WEBVTT 00:00:00.000 --> 00:00:11.590 -- and lift-off on differential equations. 00:00:11.590 --> 00:00:16.660 So, this section is about how to solve 00:00:16.660 --> 00:00:22.760 a system of first order, first derivative, constant 00:00:22.760 --> 00:00:26.500 coefficient linear equations. 00:00:26.500 --> 00:00:32.740 And if we do it right, it turns directly into linear algebra. 00:00:32.740 --> 00:00:37.690 The key idea is the solutions to constant coefficient 00:00:37.690 --> 00:00:41.730 linear equations are exponentials. 00:00:41.730 --> 00:00:44.520 So if you look for an exponential, 00:00:44.520 --> 00:00:48.230 then all you have to find is what's up there in the exponent 00:00:48.230 --> 00:00:51.060 and what multiplies the exponential 00:00:51.060 --> 00:00:53.130 and that's the linear algebra. 00:00:53.130 --> 00:00:56.950 So -- and the result -- one thing we will fine -- 00:00:56.950 --> 00:01:01.380 it's completely parallel to powers of a matrix. 00:01:01.380 --> 00:01:06.090 So the last lecture was about how would you compute 00:01:06.090 --> 00:01:08.680 A to the K or A to the 100? 00:01:08.680 --> 00:01:12.120 How do you compute high powers of a matrix? 00:01:12.120 --> 00:01:18.590 Now it's not powers anymore, but it's exponentials. 00:01:18.590 --> 00:01:22.310 That's the natural thing for differential equation. 00:01:22.310 --> 00:01:22.810 Okay. 00:01:22.810 --> 00:01:26.730 But can I begin with an example? 00:01:26.730 --> 00:01:28.630 And I'll just go through the mechanics. 00:01:28.630 --> 00:01:31.700 How would I solve the differential -- 00:01:31.700 --> 00:01:33.707 two differential equations? 00:01:33.707 --> 00:01:34.790 So I'm going to make it -- 00:01:34.790 --> 00:01:38.160 I'll have a two by two matrix and the coefficients 00:01:38.160 --> 00:01:42.530 are minus one two, one minus two and I'd better give you 00:01:42.530 --> 00:01:44.200 some initial condition. 00:01:44.200 --> 00:01:50.570 So suppose it starts u at times zero -- this is u1, u2 -- 00:01:50.570 --> 00:01:52.810 let it -- let it -- 00:01:52.810 --> 00:01:57.150 suppose everything is in u1 at times zero. 00:01:57.150 --> 00:02:00.890 So -- at -- at the start, it's all in u1. 00:02:00.890 --> 00:02:07.640 But what happens as time goes on, du2/dt will -- 00:02:07.640 --> 00:02:10.690 will be positive, because of that u1 term, 00:02:10.690 --> 00:02:16.210 so flow will move into the u2 component and it will go out 00:02:16.210 --> 00:02:19.050 of the u1 component. 00:02:19.050 --> 00:02:22.330 So we'll just follow that movement as time 00:02:22.330 --> 00:02:28.260 goes forward by looking at the eigenvalues and eigenvectors 00:02:28.260 --> 00:02:30.220 of that matrix. 00:02:30.220 --> 00:02:31.430 That's a first job. 00:02:31.430 --> 00:02:34.210 Before you do anything else, find the -- 00:02:34.210 --> 00:02:39.510 find the matrix and its eigenvalues and eigenvectors. 00:02:39.510 --> 00:02:42.400 So let me do that. 00:02:42.400 --> 00:02:43.060 Okay. 00:02:43.060 --> 00:02:45.440 So here's our matrix. 00:02:45.440 --> 00:02:47.500 Maybe you can tell me right away what -- 00:02:47.500 --> 00:02:53.190 what are the eigenvalues and -- eigenvalues anyway. 00:02:53.190 --> 00:02:54.640 And then we can check. 00:02:54.640 --> 00:02:58.070 But can you spot any of the eigenvalues of that matrix? 00:02:58.070 --> 00:03:00.900 We're looking for two eigenvalues. 00:03:00.900 --> 00:03:02.090 Do you see -- 00:03:02.090 --> 00:03:04.530 I mean, if I just wrote that matrix down, what -- 00:03:04.530 --> 00:03:07.670 what do you notice about it? 00:03:07.670 --> 00:03:09.490 It's singular, right. 00:03:09.490 --> 00:03:11.380 That -- that's a singular matrix. 00:03:11.380 --> 00:03:14.190 That tells me right away that one of the eigenvalues 00:03:14.190 --> 00:03:18.860 is lambda equals zero. 00:03:18.860 --> 00:03:20.900 I can -- that's a singular matrix, 00:03:20.900 --> 00:03:25.440 the second column is minus two times the first column, 00:03:25.440 --> 00:03:28.600 the determinant is zero, it's -- it's singular, 00:03:28.600 --> 00:03:33.670 so zero is an eigenvalue and the other eigenvalue will be -- 00:03:33.670 --> 00:03:35.640 from the trace. 00:03:35.640 --> 00:03:37.510 I look at the trace, the sum down 00:03:37.510 --> 00:03:39.620 the diagonal is minus three. 00:03:39.620 --> 00:03:42.770 That has to agree with the sum of the eigenvalue, 00:03:42.770 --> 00:03:46.970 so that second eigenvalue better be minus three. 00:03:46.970 --> 00:03:48.610 I could, of course -- 00:03:48.610 --> 00:03:51.100 I could compute -- why don't I over here -- 00:03:51.100 --> 00:03:54.530 compute the determinant of A minus lambda I, 00:03:54.530 --> 00:04:01.220 the determinant of this minus one minus lambda two one minus 00:04:01.220 --> 00:04:03.680 two minus lambda matrix. 00:04:03.680 --> 00:04:05.930 But we know what's coming. 00:04:05.930 --> 00:04:09.700 When I do that multiplication, I get a lambda squared. 00:04:09.700 --> 00:04:14.200 I get a two lambda and a one lambda, that's a three lambda. 00:04:14.200 --> 00:04:16.750 And then -- now I'm going to get the determinant, 00:04:16.750 --> 00:04:19.910 which is two minus two which is zero. 00:04:19.910 --> 00:04:26.490 So there's my characteristic polynomial, this determinant. 00:04:26.490 --> 00:04:31.960 And of course I factor that into lambda times lambda plus three 00:04:31.960 --> 00:04:37.590 and I get the two eigenvalues that we saw coming. 00:04:37.590 --> 00:04:38.830 What else do I need? 00:04:38.830 --> 00:04:40.320 The eigenvectors. 00:04:40.320 --> 00:04:44.020 So before I even think about the differential equation or what 00:04:44.020 --> 00:04:47.780 -- how to solve it, let me find the eigenvectors for this 00:04:47.780 --> 00:04:48.910 matrix. 00:04:48.910 --> 00:04:49.430 Okay. 00:04:49.430 --> 00:04:52.970 So take lambda equals zero -- 00:04:52.970 --> 00:04:55.370 so that -- that's the first eigenvalue. 00:04:55.370 --> 00:04:58.690 Lambda one equals zero and the second eigenvalue 00:04:58.690 --> 00:05:03.130 will be lambda two equals minus three. 00:05:03.130 --> 00:05:05.210 By the way, I -- 00:05:05.210 --> 00:05:09.010 I already know something important about this. 00:05:09.010 --> 00:05:11.920 The eigenvalues are telling me something. 00:05:11.920 --> 00:05:15.820 You'll see how it comes out, but let me point to -- 00:05:15.820 --> 00:05:19.640 these numbers are -- this eigenvalue, 00:05:19.640 --> 00:05:23.700 a negative eigenvalue, is going to disappear. 00:05:23.700 --> 00:05:28.620 There's going to be an e to the minus three t in the answer. 00:05:28.620 --> 00:05:31.690 That e to the minus three t as times goes on 00:05:31.690 --> 00:05:34.070 is going to be very, very small. 00:05:34.070 --> 00:05:38.360 The other part of the answer will involve an e 00:05:38.360 --> 00:05:40.390 to the zero t. 00:05:40.390 --> 00:05:44.330 But e to the zero t is one and that's a constant. 00:05:44.330 --> 00:05:49.190 So I'm expecting that this solution'll have two parts, 00:05:49.190 --> 00:05:53.570 an e to the zero t part and an e to the minus three t part, 00:05:53.570 --> 00:05:57.900 and that -- and as time goes on, the second part'll disappear 00:05:57.900 --> 00:06:00.360 and the first part will be a steady 00:06:00.360 --> 00:06:02.030 It won't move. state. 00:06:02.030 --> 00:06:06.740 It will be -- at the end of -- as t approaches infinity, 00:06:06.740 --> 00:06:09.800 this part disappears and this is the -- 00:06:09.800 --> 00:06:13.330 the e to the zero t part is what I get. 00:06:13.330 --> 00:06:17.120 And I'm very interested in these steady states, so that's -- 00:06:17.120 --> 00:06:19.330 I get a steady state when I have a zero 00:06:19.330 --> 00:06:20.150 eigenvalue. 00:06:20.150 --> 00:06:20.650 Okay. 00:06:20.650 --> 00:06:22.950 What about those eigenvectors? 00:06:22.950 --> 00:06:26.170 So what's the eigenvector that goes with eigenvalue zero? 00:06:26.170 --> 00:06:26.670 Okay. 00:06:26.670 --> 00:06:31.280 The matrix is singular as it is, the eigenvector is -- 00:06:31.280 --> 00:06:34.940 is the guy in the null space, so what vector is in the null 00:06:34.940 --> 00:06:37.890 space of that matrix? 00:06:37.890 --> 00:06:38.860 Let's see. 00:06:38.860 --> 00:06:42.900 I guess I probably give the free variable the value one 00:06:42.900 --> 00:06:48.630 and I realize that if I want to get zero I need a two up here. 00:06:48.630 --> 00:06:49.260 Okay? 00:06:49.260 --> 00:06:52.560 So Ax1 is zero x1. 00:06:52.560 --> 00:06:55.550 A x1 is zero x1. 00:06:55.550 --> 00:06:56.370 Fine. 00:06:56.370 --> 00:06:57.520 Okay. 00:06:57.520 --> 00:06:59.520 What about the other eigenvalue? 00:06:59.520 --> 00:07:01.801 Lambda two is minus three. 00:07:01.801 --> 00:07:02.300 Okay. 00:07:02.300 --> 00:07:05.010 How do I get the other eigenvalue, then? 00:07:05.010 --> 00:07:08.080 For the moment -- can I mentally do it? 00:07:08.080 --> 00:07:12.850 I subtract minus three along the diagonal, 00:07:12.850 --> 00:07:15.070 which means I add three -- 00:07:15.070 --> 00:07:18.370 can I -- I'll just do it with an erase -- erase for the moment. 00:07:18.370 --> 00:07:20.950 So I'm going to add three to the diagonal. 00:07:20.950 --> 00:07:26.040 So this minus one will become a two and -- 00:07:26.040 --> 00:07:28.450 I'll make it in big loopy letters -- 00:07:28.450 --> 00:07:32.350 and when I add three to this guy, the minus two becomes -- 00:07:32.350 --> 00:07:36.200 well, I can't make one very loopy, but how's that? 00:07:36.200 --> 00:07:37.410 Okay. 00:07:37.410 --> 00:07:40.200 Now that's A minus three I -- 00:07:40.200 --> 00:07:41.540 A plus three I, sorry. 00:07:41.540 --> 00:07:43.400 That's A plus three I. 00:07:43.400 --> 00:07:45.510 It's supposed to be singular, right? 00:07:45.510 --> 00:07:48.260 I-- if things -- if I did it right, 00:07:48.260 --> 00:07:51.930 this matrix should be singular and the x2, 00:07:51.930 --> 00:07:55.521 the eigenvector should be in its null space. 00:07:55.521 --> 00:07:56.020 Okay. 00:07:56.020 --> 00:07:58.510 What do I get for the null space of this? 00:07:58.510 --> 00:08:02.140 Maybe minus one one, or one minus one. 00:08:02.140 --> 00:08:03.240 Doesn't matter. 00:08:03.240 --> 00:08:05.300 Those are both perfectly good. 00:08:05.300 --> 00:08:05.800 Right? 00:08:05.800 --> 00:08:07.550 Because that's in the null space of this. 00:08:07.550 --> 00:08:14.370 Now I'll -- because A times that vector is three times that 00:08:14.370 --> 00:08:15.600 vector. 00:08:15.600 --> 00:08:20.370 Ax2 is minus three x2. 00:08:20.370 --> 00:08:22.540 Good. 00:08:22.540 --> 00:08:23.040 Okay. 00:08:23.040 --> 00:08:26.380 Can I get A again so we see that correctly? 00:08:26.380 --> 00:08:29.680 That was a minus one and that was a minus two. 00:08:29.680 --> 00:08:31.730 Good. 00:08:31.730 --> 00:08:32.809 Okay. 00:08:32.809 --> 00:08:35.470 That -- that's the first job. 00:08:35.470 --> 00:08:37.970 eigenvalues and eigenvectors. 00:08:37.970 --> 00:08:41.179 And already the eigenvalues are telling me 00:08:41.179 --> 00:08:44.240 the most important information about the answer. 00:08:44.240 --> 00:08:46.540 But now, what is the answer? 00:08:46.540 --> 00:08:54.420 The answer is -- the solution will be U of T -- 00:08:54.420 --> 00:08:54.920 okay. 00:08:54.920 --> 00:08:59.750 Now, wh- now I use those eigenvalues and eigenvectors. 00:08:59.750 --> 00:09:04.210 The solution is some -- there are two eigenvalues. 00:09:04.210 --> 00:09:08.100 So I -- it -- so there're going to be two special solutions 00:09:08.100 --> 00:09:08.910 here. 00:09:08.910 --> 00:09:12.040 Two pure exponential solutions. 00:09:12.040 --> 00:09:17.300 The first one is going to be either the lambda one tx1 00:09:17.300 --> 00:09:23.440 and the -- so that solves the equation, and so does this one. 00:09:23.440 --> 00:09:29.290 They both are solutions to the differential equation. 00:09:29.290 --> 00:09:31.640 That's the general solution. 00:09:31.640 --> 00:09:33.900 The general solution is a combination 00:09:33.900 --> 00:09:37.010 of that pure exponential solution 00:09:37.010 --> 00:09:39.630 and that pure exponential solution. 00:09:39.630 --> 00:09:43.150 Can I just see that those guys do solve the equation? 00:09:43.150 --> 00:09:47.540 So let me just check -- check on this one, for example. 00:09:47.540 --> 00:09:53.360 I -- I want to check that the -- my equation -- 00:09:53.360 --> 00:09:53.860 let's 00:09:53.860 --> 00:09:57.190 Check. remember, the equation -- du/dt is Au. 00:09:57.190 --> 00:10:04.340 I plug in e to the lambda one t x1 00:10:04.340 --> 00:10:08.470 and let's just see that the equation's okay. 00:10:08.470 --> 00:10:12.370 I believe this is a solution to that equation. 00:10:12.370 --> 00:10:13.980 So just plug it in. 00:10:13.980 --> 00:10:17.870 On the left-hand side, I take the time derivative -- 00:10:17.870 --> 00:10:22.700 so the left-hand side will be lambda one, e to the lambda one 00:10:22.700 --> 00:10:25.540 t x1, right? 00:10:25.540 --> 00:10:28.990 The time derivative -- this is the term that depends on time, 00:10:28.990 --> 00:10:33.250 it's just ordinary exponential, its derivative brings down 00:10:33.250 --> 00:10:34.470 a lambda one. 00:10:34.470 --> 00:10:37.460 On the other side of the equation it's A times this 00:10:37.460 --> 00:10:38.380 thing. 00:10:38.380 --> 00:10:44.790 A times either the lambda one t x one, and does that check out? 00:10:44.790 --> 00:10:47.550 Do we have equality there? 00:10:47.550 --> 00:10:52.190 Yes, because either the lambda one t appears on both sides 00:10:52.190 --> 00:10:58.030 and the other one is Ax1 equal lambda one x1 -- check. 00:10:58.030 --> 00:11:02.590 Do you -- so, the -- we've come to the first point to remember. 00:11:02.590 --> 00:11:05.030 These pure solutions. 00:11:05.030 --> 00:11:10.760 Those pure solutions are the -- those pure exponentials are 00:11:10.760 --> 00:11:14.280 the differential equations analogue of -- 00:11:14.280 --> 00:11:17.040 last time we had pure powers. 00:11:17.040 --> 00:11:19.540 Last time -- so -- 00:11:19.540 --> 00:11:24.200 so last time, the analog was lambda -- 00:11:24.200 --> 00:11:29.020 lambda one to the K-th power x1, some amount of that, 00:11:29.020 --> 00:11:35.130 plus some amount of lambda two to the K-th power x2. 00:11:35.130 --> 00:11:37.590 That was our formula from last time. 00:11:37.590 --> 00:11:41.690 I put it up just to -- so your eye compares those two 00:11:41.690 --> 00:11:42.880 formulas. 00:11:42.880 --> 00:11:45.550 Powers of lambda in the -- 00:11:45.550 --> 00:11:48.610 in the difference equation -- that -- this was in the -- 00:11:48.610 --> 00:11:55.480 this was for the equation uk plus one equals A uk. 00:11:55.480 --> 00:11:59.890 That was for the finite step -- stepping by one. 00:11:59.890 --> 00:12:02.330 And we got powers, now this is the one 00:12:02.330 --> 00:12:04.920 we're interested in, the exponentials. 00:12:04.920 --> 00:12:08.750 So -- so that's -- that's the solution -- 00:12:08.750 --> 00:12:10.660 what are c1 and c2? 00:12:10.660 --> 00:12:12.300 Then we're through. 00:12:12.300 --> 00:12:14.060 What are c1 and c2? 00:12:14.060 --> 00:12:17.340 Well, of course we know these actual things. 00:12:17.340 --> 00:12:22.260 Let me just -- let me come back to this. 00:12:22.260 --> 00:12:26.790 c1 is -- we haven't figured out yet, but e to the lambda one t, 00:12:26.790 --> 00:12:32.890 the lambda one is zero so that's just a one times x1 which is 00:12:32.890 --> 00:12:34.260 two one. 00:12:34.260 --> 00:12:39.590 So it's c1 times this one that's not moving times the vector, 00:12:39.590 --> 00:12:44.660 the eigenvector two one and c2 times -- 00:12:44.660 --> 00:12:47.730 what's e to the lambda two t? 00:12:47.730 --> 00:12:51.830 Lambda two is minus three. 00:12:51.830 --> 00:12:54.010 So this is the term that has the minus 00:12:54.010 --> 00:12:58.360 three t and its eigenvector is this one minus one. 00:13:01.520 --> 00:13:06.640 So this vector solves the equation 00:13:06.640 --> 00:13:08.580 and any multiple of it. 00:13:08.580 --> 00:13:12.190 This vector solves the equation if it's got that factor 00:13:12.190 --> 00:13:14.620 e to the minus three t. 00:13:14.620 --> 00:13:17.980 We've got the answer except for c1 and c2. 00:13:17.980 --> 00:13:22.970 So -- so everything I've done is immediate as soon as you know 00:13:22.970 --> 00:13:25.570 the eigenvalues and eigenvectors. 00:13:25.570 --> 00:13:27.640 So how do we get c1 and c2? 00:13:27.640 --> 00:13:30.930 That has to come from the initial condition. 00:13:30.930 --> 00:13:38.740 So now I -- now I use -- u of zero is given as one zero. 00:13:41.780 --> 00:13:46.346 So this is the initial condition that will find c1 and c2. 00:13:46.346 --> 00:13:48.095 So let me do that on the board underneath. 00:13:51.080 --> 00:13:52.935 At t equals zero, then -- 00:13:56.500 --> 00:14:05.280 I get c1 times this guy plus c2 and now I'm at times zero. 00:14:05.280 --> 00:14:08.320 So that's a one and this is a one minus one 00:14:08.320 --> 00:14:12.920 and that's supposed to agree with u of zero one zero. 00:14:19.550 --> 00:14:20.850 Okay. 00:14:20.850 --> 00:14:23.090 That should be two equations. 00:14:23.090 --> 00:14:26.500 That should give me c1 and c2 and then I'm through. 00:14:26.500 --> 00:14:28.540 So what are c1 and c2? 00:14:28.540 --> 00:14:30.430 Let's see. 00:14:30.430 --> 00:14:33.000 I guess we could actually spot them by eye 00:14:33.000 --> 00:14:36.800 or we could solve two equations in two unknowns. 00:14:36.800 --> 00:14:38.090 Let's see. 00:14:38.090 --> 00:14:40.940 If these were both ones -- so I'm just adding -- 00:14:40.940 --> 00:14:43.630 then I would get three zero. 00:14:43.630 --> 00:14:46.740 So what's the -- what's the solution, then? 00:14:49.970 --> 00:14:53.910 If -- if c1 and c2 are both ones, I get three zero, 00:14:53.910 --> 00:14:55.990 so I want, like, one third of that, 00:14:55.990 --> 00:14:57.750 because I want to get one zero. 00:14:57.750 --> 00:15:02.220 So I think it's c1 equals a third, c2 equals a third. 00:15:05.460 --> 00:15:08.030 So finally I have the answer. 00:15:08.030 --> 00:15:11.000 Let me keep it in the -- in this board here. 00:15:11.000 --> 00:15:20.530 Finally the answer is one third of this plus one third of this. 00:15:24.450 --> 00:15:27.990 Do you see what -- what's actually happening with this 00:15:27.990 --> 00:15:28.880 flow? 00:15:28.880 --> 00:15:32.630 This flow started out at -- the solution started out at one 00:15:32.630 --> 00:15:34.140 zero. 00:15:34.140 --> 00:15:36.790 Started at one zero. 00:15:36.790 --> 00:15:41.290 Then as time went on, people moved, essentially. 00:15:41.290 --> 00:15:46.190 Some fraction of this one moved here. 00:15:46.190 --> 00:15:52.030 And -- and in the limit, there's -- there's the limit, as -- 00:15:52.030 --> 00:15:52.530 right? 00:15:52.530 --> 00:15:55.540 As t goes to infinity, as t gets very large, 00:15:55.540 --> 00:15:59.110 this disappears and this is the steady state. 00:15:59.110 --> 00:16:02.550 So the steady state is -- 00:16:02.550 --> 00:16:04.272 so the steady state -- 00:16:08.700 --> 00:16:14.190 u -- we could call it u at infinity is one third of two 00:16:14.190 --> 00:16:15.040 and one. 00:16:15.040 --> 00:16:17.110 It's -- it's two thirds of one third. 00:16:19.970 --> 00:16:22.790 So that's the -- we really -- 00:16:22.790 --> 00:16:25.280 I mean, you're getting, like, total, 00:16:25.280 --> 00:16:29.790 insight into the behavior of the solution, 00:16:29.790 --> 00:16:32.050 what the differential equation does. 00:16:32.050 --> 00:16:37.480 Of course, we don't -- wouldn't always have a steady state. 00:16:37.480 --> 00:16:40.580 Sometimes we would approach zero. 00:16:40.580 --> 00:16:42.320 Sometimes we would blow up. 00:16:42.320 --> 00:16:45.250 Can we straighten out those cases? 00:16:45.250 --> 00:16:47.510 The eigenvalue should tell us. 00:16:47.510 --> 00:16:50.170 So when do we get -- 00:16:50.170 --> 00:16:54.440 so -- so let me ask first, when do we get stability? 00:16:57.220 --> 00:17:00.250 That's u of t going to zero. 00:17:03.070 --> 00:17:05.660 When would the solution go to zero no matter 00:17:05.660 --> 00:17:09.230 what the initial condition is? 00:17:09.230 --> 00:17:11.140 Negative eigenvalues, right. 00:17:11.140 --> 00:17:12.609 Negative eigenvalues. 00:17:12.609 --> 00:17:13.630 But now I have to -- 00:17:13.630 --> 00:17:16.950 I have to ask you for one more step. 00:17:16.950 --> 00:17:20.420 Suppose the eigenvalues are complex numbers? 00:17:20.420 --> 00:17:22.680 Because we know they could be. 00:17:22.680 --> 00:17:27.760 Then we want stability -- this -- this -- we want -- 00:17:27.760 --> 00:17:35.260 we need all these e to the lambda t-s all going to zero 00:17:35.260 --> 00:17:40.920 and somehow that asks us to have lambda negative. 00:17:40.920 --> 00:17:43.470 But suppose lambda is a complex number? 00:17:43.470 --> 00:17:45.690 Then what's the test? 00:17:45.690 --> 00:17:50.340 What -- if lambda's a complex number like, oh, 00:17:50.340 --> 00:17:54.730 suppose lambda is negative plus an imaginary part? 00:17:54.730 --> 00:17:59.810 Say lambda is minus three plus six i? 00:17:59.810 --> 00:18:01.120 What -- what happens then? 00:18:01.120 --> 00:18:03.530 Can we just, like, do a -- a case here? 00:18:03.530 --> 00:18:11.550 If -- if this lambda is minus three plus six it, 00:18:11.550 --> 00:18:14.170 how big is that number? 00:18:14.170 --> 00:18:18.450 Does this -- does this imaginary part play a -- play a -- 00:18:18.450 --> 00:18:20.840 play a role here or not? 00:18:20.840 --> 00:18:22.850 Or how big is -- 00:18:22.850 --> 00:18:25.700 what's the absolute value of that -- of that quantity? 00:18:28.530 --> 00:18:32.670 It's just e to the minus three t, right? 00:18:32.670 --> 00:18:36.880 Because this other part, this -- the -- the magnitude -- the -- 00:18:36.880 --> 00:18:41.795 this -- e to the six it -- what -- that has absolute value one. 00:18:44.680 --> 00:18:45.180 Right? 00:18:45.180 --> 00:18:50.540 That's just this cosine of six t plus i, sine of six t. 00:18:50.540 --> 00:18:53.060 And the absolute value squared will 00:18:53.060 --> 00:18:56.230 be cos squared plus sine squared will be one. 00:18:56.230 --> 00:18:59.680 This is -- this complex number is running around the unit 00:18:59.680 --> 00:19:00.660 circle. 00:19:00.660 --> 00:19:04.770 This com- this -- the -- it's the real part that matters. 00:19:04.770 --> 00:19:07.020 This is what I'm trying to do. 00:19:07.020 --> 00:19:10.980 Real part of lambda has to be negative. 00:19:10.980 --> 00:19:14.880 If lambda's a complex number, it's the real part, 00:19:14.880 --> 00:19:19.200 the minus three, that either makes us go to zero 00:19:19.200 --> 00:19:24.940 or doesn't, or let -- or blows up. 00:19:24.940 --> 00:19:27.380 The imaginary part won't -- will just, like, 00:19:27.380 --> 00:19:30.690 oscillate between the two components. 00:19:30.690 --> 00:19:31.360 Okay. 00:19:31.360 --> 00:19:33.230 So that's stability. 00:19:33.230 --> 00:19:36.040 And what about -- 00:19:36.040 --> 00:19:37.305 what about a steady state? 00:19:42.130 --> 00:19:45.490 When would we have, a steady state, 00:19:45.490 --> 00:19:47.390 always in the same direction? 00:19:47.390 --> 00:19:48.160 So let me -- 00:19:48.160 --> 00:19:51.280 I'll take this part away -- 00:19:51.280 --> 00:19:54.280 when -- so that was, like, checking that it's -- 00:19:54.280 --> 00:19:57.830 that it's the real part that we care about. 00:19:57.830 --> 00:20:01.530 Now, we have a steady state when -- 00:20:01.530 --> 00:20:12.500 when lambda one is zero and the other eigenvalues have what? 00:20:12.500 --> 00:20:14.990 So I'm looking -- like, that example was, like, 00:20:14.990 --> 00:20:18.910 perfect for a steady state. 00:20:18.910 --> 00:20:22.760 We have a zero eigenvalue and the other eigenvalues, 00:20:22.760 --> 00:20:25.070 we want those to disappear. 00:20:25.070 --> 00:20:28.975 So the other eigenvalues have real part negative. 00:20:31.880 --> 00:20:35.700 And we blow up, for sure -- 00:20:35.700 --> 00:20:45.380 we blow up if any real part of lambda is positive. 00:20:49.090 --> 00:20:54.240 So if I -- if I reverse the sign of A -- of the matrix A -- 00:20:54.240 --> 00:20:57.420 suppose instead of the matrix I had, the A that I had, 00:20:57.420 --> 00:20:58.390 I changed it -- 00:20:58.390 --> 00:21:00.770 I changed all its sines. 00:21:00.770 --> 00:21:04.950 What would that do to the eigenvalues and eigenvectors? 00:21:04.950 --> 00:21:08.090 If I -- if -- if I know the eigenvalues and eigenvectors 00:21:08.090 --> 00:21:11.520 of A, tell me about minus A. 00:21:11.520 --> 00:21:14.780 What happens to the eigenvalues? 00:21:14.780 --> 00:21:18.410 For minus A, they'll all change sine. 00:21:18.410 --> 00:21:20.660 So I'll have blow up. 00:21:20.660 --> 00:21:23.020 This -- instead of the e to the minus three t, 00:21:23.020 --> 00:21:26.460 if I change that to minus -- if I change the sines in that 00:21:26.460 --> 00:21:30.810 matrix, I would change the trace to plus three, 00:21:30.810 --> 00:21:34.020 I would have an e to the plus three t and I would have blow 00:21:34.020 --> 00:21:36.150 up. 00:21:36.150 --> 00:21:39.430 Of course the zero eigenvalue would stay at zero, 00:21:39.430 --> 00:21:42.490 but the other guy is taking off in -- 00:21:42.490 --> 00:21:45.091 if I reversed all the sines. 00:21:45.091 --> 00:21:45.590 Okay. 00:21:45.590 --> 00:21:51.090 So this is -- this is the stability picture. 00:21:51.090 --> 00:21:56.680 And for a two by two matrix, we can actually 00:21:56.680 --> 00:22:01.220 pin down even more closely what that means. 00:22:01.220 --> 00:22:02.710 Can I -- let -- can I do that? 00:22:02.710 --> 00:22:04.410 Let me do that -- 00:22:04.410 --> 00:22:05.810 I want to -- 00:22:05.810 --> 00:22:11.230 for a two by two matrix, I can tell whether the real part 00:22:11.230 --> 00:22:14.740 of the eigenvalues is negative, I -- well, let me -- 00:22:14.740 --> 00:22:18.480 let me tell you what I have in mind for that. 00:22:18.480 --> 00:22:21.040 So two by two stability -- 00:22:21.040 --> 00:22:25.750 let me -- this is just a little comment. 00:22:25.750 --> 00:22:27.506 Two by two stability. 00:22:31.240 --> 00:22:35.930 So my matrix, now, is just a b c d 00:22:35.930 --> 00:22:41.770 and I'm looking for the real parts of both eigenvalues 00:22:41.770 --> 00:22:42.910 to be negative. 00:22:47.480 --> 00:22:47.980 Okay. 00:22:52.330 --> 00:22:55.300 What -- how can I tell from looking at the matrix, 00:22:55.300 --> 00:22:58.230 without computing its eigenvalues, 00:22:58.230 --> 00:23:02.150 whether the two eigenvalues are negative, 00:23:02.150 --> 00:23:04.930 or at least their real parts are negative? 00:23:04.930 --> 00:23:07.260 What would that tell me about the trace? 00:23:07.260 --> 00:23:10.830 So -- so the trace -- 00:23:10.830 --> 00:23:14.930 that's this a plus d -- 00:23:14.930 --> 00:23:19.470 what can you tell me about the trace in the case of a two 00:23:19.470 --> 00:23:21.760 by two stable matrix? 00:23:21.760 --> 00:23:25.320 That means the eigenvalues have -- are negative, 00:23:25.320 --> 00:23:28.660 or at least the real parts of those eigenvalues are negative 00:23:28.660 --> 00:23:33.140 -- then, when I take the -- when I look at the matrix and find 00:23:33.140 --> 00:23:36.930 its trace, what -- what do I know about that? 00:23:36.930 --> 00:23:38.360 It's negative, right. 00:23:38.360 --> 00:23:40.940 This is the sum of the -- this equals -- 00:23:40.940 --> 00:23:47.010 this equals lambda one plus lambda two, so it's negative. 00:23:47.010 --> 00:23:49.590 The two eigenvalues, by the way, will have -- 00:23:49.590 --> 00:23:54.990 if they're complex -- will have plus six i and minus six i. 00:23:54.990 --> 00:23:59.860 The complex parts will -- will be conjugates of each other 00:23:59.860 --> 00:24:04.720 and disappear when we add and the trace will be negative. 00:24:04.720 --> 00:24:06.710 Okay, the trace has to be negative. 00:24:06.710 --> 00:24:09.030 Is that enough -- 00:24:09.030 --> 00:24:14.670 is a negative trace enough to make the matrix stable? 00:24:14.670 --> 00:24:16.180 Shouldn't be enough, right? 00:24:16.180 --> 00:24:19.270 Can I -- can you make -- what's a matrix that has a negative 00:24:19.270 --> 00:24:24.040 trace but still it's not stable? 00:24:24.040 --> 00:24:27.500 So it -- it has a blow -- it still has a blow-up factor 00:24:27.500 --> 00:24:30.900 and a -- and a -- and a decaying one. 00:24:30.900 --> 00:24:33.820 So what would be a -- so just -- just to see -- 00:24:33.820 --> 00:24:35.920 maybe I just put that here. 00:24:35.920 --> 00:24:40.790 This -- now I'm looking for an example where the trace could 00:24:40.790 --> 00:24:48.080 be negative but still blow up. 00:24:48.080 --> 00:24:52.830 Of course -- yeah, let's just take one. 00:24:52.830 --> 00:24:57.990 Oh, look, let me -- let me make it minus two zero zero one. 00:24:57.990 --> 00:25:00.140 Okay. 00:25:00.140 --> 00:25:04.810 There's a case where that matrix has negative trace -- 00:25:04.810 --> 00:25:06.390 I know its eigenvalues of course. 00:25:06.390 --> 00:25:09.750 They're minus two and one and it blows up. 00:25:09.750 --> 00:25:12.780 It's got -- it's got a plus one eigenvalue here, 00:25:12.780 --> 00:25:17.280 so there would be an e to the plus t in the solution 00:25:17.280 --> 00:25:21.170 and it'll blow up if it has any second component at all. 00:25:21.170 --> 00:25:23.900 I need another condition. 00:25:23.900 --> 00:25:25.615 And it's a condition on the determinant. 00:25:28.240 --> 00:25:29.560 And what's that condition? 00:25:29.560 --> 00:25:32.730 If I know that the two eigenvalues -- 00:25:32.730 --> 00:25:36.090 suppose I know they're negative, both negative. 00:25:36.090 --> 00:25:39.790 What does that tell me about the determinant? 00:25:39.790 --> 00:25:41.260 Let me ask again. 00:25:41.260 --> 00:25:44.990 If I know both the eigenvalues are negative, 00:25:44.990 --> 00:25:47.760 then I know the trace is negative 00:25:47.760 --> 00:25:53.170 but the determinant is positive, because it's 00:25:53.170 --> 00:25:56.450 the product of the two eigenvalues. 00:25:56.450 --> 00:26:00.580 So this determinant is lambda one times lambda two. 00:26:00.580 --> 00:26:04.540 This is -- this is lambda one times lambda two 00:26:04.540 --> 00:26:08.060 and if they're both negative, the product is positive. 00:26:08.060 --> 00:26:11.860 So positive determinant, negative trace. 00:26:11.860 --> 00:26:17.200 I can easily track down the -- this condition also for the -- 00:26:17.200 --> 00:26:20.380 if -- if there's an imaginary part hanging around. 00:26:20.380 --> 00:26:20.880 Okay. 00:26:20.880 --> 00:26:25.550 So that's a -- like a small but quite useful, 00:26:25.550 --> 00:26:29.820 because two by two is always important -- 00:26:29.820 --> 00:26:33.100 comment on stability. 00:26:33.100 --> 00:26:33.750 Okay. 00:26:33.750 --> 00:26:40.170 So let's just look at the picture again. 00:26:40.170 --> 00:26:41.290 Okay. 00:26:41.290 --> 00:26:43.980 The main part of my lecture, the one thing 00:26:43.980 --> 00:26:46.700 you want to be able to, like, just do automatically 00:26:46.700 --> 00:26:51.750 is this step of solving the system. 00:26:51.750 --> 00:26:54.190 Find the eigenvalues, find the eigenvectors, 00:26:54.190 --> 00:26:56.000 find the coefficients. 00:26:56.000 --> 00:27:01.090 And notice -- what's the matrix -- in this linear system here, 00:27:01.090 --> 00:27:05.590 I can't help -- if I -- if I have equations like that -- 00:27:05.590 --> 00:27:08.900 that's my equations column at a time -- 00:27:08.900 --> 00:27:11.710 what's the matrix form of that equation? 00:27:11.710 --> 00:27:18.580 So -- so this -- this system of equations is -- 00:27:18.580 --> 00:27:26.710 is some matrix multiplying c1, c2 to give u of zero. 00:27:26.710 --> 00:27:29.370 One zero. 00:27:29.370 --> 00:27:30.510 What's the matrix? 00:27:30.510 --> 00:27:35.400 Well, it's obviously two one, one minus one, 00:27:35.400 --> 00:27:37.760 but we have a name, or at least a letter -- 00:27:37.760 --> 00:27:40.090 actually a name for that matrix. 00:27:40.090 --> 00:27:42.670 Wh- what matrix are we s- are we -- 00:27:42.670 --> 00:27:47.390 are we dealing with here in this step of finding the c-s? 00:27:47.390 --> 00:27:50.690 Its letter is S -- 00:27:50.690 --> 00:27:52.340 it's the eigenvector matrix. 00:27:52.340 --> 00:27:52.970 Of course. 00:27:52.970 --> 00:27:55.230 These are the eigenvectors, there 00:27:55.230 --> 00:27:57.150 in the columns of our matrix. 00:27:57.150 --> 00:28:02.520 So this is S c equals u of zero -- 00:28:02.520 --> 00:28:09.640 is the -- that step where you find the actual coefficients, 00:28:09.640 --> 00:28:14.690 you find out how much of each pure exponential is 00:28:14.690 --> 00:28:16.910 in the solution. 00:28:16.910 --> 00:28:20.660 By getting it right at the start, then it stays right 00:28:20.660 --> 00:28:21.320 forever. 00:28:21.320 --> 00:28:24.820 I -- you're seeing this picture that each -- 00:28:24.820 --> 00:28:29.090 each pure exponential goes on its own way once you start it 00:28:29.090 --> 00:28:29.620 from u of 00:28:29.620 --> 00:28:30.490 zero. 00:28:30.490 --> 00:28:33.810 So you start it by figuring out how much 00:28:33.810 --> 00:28:37.880 each one is present in u of zero and then off they go. 00:28:37.880 --> 00:28:38.740 Okay. 00:28:38.740 --> 00:28:45.110 So -- so that's a system of two equations in two unknowns 00:28:45.110 --> 00:28:48.070 coupled -- 00:28:48.070 --> 00:28:53.630 the matrix sort of couples u1 and u2 and the eigenvalues 00:28:53.630 --> 00:28:57.770 and eigenvectors uncouple it, diagonalize it. 00:28:57.770 --> 00:29:00.830 Actually -- okay, now can I -- 00:29:00.830 --> 00:29:04.600 can I think in terms of S and lambda? 00:29:04.600 --> 00:29:07.330 So I want to write the solution down, 00:29:07.330 --> 00:29:10.840 again in terms of S and lambda. 00:29:10.840 --> 00:29:11.340 Okay. 00:29:11.340 --> 00:29:14.890 I'll do that on this far board. 00:29:14.890 --> 00:29:15.890 Okay. 00:29:15.890 --> 00:29:20.540 So I'm coming back to -- 00:29:20.540 --> 00:29:26.470 I'm coming back to our equation du/dt equals Au. 00:29:26.470 --> 00:29:35.890 Now this matrix A couples them. 00:29:35.890 --> 00:29:39.010 The whole point of eigenvectors is to uncouple. 00:29:39.010 --> 00:29:46.510 So one way to see that is introduce set u equal A -- 00:29:46.510 --> 00:29:47.010 not 00:29:47.010 --> 00:29:54.580 A. It's S, the eigenvector matrix that uncouples it. 00:29:54.580 --> 00:29:58.890 So I'm -- I'm taking this equation as I'm given, 00:29:58.890 --> 00:30:04.150 coupled with -- with A has -- is probably full of non-zeroes, 00:30:04.150 --> 00:30:08.130 but I'm -- by uncoupling it, I mean I'm diagonalizing it. 00:30:08.130 --> 00:30:11.630 If I can get a diagonal matrix, I'm -- 00:30:11.630 --> 00:30:12.470 I'm in. 00:30:12.470 --> 00:30:13.120 Okay. 00:30:13.120 --> 00:30:14.950 So I plug that in. 00:30:14.950 --> 00:30:18.720 This is A S v. 00:30:18.720 --> 00:30:20.813 And this is S dv/dt. 00:30:25.810 --> 00:30:26.890 S is a constant. 00:30:26.890 --> 00:30:30.050 It's -- this it the eigenvector matrix. 00:30:30.050 --> 00:30:31.635 This is the eigenvector matrix. 00:30:34.820 --> 00:30:35.330 Okay. 00:30:35.330 --> 00:30:37.550 Now I'm going to bring S inverse over here. 00:30:40.390 --> 00:30:41.285 And what have I got? 00:30:45.160 --> 00:30:53.470 That combination S inverse A S is lambda, the diagonal matrix. 00:30:53.470 --> 00:30:56.940 That's -- that's the point, that in -- 00:30:56.940 --> 00:31:01.400 if I'm using the eigenvectors as my basis, 00:31:01.400 --> 00:31:06.700 then my system of equations is just diagonal. 00:31:06.700 --> 00:31:10.420 I -- each -- there's no coupling anymore -- 00:31:10.420 --> 00:31:13.130 dv1/dt is lambda one v1. 00:31:13.130 --> 00:31:20.890 So let's just write that down. dv1/ dt is lambda one v1 00:31:20.890 --> 00:31:26.470 and so on for all n of the equations. 00:31:26.470 --> 00:31:29.610 It's a system of equations but they're not connected, 00:31:29.610 --> 00:31:32.865 so they're easy to solve and why don't I just 00:31:32.865 --> 00:31:33.865 write down the solution? 00:31:36.880 --> 00:31:47.160 v -- well, v is now some e to the lambda one t -- 00:31:47.160 --> 00:31:49.600 well, okay -- 00:31:49.600 --> 00:31:56.330 I guess my idea here now is to use, the natural notation -- 00:31:56.330 --> 00:32:04.740 v of T is e to the lambda tv of zero. 00:32:04.740 --> 00:32:16.150 And u of t will be Se to the lambda t S inverse, u of zero. 00:32:16.150 --> 00:32:20.990 This is the -- this is the, formula I'm headed for. 00:32:25.220 --> 00:32:29.530 This -- this matrix, S e to the lambda t S inverse, 00:32:29.530 --> 00:32:32.040 that's my exponential. 00:32:32.040 --> 00:32:40.483 That's my e to the A t, is this S e to the lambda t S inverse. 00:32:43.600 --> 00:32:47.310 So my -- my job really now is to explain what's going on with 00:32:47.310 --> 00:32:49.620 this matrix up in the exponential. 00:32:49.620 --> 00:32:51.610 What does that mean? 00:32:51.610 --> 00:32:54.110 What does it mean to say e to a matrix? 00:32:58.100 --> 00:33:01.680 This ought to be easier because this is e to a diagonal matrix, 00:33:01.680 --> 00:33:03.970 but still it's a matrix. 00:33:03.970 --> 00:33:07.350 What do we mean by e to the A t? 00:33:07.350 --> 00:33:13.120 Because really e to the A t is my answer here. 00:33:13.120 --> 00:33:18.620 My -- my answer to this equation is -- 00:33:18.620 --> 00:33:26.170 this u of t, this is my -- this is my e to the A t u of zero. 00:33:26.170 --> 00:33:30.907 So it's -- my job is really now to say what's -- 00:33:30.907 --> 00:33:31.740 what does that mean? 00:33:31.740 --> 00:33:33.780 What's the exponential of a matrix 00:33:33.780 --> 00:33:38.390 and why is that formula correct? 00:33:38.390 --> 00:33:38.950 Okay. 00:33:38.950 --> 00:33:42.190 So I'll put that on the board underneath. 00:33:42.190 --> 00:33:45.210 What's the exponential of a matrix? 00:33:45.210 --> 00:33:47.430 Let me come back here. 00:33:47.430 --> 00:33:49.460 So I'm talking about matrix exponentials. 00:33:55.040 --> 00:33:57.540 e to the At. 00:33:57.540 --> 00:33:58.350 Okay. 00:33:58.350 --> 00:34:01.090 How are we going to define the exponential of a -- 00:34:01.090 --> 00:34:01.735 of something? 00:34:04.490 --> 00:34:09.940 The trick -- the idea is -- the thing to go back to is 00:34:09.940 --> 00:34:15.860 exponential -- there's a power series for exponentials. 00:34:15.860 --> 00:34:19.690 That's how you -- you -- the good way to define e to the x 00:34:19.690 --> 00:34:25.469 is the power series one plus x plus one half x squared, 00:34:25.469 --> 00:34:29.489 one six x cubed and we'll do it now when the -- 00:34:29.489 --> 00:34:30.770 when we have a matrix. 00:34:30.770 --> 00:34:34.739 So the one becomes the identity, the x is At, 00:34:34.739 --> 00:34:42.620 the x squared is At squared and I divide by two. 00:34:42.620 --> 00:34:47.600 The cube, the x cube is At cubed over six, 00:34:47.600 --> 00:34:50.880 and what's the general term in here? 00:34:50.880 --> 00:34:55.929 At to the n-th power divided by -- 00:34:55.929 --> 00:34:57.710 and it goes on. 00:34:57.710 --> 00:35:01.430 But what do I divide by? 00:35:01.430 --> 00:35:05.350 So, you see the pattern -- here I divided by one, 00:35:05.350 --> 00:35:10.080 here I divided by one by two by six, those are the factorials. 00:35:10.080 --> 00:35:10.965 It's n factorial. 00:35:14.110 --> 00:35:17.445 That was, like, the one beautiful Taylor series. 00:35:20.300 --> 00:35:23.180 The one beautiful Taylor series -- well, there are two -- 00:35:23.180 --> 00:35:25.960 there are two beautiful Taylor series in this world. 00:35:25.960 --> 00:35:29.550 The Taylor series for e to the x is 00:35:29.550 --> 00:35:35.090 the s with x to the n-th over n factorial. 00:35:35.090 --> 00:35:38.680 And all I'm doing is doing the same thing for matrixes. 00:35:38.680 --> 00:35:40.850 The other beautiful Taylor series 00:35:40.850 --> 00:35:47.920 is just the sum of x to the n-th not divided by n factorial. 00:35:47.920 --> 00:35:51.300 Can you -- do you know what function that one is? 00:35:51.300 --> 00:35:53.990 So if I take -- this is the series, 00:35:53.990 --> 00:35:58.070 all these sums are going from zero to infinity. 00:35:58.070 --> 00:35:59.960 What's -- what function have I got -- 00:35:59.960 --> 00:36:02.770 this is like a side comment -- 00:36:02.770 --> 00:36:06.750 this is one plus x plus x squared plus x cubed plus x 00:36:06.750 --> 00:36:09.430 to the fourth not divided by anything, what's -- 00:36:09.430 --> 00:36:11.800 what's that function? 00:36:11.800 --> 00:36:15.380 One plus x plus x squared plus x cubed plus x fourth forever 00:36:15.380 --> 00:36:18.700 is one over one minus x. 00:36:18.700 --> 00:36:24.120 It's the geometric series, the nicest power series of all. 00:36:24.120 --> 00:36:28.850 So, actually, of course, there would be a similar thing here. 00:36:28.850 --> 00:36:36.810 If -- if I wanted, I minus A t inverse would be -- 00:36:36.810 --> 00:36:39.060 now I've got matrixes. 00:36:39.060 --> 00:36:43.150 I've got matrixes everywhere, but it's just like that series 00:36:43.150 --> 00:36:46.690 with -- and just like this one without the divisions. 00:36:46.690 --> 00:36:56.310 It's I plus At plus At squared plus At cubed and forever. 00:36:59.200 --> 00:37:02.460 So that's actually a -- a reasonable way to find 00:37:02.460 --> 00:37:04.660 the inverse of a matrix. 00:37:04.660 --> 00:37:07.500 If I chop it off -- 00:37:07.500 --> 00:37:10.120 well, it's reasonable if t is small. 00:37:10.120 --> 00:37:13.330 If t is a small number, then -- 00:37:13.330 --> 00:37:15.940 then t squared is extremely small, 00:37:15.940 --> 00:37:19.350 t cubed is even smaller, so approximately 00:37:19.350 --> 00:37:22.480 that inverse is I plus At. 00:37:22.480 --> 00:37:24.660 I can keep more terms if I like. 00:37:24.660 --> 00:37:25.830 Do you see what I'm doing? 00:37:25.830 --> 00:37:31.440 I'm just saying we can do the same thing for matrixes that we 00:37:31.440 --> 00:37:35.600 do for ordinary functions and the good thing about 00:37:35.600 --> 00:37:38.470 the exponential series -- so in a way, 00:37:38.470 --> 00:37:41.990 this series is better than this one. 00:37:41.990 --> 00:37:43.290 Why? 00:37:43.290 --> 00:37:45.410 Because this one always converges. 00:37:45.410 --> 00:37:48.440 I'm dividing by these bigger and bigger numbers, 00:37:48.440 --> 00:37:54.770 so whatever matrix A and however large t is, that series -- 00:37:54.770 --> 00:37:57.430 these terms go to zero. 00:37:57.430 --> 00:38:01.960 The series adds up to a finite sum, e to the At is a -- is -- 00:38:01.960 --> 00:38:04.390 is completely defined. 00:38:04.390 --> 00:38:08.710 Whereas this second guy could fail, right? 00:38:08.710 --> 00:38:11.730 If At is big -- 00:38:11.730 --> 00:38:15.190 somehow if At has an eigenvalue larger than one, 00:38:15.190 --> 00:38:18.690 then when I square it it'll have that eigenvalue squared, 00:38:18.690 --> 00:38:21.820 when I cube it the eigenvalue will be cubed -- 00:38:21.820 --> 00:38:26.560 that series will blow up unless the eigenvalues of At 00:38:26.560 --> 00:38:28.500 are smaller than one. 00:38:28.500 --> 00:38:32.150 So when the eigenvalues of At are smaller than one -- 00:38:32.150 --> 00:38:33.610 so I'd better put that in. 00:38:33.610 --> 00:38:38.260 The -- all eigenvalues of At below one -- 00:38:38.260 --> 00:38:42.230 then that series converges and gives me the inverse. 00:38:42.230 --> 00:38:42.970 Okay. 00:38:42.970 --> 00:38:47.590 So this is the guy I'm chiefly interested in, and I wanted 00:38:47.590 --> 00:38:51.820 to connect it to -- 00:38:51.820 --> 00:38:52.400 oh, okay. 00:38:52.400 --> 00:38:55.810 What's -- how do I -- how do I get -- this is my, like, 00:38:55.810 --> 00:38:58.160 main thing now to do -- 00:38:58.160 --> 00:39:02.270 how do I get e to the At -- 00:39:02.270 --> 00:39:05.760 how do I see that e to the At is the same as this? 00:39:10.450 --> 00:39:16.410 In other words, I can find e to the At by finding S and lambda, 00:39:16.410 --> 00:39:18.710 because now e to the lambda t 00:39:18.710 --> 00:39:21.350 is easy. 00:39:21.350 --> 00:39:24.820 Lambda's a diagonal matrix and we can write down either 00:39:24.820 --> 00:39:27.250 the lambda t -- and will right -- in a minute. 00:39:27.250 --> 00:39:29.810 But how -- do you see what -- 00:39:29.810 --> 00:39:33.290 do you see that we're hoping for a -- 00:39:33.290 --> 00:39:38.540 we're hoping that we can compute either the A T from S 00:39:38.540 --> 00:39:41.450 and lambda -- 00:39:41.450 --> 00:39:44.980 and I have to look at that definition and say, okay, 00:39:44.980 --> 00:39:48.660 how do -- how do I get an S and the lambda to come out of that? 00:39:48.660 --> 00:39:50.670 Okay, can -- do you see how I -- 00:39:50.670 --> 00:39:54.830 I want to connect that to that, from that definition. 00:39:54.830 --> 00:39:59.090 So let me erase this -- the geometric series, 00:39:59.090 --> 00:40:08.250 which isn't part of the differential equations case 00:40:08.250 --> 00:40:14.200 and get the S and the lambda into this picture. 00:40:14.200 --> 00:40:15.900 Oh, okay. 00:40:15.900 --> 00:40:16.420 Here we go. 00:40:19.210 --> 00:40:22.930 So identity is fine. 00:40:22.930 --> 00:40:26.880 Now -- all right, you -- you -- you'll see how I'm -- 00:40:26.880 --> 00:40:31.900 how I'm -- how I going to get A replaced by S and S is 00:40:31.900 --> 00:40:32.550 in lambda's? 00:40:32.550 --> 00:40:36.520 Well I use the fundamental formula of this whole chapter. 00:40:36.520 --> 00:40:43.540 A is S lambda S inverse and then times t. 00:40:43.540 --> 00:40:45.451 That's At. 00:40:45.451 --> 00:40:45.950 Okay. 00:40:45.950 --> 00:40:48.910 What's A squared t? 00:40:48.910 --> 00:40:51.450 I can -- I've got the divide by two, 00:40:51.450 --> 00:40:56.580 I've got the t squared and I've got an A squared. 00:40:56.580 --> 00:41:02.580 All right, I -- so I've got a -- there's A -- there's A. 00:41:02.580 --> 00:41:04.390 Now square it. 00:41:04.390 --> 00:41:05.880 So what happens when I square it? 00:41:05.880 --> 00:41:08.050 We've seen that before. 00:41:08.050 --> 00:41:16.120 When I square it, I get S lambda squared S inverse, right? 00:41:16.120 --> 00:41:20.070 When I square that thing, the -- there's an S and a -- 00:41:20.070 --> 00:41:23.840 an S cancels out an S inverse. 00:41:23.840 --> 00:41:25.900 I'm left with the S on the left, the S 00:41:25.900 --> 00:41:29.140 inverse on the right and lambda squared in the middle. 00:41:29.140 --> 00:41:31.660 And so on. 00:41:31.660 --> 00:41:36.680 The next one'll be S lambda cubed, S inverse -- 00:41:36.680 --> 00:41:39.590 times t cubed over three factorial. 00:41:39.590 --> 00:41:45.190 And now -- what do I do now? 00:41:45.190 --> 00:41:48.440 I want to pull an S out from everything. 00:41:48.440 --> 00:41:53.010 I want an S out of the whole thing. 00:41:53.010 --> 00:41:57.020 Well, look, I'd better write the identity how? 00:41:57.020 --> 00:42:01.250 I -- I want to be able to pull an S out and an S inverse out 00:42:01.250 --> 00:42:04.840 from the other side, so I just write the identity as S times S 00:42:04.840 --> 00:42:05.820 inverse. 00:42:05.820 --> 00:42:11.120 So I have an S there and an S inverse from this side 00:42:11.120 --> 00:42:13.240 and what have I got in the middle? 00:42:16.170 --> 00:42:18.241 If I pull out an S and an S inverse, 00:42:18.241 --> 00:42:19.490 what have I got in the middle? 00:42:19.490 --> 00:42:23.260 I've got the identity, a lambda t, 00:42:23.260 --> 00:42:26.650 a lambda squared t squared over two -- 00:42:26.650 --> 00:42:30.780 I've got e to the lambda t. 00:42:30.780 --> 00:42:32.600 That's what's in the middle. 00:42:32.600 --> 00:42:36.630 That's my formula for e to the At. 00:42:36.630 --> 00:42:39.120 Oh, now I have to ask you. 00:42:39.120 --> 00:42:42.290 Does this formula always work? 00:42:42.290 --> 00:42:45.420 This formula always works -- 00:42:45.420 --> 00:42:48.540 well, except it's an infinite series. 00:42:48.540 --> 00:42:51.800 But what do I mean by always work? 00:42:51.800 --> 00:42:55.300 And this one doesn't always work and I just 00:42:55.300 --> 00:42:58.460 have to remind you of what assumption 00:42:58.460 --> 00:43:00.660 is built into this formula that's 00:43:00.660 --> 00:43:03.580 not built into the original. 00:43:03.580 --> 00:43:07.900 The assumption that A can be diagonalized. 00:43:07.900 --> 00:43:11.660 You'll remember that there are some small -- 00:43:11.660 --> 00:43:14.770 sm- some subset of matrixes that don't 00:43:14.770 --> 00:43:18.000 have n independent eigenvectors, so we 00:43:18.000 --> 00:43:20.590 don't have an S inverse for those matrixes 00:43:20.590 --> 00:43:24.780 and the whole diagonalization breaks down. 00:43:24.780 --> 00:43:26.860 We could still make it triangular. 00:43:26.860 --> 00:43:28.010 I'll tell you about that. 00:43:28.010 --> 00:43:32.930 But diagonal we can't do for those particular degenerate 00:43:32.930 --> 00:43:37.310 matrixes that don't have enough independent eigenvectors. 00:43:37.310 --> 00:43:40.240 But otherwise, this is golden. 00:43:40.240 --> 00:43:40.970 Okay. 00:43:40.970 --> 00:43:44.780 So that's the formula -- that's the matrix exponential. 00:43:44.780 --> 00:43:48.680 Now it just remains for me to say what is e to the lambda t? 00:43:48.680 --> 00:43:50.460 Can I just do that? 00:43:50.460 --> 00:43:55.280 Let me just put that in the corner here. 00:43:55.280 --> 00:44:02.180 What is the exponential of a diagonal matrix? 00:44:02.180 --> 00:44:10.140 Remember lambda is diagonal, lambda one down to lambda n. 00:44:10.140 --> 00:44:14.520 What's the exponential of that diagonal matrix? 00:44:14.520 --> 00:44:19.550 Because our whole point is that this ought to be simple. 00:44:19.550 --> 00:44:23.240 Our whole point is that to take the exponential of a diagonal 00:44:23.240 --> 00:44:27.560 matrix ought to be completely decoupled -- 00:44:27.560 --> 00:44:30.300 it ought to be diagonal, in other words, and it is. 00:44:30.300 --> 00:44:37.010 It's just e to the lambda one t, e to the lambda two t, 00:44:37.010 --> 00:44:41.070 e to the lambda n t, all zeroes. 00:44:41.070 --> 00:44:47.620 So -- so if we have a diagonal matrix and I plug it into this 00:44:47.620 --> 00:44:52.140 exponential formula, everything's diagonal 00:44:52.140 --> 00:44:55.710 and the diagonal terms are just the ordinary scaler 00:44:55.710 --> 00:44:58.930 exponentials e to the lambda one t. 00:44:58.930 --> 00:45:01.840 Okay, so that's -- that's -- 00:45:01.840 --> 00:45:06.050 in a sense, I'm doing here, on this board, with -- with, like, 00:45:06.050 --> 00:45:11.100 formulas what I did on the -- 00:45:11.100 --> 00:45:15.940 in the first half of the lecture with specific matrix A 00:45:15.940 --> 00:45:19.270 and specific eigenvalues and eigenvectors. 00:45:19.270 --> 00:45:21.990 The -- so let me show you the formulas again. 00:45:21.990 --> 00:45:24.770 But the -- so you -- 00:45:24.770 --> 00:45:27.300 I guess -- what should you know from this 00:45:27.300 --> 00:45:28.300 lecture? 00:45:28.300 --> 00:45:34.180 You should know what this matrix exponential is and, like, 00:45:34.180 --> 00:45:36.670 when does it go to zero? 00:45:36.670 --> 00:45:38.390 Tell me again, now, the answer to that. 00:45:38.390 --> 00:45:41.350 When does e to the At approach -- 00:45:41.350 --> 00:45:45.510 get smaller and smaller as t increases? 00:45:45.510 --> 00:45:49.070 Well, the S and the S inverse aren't moving. 00:45:49.070 --> 00:45:51.780 It's this one that has to get smaller and smaller 00:45:51.780 --> 00:45:57.160 and that one has this simple diagonal form. 00:45:57.160 --> 00:46:01.740 And it goes to zero provided every one of these lambdas -- 00:46:01.740 --> 00:46:04.840 I -- I need to have each one of these guys go to zero, 00:46:04.840 --> 00:46:09.930 so I need every real part of every eigenvalue negative. 00:46:12.650 --> 00:46:13.230 Right? 00:46:13.230 --> 00:46:15.690 If the real part is negative, that's -- 00:46:15.690 --> 00:46:19.160 that takes the exponential -- that forces -- 00:46:19.160 --> 00:46:21.420 the exponential goes to zero. 00:46:21.420 --> 00:46:24.480 Okay, so that -- that's really the difference. 00:46:24.480 --> 00:46:33.250 If I can just draw the -- here's a picture of the -- of the -- 00:46:33.250 --> 00:46:36.780 this is the complex plain. 00:46:36.780 --> 00:46:42.080 Here's the real axis and here's the imaginary axis. 00:46:42.080 --> 00:46:43.960 And where do the eigenvalues have 00:46:43.960 --> 00:46:47.570 to be for stability in differential equations? 00:46:47.570 --> 00:46:52.470 They have to be over here, in the left half plain. 00:46:52.470 --> 00:46:55.910 So the left half plain is this plain, real part of lambda, 00:46:55.910 --> 00:46:58.820 less than zero. 00:46:58.820 --> 00:47:01.190 Where do the ma- where do the eigenvalues have 00:47:01.190 --> 00:47:06.500 to be for powers of the matrix to go to zero? 00:47:06.500 --> 00:47:11.960 Powers of the matrix go to zero if the eigenvalues are in here. 00:47:11.960 --> 00:47:17.800 So this is the stability region for powers -- 00:47:17.800 --> 00:47:22.490 this is the region -- absolute value of lambda, less than one. 00:47:22.490 --> 00:47:26.740 That's the stability for -- that tells us that the powers of A 00:47:26.740 --> 00:47:30.260 go to zero, this tells us that the exponential of A goes 00:47:30.260 --> 00:47:31.220 to zero. 00:47:31.220 --> 00:47:31.790 Okay. 00:47:31.790 --> 00:47:33.700 One final example. 00:47:33.700 --> 00:47:38.580 Let me just write down how to deal with a final example. 00:47:38.580 --> 00:47:39.980 Let's see. 00:47:44.480 --> 00:47:48.930 So my final example will be a single equation, u''+bu'+Ku=0. 00:47:57.040 --> 00:48:01.000 One equation, second order. 00:48:01.000 --> 00:48:03.490 How do I -- 00:48:03.490 --> 00:48:05.290 and maybe I should have used -- 00:48:05.290 --> 00:48:08.170 I'll use -- I prefer to use y here, 00:48:08.170 --> 00:48:12.170 because that's what you see in differential equations. 00:48:12.170 --> 00:48:14.790 And I want u to be a vector. 00:48:14.790 --> 00:48:23.730 So how do I change one second order equation into a two 00:48:23.730 --> 00:48:28.250 by two first order system? 00:48:28.250 --> 00:48:30.540 Just the way I did for Fibonacci. 00:48:30.540 --> 00:48:38.180 I'll let u be y prime and y. 00:48:38.180 --> 00:48:43.210 What I'm going to do is I'm going to add an extra equation, 00:48:43.210 --> 00:48:46.620 y prime equals y prime. 00:48:46.620 --> 00:48:50.800 So I take this -- so by -- 00:48:50.800 --> 00:48:55.110 so using this as the vector unknown, 00:48:55.110 --> 00:48:58.830 now my equation is u prime. 00:48:58.830 --> 00:49:00.800 My first order system is u prime, 00:49:00.800 --> 00:49:06.160 that'll be y double prime y prime, the derivative of u, 00:49:06.160 --> 00:49:10.940 okay, now the differential equation is telling me that y 00:49:10.940 --> 00:49:14.430 double prime is m- so I'm just looking for -- 00:49:14.430 --> 00:49:17.160 what's this matrix? 00:49:17.160 --> 00:49:19.410 y prime y. 00:49:19.410 --> 00:49:23.220 I'm looking for the matrix A. 00:49:23.220 --> 00:49:28.460 What's the matrix in case I have a single first order equation 00:49:28.460 --> 00:49:31.140 and I want to make it into a two by two system? 00:49:31.140 --> 00:49:32.270 Okay, simple. 00:49:32.270 --> 00:49:35.920 The first row of the matrix is given by the equation. 00:49:35.920 --> 00:49:43.800 So y''-by'-ky -- no problem. 00:49:43.800 --> 00:49:47.240 And what's the second row on the matrix? 00:49:47.240 --> 00:49:48.660 Then we're done. 00:49:48.660 --> 00:49:50.710 The second row of the matrix I want just 00:49:50.710 --> 00:49:54.490 to be the trivial equation y prime equals y prime, 00:49:54.490 --> 00:49:56.340 so I need a one and a zero there. 00:49:59.240 --> 00:50:03.950 So matrixes like these, the gen- the general case, 00:50:03.950 --> 00:50:09.050 if I had a five by five -- if I had a fifth order equation 00:50:09.050 --> 00:50:11.590 and I wanted a five by five matrix, 00:50:11.590 --> 00:50:15.850 I would see the coefficients of the equation up there and then 00:50:15.850 --> 00:50:21.260 my four trivial equations would put ones here. 00:50:21.260 --> 00:50:27.110 This is the kind of matrix that goes from a fifth order 00:50:27.110 --> 00:50:32.060 to a five by five first order. 00:50:35.140 --> 00:50:40.010 So the -- and the eigenvalues will come out in a natural way 00:50:40.010 --> 00:50:41.350 connected to the differential 00:50:41.350 --> 00:50:41.970 equation. 00:50:41.970 --> 00:50:45.840 Okay, that's differential equations. 00:50:45.840 --> 00:50:49.890 The -- a parallel lecture compared to powers of a matrix 00:50:49.890 --> 00:50:52.060 we can now do exponentials. 00:50:52.060 --> 00:50:53.610 Thanks.