r- its column space find the projection 00:41:07.030 --> 00:41:12.330 matrix onto the column space of A3, for example, is in here. 00:41:12.330 --> 00:41:21.730 So -- so one -- so A3 is zero, one, zero, one, zero, two, 00:41:21.730 --> 00:41:24.100 zero, two, zero. 00:41:32.400 --> 00:41:36.900 OK, find the projection matrix onto the column space. 00:41:36.900 --> 00:41:41.540 By the way, is that matrix singular or invertible? 00:41:41.540 --> 00:41:42.310 Singular. 00:41:42.310 --> 00:41:45.750 Why do we know it's singular? 00:41:45.750 --> 00:41:50.500 I see that column three is a multiple of column one. 00:41:50.500 --> 00:41:53.090 Or we could take its determinant. 00:41:53.090 --> 00:41:55.730 So it's certainly singular. 00:41:55.730 --> 00:42:00.200 The projection will be matrix will be three-by-three 00:42:00.200 --> 00:42:03.190 but it will project onto the column space, 00:42:03.190 --> 00:42:05.610 it'll project onto this plane. 00:42:05.610 --> 00:42:09.240 The column space of A3, and I guess I would find it from 00:42:09.240 --> 00:42:11.100 the formula AA -- 00:42:11.100 --> 00:42:15.120 AA transpose A inverse, I would have to -- 00:42:15.120 --> 00:42:17.120 I would -- I guess I would do all this. 00:42:20.970 --> 00:42:23.050 There may be a better way, perhaps 00:42:23.050 --> 00:42:25.700 I could think there might be a slightly quicker way, 00:42:25.700 --> 00:42:27.390 but that would come out pretty fast. 00:42:27.390 --> 00:42:28.450 OK. 00:42:28.450 --> 00:42:31.260 So that's be the projection matrix. 00:42:31.260 --> 00:42:32.360 Next question. 00:42:32.360 --> 00:42:35.010 Find the eigenvalues and eigenvectors of that matrix. 00:42:35.010 --> 00:42:36.020 OK. 00:42:36.020 --> 00:42:38.590 There's a three-by-three matrix, oh, yeah, 00:42:38.590 --> 00:42:40.560 so what are its eigenvalues and eigenvectors, 00:42:40.560 --> 00:42:42.840 we haven't done any three-by-threes. 00:42:42.840 --> 00:42:45.070 Let's do one. 00:42:45.070 --> 00:42:48.770 I want to find, so how do I find eigenvalues? 00:42:48.770 --> 00:42:53.280 I take the determinant of A3 minus lambda I. 00:42:53.280 --> 00:42:57.200 So this is you just have to -- so I'm subtracting lambda from 00:42:57.200 --> 00:43:02.650 the diagonal, and I have a one, one, zero, zero, two, 00:43:02.650 --> 00:43:07.200 two there, and I just have to find that determinant. 00:43:07.200 --> 00:43:11.030 OK, since it's three-by-three I'll just go for it. 00:43:11.030 --> 00:43:16.850 This way gives me minus lambda cubed and a zero and zero. 00:43:16.850 --> 00:43:20.060 Then in this direction which has the minus sign, that's 00:43:20.060 --> 00:43:25.090 a zero, four lambdas, I mean minus four lambdas, 00:43:25.090 --> 00:43:29.110 and minus another lambda, so that's minus five lambdas, 00:43:29.110 --> 00:43:32.550 but that direction goes with a minus sign, 00:43:32.550 --> 00:43:36.850 so I think it's plus five lambda. 00:43:36.850 --> 00:43:40.330 That looks like the determinant of A3 minus lambda I, 00:43:40.330 --> 00:43:43.030 so I set it to zero. 00:43:43.030 --> 00:43:45.520 So what are the eigenvalues? 00:43:45.520 --> 00:43:48.950 Well, lambda equals zero -- lambda factors out of this, 00:43:48.950 --> 00:43:52.790 times minus lambda squared plus four, 00:43:52.790 --> 00:44:03.610 so the eigenvalues are five, thanks, thanks, 00:44:03.610 --> 00:44:08.470 so the eigenvalues are zero, square root of five, 00:44:08.470 --> 00:44:11.410 and minus square root of five. 00:44:11.410 --> 00:44:13.970 And I would never write down those three eigenvalues 00:44:13.970 --> 00:44:17.120 without checking the trace to tell the truth. 00:44:17.120 --> 00:44:19.520 Because -- because we did a bunch of calculations here 00:44:19.520 --> 00:44:22.990 but then I can quickly add up the eigenvalues to get zero, 00:44:22.990 --> 00:44:27.270 add up the trace to get zero, and feel that I'm -- 00:44:27.270 --> 00:44:30.510 well, I guess that wouldn't have caught my error if I'd made it 00:44:30.510 --> 00:44:34.280 -- if -- if that had been a four I wouldn't have noticed,the 00:44:34.280 --> 00:44:37.660 determinant isn't anything greatly useful here, right, 00:44:37.660 --> 00:44:41.310 because the determinant is just zero. 00:44:41.310 --> 00:44:44.190 And so I never would know whether that five 00:44:44.190 --> 00:44:48.990 was right or wrong, but thanks for making it right. 00:44:48.990 --> 00:44:49.490 OK. 00:44:52.690 --> 00:44:54.380 Ha. 00:44:54.380 --> 00:45:00.760 Question two c, whoever wrote this, probably me, 00:45:00.760 --> 00:45:03.270 said this is not difficult. 00:45:03.270 --> 00:45:06.570 I don't know why I put that in. 00:45:06.570 --> 00:45:10.640 just -- it asks for the projection matrix onto 00:45:10.640 --> 00:45:12.050 the column space of A4. 00:45:15.402 --> 00:45:17.360 How could I have thought that wasn't difficult? 00:45:17.360 --> 00:45:25.580 It looks extremely difficult. what's the projection matrix 00:45:25.580 --> 00:45:27.505 onto the column space of A4? 00:45:32.250 --> 00:45:34.680 I don't know whether that -- this is not difficult is just 00:45:34.680 --> 00:45:37.650 like helpful or -- or insulting. 00:45:37.650 --> 00:45:41.380 Uh, what do you think? 00:45:41.380 --> 00:45:43.870 The -- what's the column space of A4 here? 00:45:46.910 --> 00:45:52.800 Well, what's our first question is is the matrix singular 00:45:52.800 --> 00:45:54.280 or invertible? 00:45:54.280 --> 00:45:58.210 If the answer is invertible, then what's the column space? 00:46:00.770 --> 00:46:03.820 If -- if this matrix A4 is invertible, so that's my guess, 00:46:03.820 --> 00:46:07.200 if this problem's easy it has to be because this matrix is 00:46:07.200 --> 00:46:08.600 probably invertible. 00:46:08.600 --> 00:46:12.940 Then its column space is R^4, good, 00:46:12.940 --> 00:46:15.260 the column space is the whole space, 00:46:15.260 --> 00:46:18.320 and the answer to this easy question is the projection 00:46:18.320 --> 00:46:23.180 matrix is the identity, it's the four-by-four identity matrix. 00:46:23.180 --> 00:46:26.130 If this matrix is invertible. 00:46:26.130 --> 00:46:27.510 Shall we check invertibility? 00:46:27.510 --> 00:46:29.840 How would you find its determinant? 00:46:29.840 --> 00:46:33.770 Can we just like take the determinant of that matrix? 00:46:33.770 --> 00:46:37.010 I could ask you how -- so there -- there are twenty-four terms, 00:46:37.010 --> 00:46:39.950 do we want to write all twenty-four terms down? 00:46:39.950 --> 00:46:42.470 not in the remaining ten seconds. 00:46:42.470 --> 00:46:43.930 Better to use cofactors. 00:46:43.930 --> 00:46:47.890 So I go along row one, I see one -- 00:46:47.890 --> 00:46:52.060 the only nonzero is this guy, so I should take that one times 00:46:52.060 --> 00:46:53.380 the cofactor. 00:46:53.380 --> 00:46:55.680 Now so I'm down to this determinant. 00:46:55.680 --> 00:46:56.640 OK. 00:46:56.640 --> 00:47:00.970 So now I'm -- look at this first column, I see one times this, 00:47:00.970 --> 00:47:05.640 there's the cofactor of the one, so I'm using up row one -- 00:47:05.640 --> 00:47:09.530 row one and column one of this three-by-three matrix, 00:47:09.530 --> 00:47:12.060 I'm down to this cofactor, and by the way, 00:47:12.060 --> 00:47:14.370 those were both plus signs, right? 00:47:14.370 --> 00:47:15.410 No, they weren't. 00:47:15.410 --> 00:47:17.240 That was a minus sign. 00:47:17.240 --> 00:47:18.490 That was a -- 00:47:18.490 --> 00:47:22.540 that was a minus, and then that was a plus, and then this, so 00:47:22.540 --> 00:47:24.670 what's the determinant? 00:47:24.670 --> 00:47:25.400 Nine. 00:47:25.400 --> 00:47:25.900 Nine. 00:47:25.900 --> 00:47:27.960 Determinant is nine. 00:47:27.960 --> 00:47:31.190 Determinant of A4 is nine. 00:47:31.190 --> 00:47:32.580 OK. 00:47:32.580 --> 00:47:38.640 Where A3, so my guess is I'll put that on the final this 00:47:38.640 --> 00:47:43.320 year, the -- probably the odd- numbered ones are singular 00:47:43.320 --> 00:47:47.830 and the even-numbered ones are invertible. 00:47:47.830 --> 00:47:50.710 And I don't know what the determinants 00:47:50.710 --> 00:47:55.750 are but I'm betting that they have some nice formula. 00:47:55.750 --> 00:47:56.290 OK. 00:47:56.290 --> 00:48:02.770 So, recitations this week will also be quiz review 00:48:02.770 --> 00:48:08.740 and then the quiz is Wednesday at one o'clock. 00:48:08.740 --> 00:48:10.290 Thanks.