WEBVTT 00:00:00.030 --> 00:00:02.330 The following content is provided under a Creative 00:00:02.330 --> 00:00:03.610 Commons license. 00:00:03.610 --> 00:00:05.970 Your support will help MIT OpenCourseWare 00:00:05.970 --> 00:00:09.960 continue to offer high-quality educational resources for free. 00:00:09.960 --> 00:00:12.590 To make a donation, or to view additional materials 00:00:12.590 --> 00:00:15.210 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:15.210 --> 00:00:19.819 at ocw.mit.edu. 00:00:19.819 --> 00:00:21.610 PROFESSOR STRANG: This is my second lecture 00:00:21.610 --> 00:00:27.430 on the example, the model of graphs and networks. 00:00:27.430 --> 00:00:32.220 Really perfect, beautiful model for the whole framework. 00:00:32.220 --> 00:00:37.230 And important in itself. 00:00:37.230 --> 00:00:42.700 So, I guess a major thing that I still have to do 00:00:42.700 --> 00:00:45.080 is discuss A transpose. 00:00:45.080 --> 00:00:48.140 See how A transpose just naturally appears 00:00:48.140 --> 00:00:50.800 in the balance law. 00:00:50.800 --> 00:00:55.500 Kirchhoff's current law, KCL, is just 00:00:55.500 --> 00:00:58.290 like a model for balance equations. 00:00:58.290 --> 00:01:02.700 By balance I mean, in equals out, essentially. 00:01:02.700 --> 00:01:05.030 Flow in equals flow out because we're 00:01:05.030 --> 00:01:06.860 talking about steady state. 00:01:06.860 --> 00:01:11.550 So in each node, at node three, for example, 00:01:11.550 --> 00:01:14.500 I have three edges. 00:01:14.500 --> 00:01:18.210 So the law, Kirchhoff's current law at that point 00:01:18.210 --> 00:01:29.810 is going to tell me that the flow w_2 plus w_3 minus w_5 00:01:29.810 --> 00:01:33.150 would be zero if there's no current source, 00:01:33.150 --> 00:01:37.850 or if I'm feeding current into there, some current f_3, 00:01:37.850 --> 00:01:41.000 then it would match f_3. 00:01:41.000 --> 00:01:43.330 So maybe having just said those words, 00:01:43.330 --> 00:01:46.790 let me just say, w_2, I'll say it again, 00:01:46.790 --> 00:01:52.160 w_2 plus w_3 minus w_5, and I'm hoping 00:01:52.160 --> 00:01:56.860 that I'll find that in the third column here. 00:01:56.860 --> 00:01:58.790 And I do. 00:01:58.790 --> 00:02:02.850 Because I'm thinking, and we'll write it down, 00:02:02.850 --> 00:02:04.490 I'm thinking to take a transpose, 00:02:04.490 --> 00:02:07.040 so it'll be the third row, and here I 00:02:07.040 --> 00:02:10.810 see the w_2, w_3 and minus w_5 that 00:02:10.810 --> 00:02:15.780 will show up in the third equation there, 00:02:15.780 --> 00:02:18.900 the in equals out at node three. 00:02:18.900 --> 00:02:23.630 Well I'll write that down, because it's-- So, 00:02:23.630 --> 00:02:26.400 two or three jobs for today. 00:02:26.400 --> 00:02:32.390 And then Monday I plan to spend a part of the lecture 00:02:32.390 --> 00:02:38.620 and all of the review session in review, 00:02:38.620 --> 00:02:43.650 it's a great chance to go back to the things we've done 00:02:43.650 --> 00:02:49.860 and collect them, assemble them, organize them and see them 00:02:49.860 --> 00:02:50.480 again. 00:02:50.480 --> 00:02:54.950 So that's my goal for both sessions on Monday. 00:02:54.950 --> 00:02:59.050 And questions, then, about every aspect. 00:02:59.050 --> 00:03:05.600 So, before I get to A transpose, this part 00:03:05.600 --> 00:03:09.490 I had written down last time, but there's a little more 00:03:09.490 --> 00:03:10.880 to say. 00:03:10.880 --> 00:03:15.840 And what I was saying and want to continue with 00:03:15.840 --> 00:03:20.750 is the source terms. 00:03:20.750 --> 00:03:28.160 I put in this b but I didn't draw anything on the network. 00:03:28.160 --> 00:03:34.010 So let me draw what these b's represent. 00:03:34.010 --> 00:03:41.140 So this second, lower, row is about edge equations, 00:03:41.140 --> 00:03:42.610 edge variables. 00:03:42.610 --> 00:03:48.020 So those batteries b are on the edges. 00:03:48.020 --> 00:03:51.120 And there will be, b has length five. 00:03:51.120 --> 00:03:54.220 There are five edges. 00:03:54.220 --> 00:03:57.320 This is a vector of length five, this matrix, A, 00:03:57.320 --> 00:04:00.690 you remember was five by four. 00:04:00.690 --> 00:04:05.410 It produced from four inputs, from four potentials 00:04:05.410 --> 00:04:13.370 at the four nodes, A produces Au, five potential differences. 00:04:13.370 --> 00:04:16.010 So those are the differences in potentials. 00:04:16.010 --> 00:04:18.920 But then also there can be source terms 00:04:18.920 --> 00:04:22.270 from batteries in the edges, and the minus sign 00:04:22.270 --> 00:04:25.280 is there because I'm talking about voltage drops. 00:04:25.280 --> 00:04:28.440 So when I say differences I'm really 00:04:28.440 --> 00:04:32.280 speaking about voltage drops. 00:04:32.280 --> 00:04:35.720 Because that's the way current flows. 00:04:35.720 --> 00:04:40.560 OK, this is the moment that I hate. 00:04:40.560 --> 00:04:44.480 Putting the batteries in. 00:04:44.480 --> 00:04:49.580 I think you draw a battery with a long and a short? 00:04:49.580 --> 00:04:50.650 Is that right? 00:04:50.650 --> 00:04:55.460 And then you put a plus and a minus? 00:04:55.460 --> 00:05:04.340 Well, can I just say, life is too short to get those, 00:05:04.340 --> 00:05:10.930 to get this sign right. 00:05:10.930 --> 00:05:13.210 You may say when my car battery stalls 00:05:13.210 --> 00:05:15.880 how do I, because it's important at that moment, right? 00:05:15.880 --> 00:05:17.930 When you start it up you're supposed 00:05:17.930 --> 00:05:23.330 to put the right battery, the right lead on the positive 00:05:23.330 --> 00:05:26.250 and the right on the negative, or your blow yourself up. 00:05:26.250 --> 00:05:27.420 OK. 00:05:27.420 --> 00:05:28.670 So what's my solution? 00:05:28.670 --> 00:05:32.330 Because I literally refuse to deal with these signs. 00:05:32.330 --> 00:05:35.220 So my solution is call AAA. 00:05:35.220 --> 00:05:37.710 OK, they're paid to blow themselves up. 00:05:37.710 --> 00:05:39.090 They can do it. 00:05:39.090 --> 00:05:45.360 But so this is serious now, I don't want 00:05:45.360 --> 00:05:48.820 to be asked about these signs. 00:05:48.820 --> 00:05:52.040 And I forgive you for messing up the signs. 00:05:52.040 --> 00:05:56.330 So possibly that's plus, possibly minus, I don't know. 00:05:56.330 --> 00:06:00.290 But there's a battery in there of length b_1, 00:06:00.290 --> 00:06:04.230 of voltage-- A nine-volt battery, b_1 would be nine. 00:06:04.230 --> 00:06:06.780 And depending how it was placed in there, 00:06:06.780 --> 00:06:10.510 the b here would be a plus nine or a minus nine 00:06:10.510 --> 00:06:12.000 in the first component. 00:06:12.000 --> 00:06:15.020 And then if I had another battery here, 00:06:15.020 --> 00:06:18.990 a b that's on edge four, there would be a battery b_4, 00:06:18.990 --> 00:06:24.230 and that would show up there in Ohm's law. 00:06:24.230 --> 00:06:27.330 Because Ohm's law will look at the difference 00:06:27.330 --> 00:06:30.400 in these potentials, but then it'll also 00:06:30.400 --> 00:06:33.790 account for the battery, right? 00:06:33.790 --> 00:06:39.130 The voltage that comes from the battery, and somehow 00:06:39.130 --> 00:06:42.730 combining the Au, which is the difference in these guys, 00:06:42.730 --> 00:06:46.760 with the b_4, I'll know what is the voltage drop 00:06:46.760 --> 00:06:48.010 across the resistor. 00:06:48.010 --> 00:06:50.060 And that's what Ohm's law applies to. 00:06:50.060 --> 00:06:53.870 Ohm's law says the voltage drop across the resistor 00:06:53.870 --> 00:07:00.300 times the conductance, so this is Ohm's law, that on that edge 00:07:00.300 --> 00:07:04.220 the voltage drop across the resistor times the conductance 00:07:04.220 --> 00:07:05.570 gives the current. 00:07:05.570 --> 00:07:07.410 So that's the physical law. 00:07:07.410 --> 00:07:09.470 Is that OK? 00:07:09.470 --> 00:07:10.850 For batteries. 00:07:10.850 --> 00:07:12.950 Now a comment on current sources. 00:07:12.950 --> 00:07:14.530 So what's with current sources? 00:07:14.530 --> 00:07:16.180 How would I draw those? 00:07:16.180 --> 00:07:22.990 Well, very often maybe I might draw a current going 00:07:22.990 --> 00:07:28.590 into node one, so that would be in f_1 that goes into node one 00:07:28.590 --> 00:07:30.980 and maybe comes out at ground. 00:07:30.980 --> 00:07:34.010 So that would be a typical f. 00:07:34.010 --> 00:07:37.850 That if I imposed a current source, 00:07:37.850 --> 00:07:40.380 if I sent a current source through there 00:07:40.380 --> 00:07:44.490 it would go down and come out at ground, 00:07:44.490 --> 00:07:49.480 and our question is what are the currents in the five edges? 00:07:49.480 --> 00:07:51.740 What are the potentials at the four nodes? 00:07:51.740 --> 00:07:53.890 Well, I'm making this one ground. 00:07:53.890 --> 00:07:58.240 So I'm grounding this one to be u_4=0. 00:07:58.240 --> 00:08:00.370 And you remember of course why I had 00:08:00.370 --> 00:08:04.980 to do that, because this matrix A transpose A, as it stands 00:08:04.980 --> 00:08:10.060 is-- What's the matter with A transpose A as it stands? 00:08:10.060 --> 00:08:11.960 It's singular, right? 00:08:11.960 --> 00:08:18.360 And as I, looking ahead, just a small comment, 00:08:18.360 --> 00:08:20.640 that this will have exactly the same thing 00:08:20.640 --> 00:08:24.030 in so many other applications in big, finite element codes, 00:08:24.030 --> 00:08:28.030 you produce a stiffness matrix that's initially singular. 00:08:28.030 --> 00:08:30.970 And then you impose the boundary conditions. 00:08:30.970 --> 00:08:32.780 That's the efficient way to do it, 00:08:32.780 --> 00:08:36.080 is create the matrix first, that's the big job, 00:08:36.080 --> 00:08:38.010 then impose the boundary conditions, 00:08:38.010 --> 00:08:42.100 that's the small but occasionally tricky part. 00:08:42.100 --> 00:08:45.540 Now I indicated what happened here. 00:08:45.540 --> 00:08:50.770 When u_4 was zero, that means that A times u, the u_4 00:08:50.770 --> 00:08:54.960 there, the zero is multiplying this, is not unknown any more. 00:08:54.960 --> 00:08:56.150 It's known. 00:08:56.150 --> 00:09:03.440 And so that column is not really any more needed in A. 00:09:03.440 --> 00:09:11.590 Because u_4 is gone from my list of unknown potentials. 00:09:11.590 --> 00:09:18.500 Now, at the same time, when I went over to A transpose A-- 00:09:18.500 --> 00:09:20.620 I'm making this comment because there 00:09:20.620 --> 00:09:22.680 were several good questions about it. 00:09:22.680 --> 00:09:30.820 I claim that also, that row, coming, 00:09:30.820 --> 00:09:34.030 which of course comes from the fourth row of A transpose. 00:09:34.030 --> 00:09:38.640 The fourth column of A is gone, so the fourth row 00:09:38.640 --> 00:09:40.800 of A transpose should be gone. 00:09:40.800 --> 00:09:44.150 And we might just think why's that, what's going on? 00:09:44.150 --> 00:09:47.200 Of course, it produces exactly what we want. 00:09:47.200 --> 00:09:49.870 It leaves us with a three by three matrix 00:09:49.870 --> 00:09:52.000 that's not singular any more. 00:09:52.000 --> 00:09:57.140 I've removed that [1, 1, 1, 1] from the null space by fixing 00:09:57.140 --> 00:09:58.910 a potential. 00:09:58.910 --> 00:10:03.850 Grounding a node, and the problem 00:10:03.850 --> 00:10:05.990 becomes just what we want. 00:10:05.990 --> 00:10:09.350 And I'll write down the equations 00:10:09.350 --> 00:10:15.080 when I get the Kirchhoff current law to complete the loop. 00:10:15.080 --> 00:10:17.880 Now, what's going on? 00:10:17.880 --> 00:10:21.910 Let me remove this current source, 00:10:21.910 --> 00:10:28.690 just so we focus on what I'm speaking about now. 00:10:28.690 --> 00:10:34.360 On this type of boundary condition, 00:10:34.360 --> 00:10:38.590 this is like fixing a support, right? 00:10:38.590 --> 00:10:43.060 It's like fixing a support in our spring mass problem. 00:10:43.060 --> 00:10:46.070 Can I squeeze in a little spring mass problem, 00:10:46.070 --> 00:10:56.760 so I have a spring, and then I'm fixing this displacement, 00:10:56.760 --> 00:10:58.840 u_4 to be zero. 00:10:58.840 --> 00:11:06.220 Now, I want to try to think through what 00:11:06.220 --> 00:11:11.660 happens in a balance equation, A transpose w, 00:11:11.660 --> 00:11:14.850 let me make it A transpose w equal f, 00:11:14.850 --> 00:11:23.410 because that's the case with right hand side allowed. 00:11:23.410 --> 00:11:28.730 And now what happens when I fix this, 00:11:28.730 --> 00:11:34.780 I'm asking you to think back about force balance, which 00:11:34.780 --> 00:11:41.370 we certainly saw was an A transpose w equal f business. 00:11:41.370 --> 00:11:47.480 And then parallel will be the current balance, at that node. 00:11:47.480 --> 00:11:54.240 OK, so what was the deal on force balance? 00:11:54.240 --> 00:12:00.480 A transpose with this fixed in here. 00:12:00.480 --> 00:12:04.600 What was the thing with A transpose that that fixed in? 00:12:04.600 --> 00:12:07.880 We did not write the equation A transpose w equal 00:12:07.880 --> 00:12:09.790 f at this point. 00:12:09.790 --> 00:12:14.120 We did not write the force balance equation at that point. 00:12:14.120 --> 00:12:19.960 When I fixed u_4, in this case it was the displacement so I'm 00:12:19.960 --> 00:12:24.130 fixing it at zero displacement, I 00:12:24.130 --> 00:12:27.390 didn't have a force balance in the A transpose part 00:12:27.390 --> 00:12:28.680 at this node. 00:12:28.680 --> 00:12:30.670 Now, you could say why? 00:12:30.670 --> 00:12:33.310 Because of course forces have to balance. 00:12:33.310 --> 00:12:35.020 But what's going on? 00:12:35.020 --> 00:12:39.430 What's really happening here is I don't have to write out, 00:12:39.430 --> 00:12:43.680 I don't have to-- the displacement here is known. 00:12:43.680 --> 00:12:45.700 It's not unknown. 00:12:45.700 --> 00:12:50.260 And let me say it in a sentence. 00:12:50.260 --> 00:12:54.530 Force balance does hold because the support 00:12:54.530 --> 00:13:01.890 supplies whatever force it takes to balance the internal forces. 00:13:01.890 --> 00:13:09.120 So, in other words, let me say that again, the force balance 00:13:09.120 --> 00:13:11.720 will hold and it will tell me, after I've 00:13:11.720 --> 00:13:14.120 solved the rest of the problem, it'll tell me 00:13:14.120 --> 00:13:18.850 what the support has to supply, how much force the support is 00:13:18.850 --> 00:13:19.720 actually supplying. 00:13:19.720 --> 00:13:22.430 It's a reaction force, it would be called. 00:13:22.430 --> 00:13:26.710 So a reaction force is whatever the support has 00:13:26.710 --> 00:13:29.530 to do to fix that displacement. 00:13:29.530 --> 00:13:35.810 And so I solved the fixed problems 00:13:35.810 --> 00:13:40.020 for all the other displacements and all the spring forces. 00:13:40.020 --> 00:13:42.320 And then I could come back at the end 00:13:42.320 --> 00:13:46.060 and say OK, what was the force in that spring and therefore 00:13:46.060 --> 00:13:51.340 how much is that support, what's the force being supplied 00:13:51.340 --> 00:13:52.770 by that support. 00:13:52.770 --> 00:13:53.920 See what I'm saying? 00:13:53.920 --> 00:14:05.270 That the force balance at this node comes afterwards. 00:14:05.270 --> 00:14:12.020 That equation is like knocking that one out of this problem. 00:14:12.020 --> 00:14:17.860 It would be the same here, I fix the potential at zero. 00:14:17.860 --> 00:14:19.760 I fix ground at zero. 00:14:19.760 --> 00:14:21.960 Current flows. 00:14:21.960 --> 00:14:26.540 Maybe some, maybe I might fix that potential at one. 00:14:26.540 --> 00:14:28.350 I fix that potential at zero. 00:14:28.350 --> 00:14:29.770 Current flows. 00:14:29.770 --> 00:14:31.010 OK. 00:14:31.010 --> 00:14:33.730 Ta-da, I find out, I compute what they are, 00:14:33.730 --> 00:14:40.200 and this row and column will be gone. 00:14:40.200 --> 00:14:41.620 Find out what they are. 00:14:41.620 --> 00:14:46.840 At this node, what's happened? 00:14:46.840 --> 00:14:50.010 What's happening to the balance of currents? 00:14:50.010 --> 00:14:52.820 Does this current have to balance that current 00:14:52.820 --> 00:14:54.220 at the ground? 00:14:54.220 --> 00:14:55.070 No. 00:14:55.070 --> 00:15:00.700 The ground, whatever current comes here 00:15:00.700 --> 00:15:04.110 plus whatever current comes here, goes into ground. 00:15:04.110 --> 00:15:07.920 Do you see the point? 00:15:07.920 --> 00:15:10.200 Kirchhoff's current law, the balance 00:15:10.200 --> 00:15:14.840 of currents that in equals out, is true 00:15:14.840 --> 00:15:18.000 but it's not an equation I have to solve 00:15:18.000 --> 00:15:20.920 in finding the currents, it's something 00:15:20.920 --> 00:15:22.540 I can discover at the end. 00:15:22.540 --> 00:15:25.860 I can say OK, how much current flowed from ground? 00:15:25.860 --> 00:15:27.280 And similarly up here. 00:15:27.280 --> 00:15:32.167 If I fix u_1 to be zero, then you, no, I 00:15:32.167 --> 00:15:34.500 don't want to fix it to be zero too, that would be a way 00:15:34.500 --> 00:15:36.850 to make very little happen. 00:15:36.850 --> 00:15:41.940 Let me fix u_1 to be one, so this is 00:15:41.940 --> 00:15:44.810 a standard important problem. 00:15:44.810 --> 00:15:48.210 It's like what's the resistance in the net-- what's 00:15:48.210 --> 00:15:50.870 the net system resistance? 00:15:50.870 --> 00:15:55.500 If I fix this at one fix this at zero, some current will flow, 00:15:55.500 --> 00:15:59.740 it'll come out here. 00:15:59.740 --> 00:16:01.800 And that'll be the current going into here. 00:16:01.800 --> 00:16:06.680 Somehow there'll be a balance there, and a balance there. 00:16:06.680 --> 00:16:10.680 But it's found later. 00:16:10.680 --> 00:16:14.220 Just the way the force in the support is found later. 00:16:14.220 --> 00:16:16.430 OK, that takes a little thinking. 00:16:16.430 --> 00:16:23.400 I just wanted to, because I had blithely knocked this row out. 00:16:23.400 --> 00:16:26.000 And you could do it on the basis, 00:16:26.000 --> 00:16:28.740 well if you knock out this column of A 00:16:28.740 --> 00:16:31.360 then you're knocking out the row of A transpose, 00:16:31.360 --> 00:16:35.050 and therefore A transpose A will be three by three. 00:16:35.050 --> 00:16:40.340 And it'll go down to two by two if I fix that potential. 00:16:40.340 --> 00:16:43.730 And if I don't fix it at zero, if I fix it at one 00:16:43.730 --> 00:16:47.180 then something will move to the right hand side. 00:16:47.180 --> 00:16:57.590 OK. that's that point I hope at least discussed. 00:16:57.590 --> 00:17:03.220 So now we have the whole pattern here. 00:17:03.220 --> 00:17:09.480 Except that I really have still to justify the fact that it 00:17:09.480 --> 00:17:14.650 truly is A transpose, the transpose of that matrix, 00:17:14.650 --> 00:17:17.010 that comes into Kirchhoff's current law. 00:17:17.010 --> 00:17:20.200 I guess in the first minute of the lecture I 00:17:20.200 --> 00:17:21.800 looked at that particular node. 00:17:21.800 --> 00:17:23.410 Let's look at all the nodes. 00:17:23.410 --> 00:17:28.100 Let me look at A transpose w equals zero. 00:17:28.100 --> 00:17:28.960 OK. 00:17:28.960 --> 00:17:32.150 So I'll copy a transpose because I truly 00:17:32.150 --> 00:17:35.790 believe that Kirchhoff would want me to do it. 00:17:35.790 --> 00:17:45.080 OK, so that becomes a row, that becomes the next row, of course 00:17:45.080 --> 00:17:50.500 I see three guys going into node one, 00:17:50.500 --> 00:17:53.370 then the one that I looked at before. 00:17:53.370 --> 00:17:59.310 That's the three edges, node three, and then the last one. 00:17:59.310 --> 00:18:02.340 Let's keep the last one in for now. 00:18:02.340 --> 00:18:08.310 And because if I didn't fix that I'd have that last one. 00:18:08.310 --> 00:18:12.290 There we go, so that multiplies, what does that multiply now 00:18:12.290 --> 00:18:14.020 in Kirchhoff's current law? 00:18:14.020 --> 00:18:16.680 Multiplies w, so currents. 00:18:16.680 --> 00:18:22.480 So here the currents, one, two, three, four and five. 00:18:22.480 --> 00:18:26.300 All I'm doing now is just, like, convincing you 00:18:26.300 --> 00:18:29.230 that it really is a transpose, that if I look at, 00:18:29.230 --> 00:18:32.250 let me pick that node now, if I look 00:18:32.250 --> 00:18:38.840 at that node I see edge one coming in, 00:18:38.840 --> 00:18:42.820 I see edge three going out, I see edge four going out 00:18:42.820 --> 00:18:45.410 and when I look at that second node, 00:18:45.410 --> 00:18:48.980 I look here at the second row, I see edge one coming in, 00:18:48.980 --> 00:18:51.020 edge three and edge four going out, 00:18:51.020 --> 00:18:54.280 multiplying those currents, and that will give, 00:18:54.280 --> 00:18:58.580 that current balance there gives that zero. 00:18:58.580 --> 00:18:59.080 Right? 00:18:59.080 --> 00:19:02.510 That current balance at node two gives that 00:19:02.510 --> 00:19:05.770 zero in the right hand side. 00:19:05.770 --> 00:19:09.720 And then, of course, other zeroes are here too. 00:19:09.720 --> 00:19:14.860 This is minus w_1, minus w_2, so I have a zero. 00:19:14.860 --> 00:19:19.240 This is with no current sources. 00:19:19.240 --> 00:19:19.930 OK. 00:19:19.930 --> 00:19:23.890 And this one, of course. 00:19:23.890 --> 00:19:29.070 I guess I'm hoping that you say yes, the A transpose really 00:19:29.070 --> 00:19:35.010 was the right matrix to express in equals out, Kirchhoff's 00:19:35.010 --> 00:19:38.040 current law. 00:19:38.040 --> 00:19:41.020 OK. 00:19:41.020 --> 00:19:43.620 Of course, by now you're probably getting blasé. 00:19:43.620 --> 00:19:47.600 You expect it to be A transpose, you don't need any convincing. 00:19:47.600 --> 00:19:50.980 But it's kind of good to see each time because it's 00:19:50.980 --> 00:19:53.620 like, well, it's not a miracle. 00:19:53.620 --> 00:19:56.100 But it's like a miracle. 00:19:56.100 --> 00:19:57.930 It's as good as a miracle. 00:19:57.930 --> 00:20:01.930 Because to get A transpose is just 00:20:01.930 --> 00:20:04.590 what makes everything right. 00:20:04.590 --> 00:20:11.340 Now, here's a question. 00:20:11.340 --> 00:20:15.830 This is a question worth thinking about. 00:20:15.830 --> 00:20:17.120 What are the solutions? 00:20:17.120 --> 00:20:21.540 If I only look at this piece of the framework, 00:20:21.540 --> 00:20:25.330 if I just look at Kirchhoff's current law, it's telling me, 00:20:25.330 --> 00:20:27.950 and I have zero current sources. 00:20:27.950 --> 00:20:30.410 Well let's take that, let me take 00:20:30.410 --> 00:20:35.210 just this piece of the whole framework and ask you, 00:20:35.210 --> 00:20:41.210 how many vectors w, how many solutions? 00:20:41.210 --> 00:20:42.780 Are there any solutions? 00:20:42.780 --> 00:20:45.510 Well, of course, there's always the zero solution. 00:20:45.510 --> 00:20:50.300 But I always ask you, what are the solutions when 00:20:50.300 --> 00:20:52.790 zero is on the right hand side? 00:20:52.790 --> 00:20:56.770 So, what are the w's that solve Kirchhoff's current law. 00:20:56.770 --> 00:20:58.390 Now that's a new question. 00:20:58.390 --> 00:20:59.930 We haven't asked that before. 00:20:59.930 --> 00:21:04.880 What we asked before was the question Au, Au=0, 00:21:04.880 --> 00:21:09.590 remember it was then A. This five by four matrix, 00:21:09.590 --> 00:21:11.880 u had four components. 00:21:11.880 --> 00:21:16.720 Just remind me, and I'm putting that column back in, 00:21:16.720 --> 00:21:19.840 so this is still here. 00:21:19.840 --> 00:21:23.760 In the un-reduced, un-grounded case. 00:21:23.760 --> 00:21:27.090 Well, just so we get started, remind me 00:21:27.090 --> 00:21:35.120 what the solutions u were for Au equals zero. [1, 1, 1, 1]. 00:21:35.120 --> 00:21:36.330 Or any multiple of it. 00:21:36.330 --> 00:21:39.280 A whole line of vectors. 00:21:39.280 --> 00:21:42.090 [c, c, c, c], any constant c. 00:21:42.090 --> 00:21:47.040 OK, so this had, there were four columns. 00:21:47.040 --> 00:21:49.140 But only three were independent. 00:21:49.140 --> 00:21:49.900 OK. 00:21:49.900 --> 00:21:56.580 Now, now I've made them into rows. 00:21:56.580 --> 00:21:59.170 And I made the rows into columns. 00:21:59.170 --> 00:22:02.640 So now I have five columns. 00:22:02.640 --> 00:22:08.240 What I'm leading to is, I want to count in advance 00:22:08.240 --> 00:22:12.900 how many w's I should be looking for, and then look for them. 00:22:12.900 --> 00:22:18.420 OK, so the first question is how many different solutions w. 00:22:18.420 --> 00:22:22.280 First of all, are there some solutions? 00:22:22.280 --> 00:22:25.880 Is there is a solution w, other than zero of course, 00:22:25.880 --> 00:22:27.980 to this system? 00:22:27.980 --> 00:22:30.950 Well, as we said last time, we've 00:22:30.950 --> 00:22:33.180 only got four equations here. 00:22:33.180 --> 00:22:35.810 We've got five unknowns. 00:22:35.810 --> 00:22:38.490 Of course there's at least one solution. 00:22:38.490 --> 00:22:42.190 Four equations, five unknowns, I can do elimination, 00:22:42.190 --> 00:22:45.490 whatever systematic procedure you want me to do. 00:22:45.490 --> 00:22:48.030 In the end, I'm going to find a solution. 00:22:48.030 --> 00:22:50.100 Now, I might find more solutions. 00:22:50.100 --> 00:22:52.590 So that's the question. 00:22:52.590 --> 00:22:57.840 So what's the, now this is the key fact of linear algebra. 00:22:57.840 --> 00:23:02.930 Which is, it just tells us the numbers of everything. 00:23:02.930 --> 00:23:07.950 So this told us that there were three independent columns. 00:23:07.950 --> 00:23:15.130 Of A. Now, for that key theorem, which tells me that, 00:23:15.130 --> 00:23:19.040 how many independent rows of A are there? 00:23:19.040 --> 00:23:19.970 Three. 00:23:19.970 --> 00:23:22.940 That number is equal. 00:23:22.940 --> 00:23:25.860 I just want to say that's a pretty remarkable fact. 00:23:25.860 --> 00:23:32.930 If I have a 50 by 80 matrix and that 50 by 80 matrix 00:23:32.930 --> 00:23:38.240 has 17 independent columns, then this great fact 00:23:38.240 --> 00:23:41.330 tells me that there are 17 independent rows. 00:23:41.330 --> 00:23:47.960 And if those 50 times 80, 4,000 numbers are random, 00:23:47.960 --> 00:23:52.690 boy, you can't look at it and see what are they. 00:23:52.690 --> 00:23:53.740 Independent rows. 00:23:53.740 --> 00:23:56.660 But if you know there are 17 independent columns then 00:23:56.660 --> 00:23:58.570 there are 17 independent rows. 00:23:58.570 --> 00:24:01.990 So, what does that mean here? 00:24:01.990 --> 00:24:04.450 That means that out of the five columns 00:24:04.450 --> 00:24:10.810 of A transpose, which were rows of A, three are independent. 00:24:10.810 --> 00:24:14.220 So just tell me, how many solutions I'm looking for. 00:24:14.220 --> 00:24:16.470 Before I look for them. 00:24:16.470 --> 00:24:20.750 So the number of-- The number of independent w's, 00:24:20.750 --> 00:24:25.700 independent solutions will be what? 00:24:25.700 --> 00:24:27.770 What's your guess? 00:24:27.770 --> 00:24:28.620 Two! 00:24:28.620 --> 00:24:30.320 Two is the right guess. 00:24:30.320 --> 00:24:34.700 Two, because I have altogether five unknowns, 00:24:34.700 --> 00:24:39.650 I subtract three equations that are really there, 00:24:39.650 --> 00:24:41.670 I have three real equations there 00:24:41.670 --> 00:24:43.300 even though it looks like four. 00:24:43.300 --> 00:24:44.940 And I get two. 00:24:44.940 --> 00:24:49.460 So that general picture is n-- Oh, I'm 00:24:49.460 --> 00:24:54.070 sorry it's actually m because I'm doing the transpose here. 00:24:54.070 --> 00:25:00.890 So it's m w's minus r, the rank. 00:25:00.890 --> 00:25:03.840 So that's m is five, the rank is three 00:25:03.840 --> 00:25:07.520 and this counts the number of independent solutions. 00:25:07.520 --> 00:25:12.960 So it's a nice, it couldn't be better. 00:25:12.960 --> 00:25:17.520 It's a fundamental count of how many solutions are there. 00:25:17.520 --> 00:25:22.090 You're really taking the number of unknowns, five, 00:25:22.090 --> 00:25:24.490 and you're subtracting the number of equations 00:25:24.490 --> 00:25:27.270 that are really there, three, and that leaves you 00:25:27.270 --> 00:25:30.780 with two solutions which we will have to find in a minute. 00:25:30.780 --> 00:25:35.930 Can you say why there's really only three equations there? 00:25:35.930 --> 00:25:41.910 Why do I say that that fourth equation is not 00:25:41.910 --> 00:25:45.600 contributing anything new? 00:25:45.600 --> 00:25:50.040 I believe that that fourth equation is a consequence 00:25:50.040 --> 00:25:52.400 of the first three. 00:25:52.400 --> 00:25:55.830 And, therefore, if it's there or if it's not there, 00:25:55.830 --> 00:25:59.490 it's not telling me anything new about in=out. 00:25:59.490 --> 00:26:03.060 In other words, if I have a closed system, 00:26:03.060 --> 00:26:05.700 closed because it's zero on the right side, 00:26:05.700 --> 00:26:10.430 if I have a closed system and I have in=out at those three 00:26:10.430 --> 00:26:14.400 nodes, then I'll automatically have in=out at the fourth node, 00:26:14.400 --> 00:26:19.060 because the total in is zero. 00:26:19.060 --> 00:26:20.690 And the total out is zero. 00:26:20.690 --> 00:26:23.850 So if I'm right at three I'll be right at the fourth one. 00:26:23.850 --> 00:26:26.360 And now just tell me with the numbers, 00:26:26.360 --> 00:26:37.050 how would I get this equation w_4+w_5=0 from the first three? 00:26:37.050 --> 00:26:39.450 Well, it's probably the same way that I 00:26:39.450 --> 00:26:41.400 got that column, that that column was 00:26:41.400 --> 00:26:44.920 connected to those columns. 00:26:44.920 --> 00:26:46.940 What do I do? 00:26:46.940 --> 00:26:48.900 Add them. 00:26:48.900 --> 00:26:52.640 If you add that equation to that equation to that equation, 00:26:52.640 --> 00:26:54.930 add those three equations, what happens? 00:26:54.930 --> 00:26:59.320 the w_1's cancel, the w_2's cancel, the w_3's cancel, 00:26:59.320 --> 00:27:02.940 this says there's a minus w_4 and a minus w_5 00:27:02.940 --> 00:27:07.280 that adds to zero plus zero plus zero, minus w_4, 00:27:07.280 --> 00:27:11.850 minus w_5 equalling zero is the same as plus w_4 00:27:11.850 --> 00:27:14.930 plus w_5 equals zero. 00:27:14.930 --> 00:27:18.380 The four equations add to zero equals zero. 00:27:18.380 --> 00:27:21.530 That's what the central thing is. 00:27:21.530 --> 00:27:24.210 The four equations add to zero equals zero just 00:27:24.210 --> 00:27:28.740 the way the four columns up here added to the zero column. 00:27:28.740 --> 00:27:29.730 OK. 00:27:29.730 --> 00:27:32.070 So we got the count. 00:27:32.070 --> 00:27:37.010 Now, this is the interesting part, always. 00:27:37.010 --> 00:27:38.860 What are the solutions? 00:27:38.860 --> 00:27:40.900 What are the actual w's? 00:27:40.900 --> 00:27:43.570 OK. 00:27:43.570 --> 00:27:45.890 You could say, wait a minute, you're 00:27:45.890 --> 00:27:52.210 asking me to solve four equations and five unknowns, 00:27:52.210 --> 00:27:53.710 and just say what the answer is. 00:27:53.710 --> 00:27:57.010 Well, normally that's not reasonable. 00:27:57.010 --> 00:28:02.140 But here we can get help from the graph. 00:28:02.140 --> 00:28:04.570 Let me give you an idea here. 00:28:04.570 --> 00:28:09.380 So what are we looking for on the graph, that solves it? 00:28:09.380 --> 00:28:11.760 We're looking for a bunch of currents 00:28:11.760 --> 00:28:14.860 that balance themselves. 00:28:14.860 --> 00:28:15.360 Right? 00:28:15.360 --> 00:28:17.490 We've got zero on the right hand side. 00:28:17.490 --> 00:28:20.810 So we're not getting any help from outside. 00:28:20.810 --> 00:28:24.200 How could you send current in this loop 00:28:24.200 --> 00:28:27.750 in a way that would satisfy Kirchhoff. 00:28:27.750 --> 00:28:29.650 He'd be happy. 00:28:29.650 --> 00:28:32.270 The balance law would be true. 00:28:32.270 --> 00:28:37.810 OK. b-- I'm just looking at currents. 00:28:37.810 --> 00:28:41.380 Current w_1, w_2, w_3. 00:28:41.380 --> 00:28:44.010 Is there any combination of w_1, w_2, 00:28:44.010 --> 00:28:52.510 w_3 that would balance itself, that would make Kirchhoff OK? 00:28:52.510 --> 00:28:55.820 Well, here's the idea. 00:28:55.820 --> 00:28:58.620 Send that current around a loop. 00:28:58.620 --> 00:29:05.120 Loop currents are solutions to Kirchhoff's balance law. 00:29:05.120 --> 00:29:09.970 If I send an amp on that edge, on that edge 00:29:09.970 --> 00:29:13.640 and backward on that edge, right? 00:29:13.640 --> 00:29:17.830 It's around a loop at every node, it's totally OK. 00:29:17.830 --> 00:29:21.310 So I believe that a particular solution will 00:29:21.310 --> 00:29:29.280 be for these things to be, let's see what did I say? w_1, 00:29:29.280 --> 00:29:34.710 I'll send one around. w_3 will be a one. w_2 was backwards, 00:29:34.710 --> 00:29:37.260 it wasn't traveling on w_4. 00:29:37.260 --> 00:29:39.790 I think that's a solution. 00:29:39.790 --> 00:29:42.180 That loop current gives me a solution so 00:29:42.180 --> 00:29:49.340 let me call this solution, that's the first solution. 00:29:49.340 --> 00:29:51.910 And if we do these multiplications, of course 00:29:51.910 --> 00:29:53.670 it's going to come out right. 00:29:53.670 --> 00:29:58.930 OK, so that's solution number one. 00:29:58.930 --> 00:30:01.270 A w that works. 00:30:01.270 --> 00:30:04.490 OK, with that hint, tell me a second w that works. 00:30:04.490 --> 00:30:08.030 In fact, since there are only two, 00:30:08.030 --> 00:30:12.820 you'll be giving me the rest of the answer. 00:30:12.820 --> 00:30:17.680 So that was a loop current that went around that loop. 00:30:17.680 --> 00:30:20.730 Tell me another one. 00:30:20.730 --> 00:30:25.640 Well, we're a big class but everybody's seen it. 00:30:25.640 --> 00:30:30.620 How about around that loop? 00:30:30.620 --> 00:30:33.540 So that would be another thing and it's 00:30:33.540 --> 00:30:37.620 pretty clearly not the same as this one. 00:30:37.620 --> 00:30:41.390 So I'm really truly finding two independent solutions. 00:30:41.390 --> 00:30:43.670 And what is that second solution? 00:30:43.670 --> 00:30:48.810 Let me maybe just put it in here. 00:30:48.810 --> 00:30:52.860 And they're both giving me [0, 0, 0, 0]. 00:30:52.860 --> 00:30:57.560 And now, what is this number two solution, the loop number two? 00:30:57.560 --> 00:31:00.790 One and two are not involved now. 00:31:00.790 --> 00:31:07.900 Number three, see I'm usually sending it counterclockwise, 00:31:07.900 --> 00:31:09.430 that's the sort of convention. 00:31:09.430 --> 00:31:13.460 But you know, of course you just have 00:31:13.460 --> 00:31:17.270 to follow some convention in connection with the arrows. 00:31:17.270 --> 00:31:20.570 So that would go backwards on three, I think. 00:31:20.570 --> 00:31:24.630 Forwards on four, and backwards on five. 00:31:24.630 --> 00:31:27.770 So that would be solution number two. 00:31:27.770 --> 00:31:29.650 OK. 00:31:29.650 --> 00:31:32.040 Now, tell me what all the solutions are. 00:31:32.040 --> 00:31:36.960 I found two particular solutions, two particular loop 00:31:36.960 --> 00:31:39.970 currents, particularly easy. 00:31:39.970 --> 00:31:45.090 What would be all the solutions to Kirchhoff's current law, 00:31:45.090 --> 00:31:47.800 A transpose w equals zero? 00:31:47.800 --> 00:31:50.980 Every w now since I've found the right number, 00:31:50.980 --> 00:31:57.840 every w will be a combination of those two. 00:31:57.840 --> 00:31:59.900 Ah, well wait a minute. 00:31:59.900 --> 00:32:01.110 Have I got them all? 00:32:01.110 --> 00:32:04.720 I should have thought, what about current 00:32:04.720 --> 00:32:07.660 around the big loop? 00:32:07.660 --> 00:32:10.870 That would certainly satisfy Kirchhoff. 00:32:10.870 --> 00:32:19.560 Plus one, one, so why is this not number three? 00:32:19.560 --> 00:32:22.690 Around the big loop I have a one and then 00:32:22.690 --> 00:32:24.760 a one on the fourth position. 00:32:24.760 --> 00:32:28.420 Backwards on five, backwards on two. 00:32:28.420 --> 00:32:32.310 And so there is number-- So I'll put number three 00:32:32.310 --> 00:32:36.130 with a question mark. 00:32:36.130 --> 00:32:41.670 What's up with that guy? 00:32:41.670 --> 00:32:46.510 You know, unless mathematics has got to close up shop, 00:32:46.510 --> 00:32:51.070 this better be a combination of those. 00:32:51.070 --> 00:32:52.170 And of course, it is. 00:32:52.170 --> 00:32:56.090 If I send something around the top loop and something 00:32:56.090 --> 00:32:58.440 around the second loop and add them together, 00:32:58.440 --> 00:33:00.740 they'll cancel on the middle edge 00:33:00.740 --> 00:33:04.000 there and produce number three. 00:33:04.000 --> 00:33:07.580 So this is probably just the sum. 00:33:07.580 --> 00:33:11.240 If I add that one to that one, I think I get number three. 00:33:11.240 --> 00:33:14.270 So number three is true. 00:33:14.270 --> 00:33:17.510 It's a solution but it's not a new one. 00:33:17.510 --> 00:33:20.590 OK. 00:33:20.590 --> 00:33:25.720 That was simple, right, once we saw that loops gave the answer. 00:33:25.720 --> 00:33:33.330 But, it's, actually it's quite important 00:33:33.330 --> 00:33:35.570 and appears everywhere. 00:33:35.570 --> 00:33:42.400 In fact the theory of electrical networks, 00:33:42.400 --> 00:33:43.880 current laws and so on. 00:33:43.880 --> 00:33:46.060 I mean that used to be a, like, basic course 00:33:46.060 --> 00:33:47.340 in electrical engineering. 00:33:47.340 --> 00:33:52.220 There was a text by Professor Ernst Guillemin, I remember. 00:33:52.220 --> 00:33:56.420 It's sort of not so central to the world any more. 00:33:56.420 --> 00:34:03.260 And now-- Bu the structure is just right somehow. 00:34:03.260 --> 00:34:08.040 And what I wanted to say is you could take, in those days 00:34:08.040 --> 00:34:11.190 you maybe took loop currents as the unknowns. 00:34:11.190 --> 00:34:13.510 You could think of currents in the loops 00:34:13.510 --> 00:34:16.300 as your principal unknowns. 00:34:16.300 --> 00:34:17.620 We don't do that now. 00:34:17.620 --> 00:34:23.400 But, oh, there's, yeah it comes up again. 00:34:23.400 --> 00:34:27.250 Knowing all the solutions to A transpose w equals zero, 00:34:27.250 --> 00:34:31.350 well, you'll see, what's ahead? 00:34:31.350 --> 00:34:36.430 It will be the continuous analog of this, where I have flows, 00:34:36.430 --> 00:34:40.970 not just around a graph, but in a region. 00:34:40.970 --> 00:34:43.510 And Laplace's equation is going to come up, 00:34:43.510 --> 00:34:46.830 and the equations of divergence and gradient, 00:34:46.830 --> 00:34:50.830 all that great stuff is coming in Chapter 3. 00:34:50.830 --> 00:34:59.160 And this is somehow the discrete case. 00:34:59.160 --> 00:35:03.000 These loop currents, that has something to do with the curl. 00:35:03.000 --> 00:35:07.560 And these differences that A takes 00:35:07.560 --> 00:35:09.800 has something to do with gradients. 00:35:09.800 --> 00:35:12.890 And this, Kirchhoff's current law 00:35:12.890 --> 00:35:15.580 has something to do with divergence. 00:35:15.580 --> 00:35:21.010 Can I just say ahead of time, what we're doing is really 00:35:21.010 --> 00:35:27.270 good to see and get it because then you have a way 00:35:27.270 --> 00:35:33.390 to understand vector calculus. 00:35:33.390 --> 00:35:36.410 This is discrete vector calculus we're doing. 00:35:36.410 --> 00:35:37.170 OK. 00:35:37.170 --> 00:35:40.810 Now, it's just right. 00:35:40.810 --> 00:35:45.290 Forgive me for my sermon here. 00:35:45.290 --> 00:35:45.910 Alright. 00:35:45.910 --> 00:35:50.980 Now, may I bring the pieces together finally? 00:35:50.980 --> 00:35:56.930 May I bring the three steps together into, 00:35:56.930 --> 00:36:01.910 well first you would say bring them into one equation. 00:36:01.910 --> 00:36:05.880 Put the three steps, combine the three steps into one. 00:36:05.880 --> 00:36:06.800 OK. 00:36:06.800 --> 00:36:10.760 So, what happens if I do that? 00:36:10.760 --> 00:36:18.250 I take that last step A transpose-- f is A transpose w. 00:36:18.250 --> 00:36:24.880 So now I'm going to get one equation. 00:36:24.880 --> 00:36:26.790 Which is the equation that's going 00:36:26.790 --> 00:36:30.320 to have the stiffness matrix in it, A transpose C A. 00:36:30.320 --> 00:36:35.380 It's the conductance matrix. 00:36:35.380 --> 00:36:38.050 And it's the equation that a big finite element 00:36:38.050 --> 00:36:43.490 code, a circuit simulation code-- It's the matrix 00:36:43.490 --> 00:36:45.870 they have to find and work with. 00:36:45.870 --> 00:36:50.450 And those codes are enormous, and SPICE by the way 00:36:50.450 --> 00:36:54.540 is the sort of grandfather of circuit simulation codes. 00:36:54.540 --> 00:36:57.750 Somebody at Berkeley had the sense to see hey, 00:36:57.750 --> 00:37:00.700 we've got giant circuits now. 00:37:00.700 --> 00:37:03.730 Modern circuits have thousands of elements. 00:37:03.730 --> 00:37:09.160 And you can't do it by eye the way we can do this one by eye. 00:37:09.160 --> 00:37:12.780 You've got to organize it and write a code, 00:37:12.780 --> 00:37:14.660 and SPICE is the start. 00:37:14.660 --> 00:37:18.530 So one way to do it is to end up with one equation. 00:37:18.530 --> 00:37:22.210 So that was f equals A transpose w. 00:37:22.210 --> 00:37:25.550 I'm now going, I'm just assembling the whole loop. 00:37:25.550 --> 00:37:28.500 But w is Ce. 00:37:28.500 --> 00:37:41.400 So that's A transpose Ce, but e is b minus Au. 00:37:41.400 --> 00:37:43.750 Nothing new there. 00:37:43.750 --> 00:37:49.290 Nothing new maybe except that it involves both the f and the b, 00:37:49.290 --> 00:37:52.180 where our earlier examples involved either 00:37:52.180 --> 00:37:56.920 an f, in masses and springs, or a b in the squares. 00:37:56.920 --> 00:37:58.060 Now they're both here. 00:37:58.060 --> 00:38:01.760 So now, that's my equation, f equals this. 00:38:01.760 --> 00:38:06.600 And now let me just move that to the, let me just recollect it, 00:38:06.600 --> 00:38:10.180 that's A transpose C A, the big thing that I wanted to see. 00:38:10.180 --> 00:38:13.220 I'll put it on the left side. 00:38:13.220 --> 00:38:15.560 And what will I have on the right side? 00:38:15.560 --> 00:38:21.740 I'll have A transpose C b. 00:38:21.740 --> 00:38:25.050 And I'll have, f is now coming over to the other side 00:38:25.050 --> 00:38:26.710 with a minus. 00:38:26.710 --> 00:38:28.360 Minus f. 00:38:28.360 --> 00:38:31.840 That's the big equation. 00:38:31.840 --> 00:38:33.840 You could say that's the fundamental equation 00:38:33.840 --> 00:38:35.440 of equilibrium. 00:38:35.440 --> 00:38:38.030 And you see how it's right? 00:38:38.030 --> 00:38:43.230 It involves the A transpose C A, which we expect. 00:38:43.230 --> 00:38:45.960 Over here was the A transpose C b. 00:38:45.960 --> 00:38:51.760 Now, which problem, before networks, 00:38:51.760 --> 00:38:56.130 produced an A transpose b or an A transpose C b? 00:38:56.130 --> 00:38:57.750 That was least squares. 00:38:57.750 --> 00:39:01.540 And now, so that's the least squares, the b term. 00:39:01.540 --> 00:39:03.570 The b is there with a couple of matrices 00:39:03.570 --> 00:39:07.660 because b entered the problem just one step around. 00:39:07.660 --> 00:39:09.930 It had two more steps to go. 00:39:09.930 --> 00:39:13.160 It had a C step and then an A transpose step. 00:39:13.160 --> 00:39:16.040 Whereas f up here is at the very end and now 00:39:16.040 --> 00:39:19.310 it appears with a minus sign. 00:39:19.310 --> 00:39:22.810 That's different from springs and masses 00:39:22.810 --> 00:39:26.510 simply because the sign conventions, you could say. 00:39:26.510 --> 00:39:28.480 OK, there is the equation. 00:39:28.480 --> 00:39:30.890 OK, fine. 00:39:30.890 --> 00:39:34.090 So that's what you have to solve. 00:39:34.090 --> 00:39:36.760 And actually I think of that as the fundamental problem 00:39:36.760 --> 00:39:38.310 of numerical analysis. 00:39:38.310 --> 00:39:40.170 How to solve that equation. 00:39:40.170 --> 00:39:44.010 More effort, more thinking goes into that than probably 00:39:44.010 --> 00:39:46.560 any other single problem. 00:39:46.560 --> 00:39:48.150 In some form. 00:39:48.150 --> 00:39:52.030 OK, and here's some part of that thinking. 00:39:52.030 --> 00:39:55.100 Part of that thinking, and another important possibility, 00:39:55.100 --> 00:40:00.840 is to keep-- This was like the one equation, the one field 00:40:00.840 --> 00:40:06.470 problem, this corresponds to the displacement method. 00:40:06.470 --> 00:40:11.940 Can I use words that I'm not going to use seriously 00:40:11.940 --> 00:40:17.120 for another few weeks? 00:40:17.120 --> 00:40:20.030 This would correspond to the displacement method 00:40:20.030 --> 00:40:22.640 in finite elements. 00:40:22.640 --> 00:40:25.710 In FEM, FEM for finite element method. 00:40:25.710 --> 00:40:27.880 That's the displacement method, it's the method 00:40:27.880 --> 00:40:30.800 that that most people use. 00:40:30.800 --> 00:40:33.040 It's the standard method. 00:40:33.040 --> 00:40:33.910 OK. 00:40:33.910 --> 00:40:36.410 But it's not the only possibility, 00:40:36.410 --> 00:40:40.860 and let me show you a second one that involves two equations. 00:40:40.860 --> 00:40:44.380 Because that's also very important, 00:40:44.380 --> 00:40:47.860 with many other applications, that we will see but 00:40:47.860 --> 00:40:49.110 haven't seen yet. 00:40:49.110 --> 00:40:54.320 So my two equations, I really should say two systems, 00:40:54.320 --> 00:40:58.330 because one equation, that's a vector equation of course. 00:40:58.330 --> 00:41:01.480 So I have a system of two vector equations, that 00:41:01.480 --> 00:41:04.820 would go into a block matrix and you'll see it. 00:41:04.820 --> 00:41:11.170 OK, so what what two unknowns am I going to keep? 00:41:11.170 --> 00:41:12.880 u, I'm keeping. 00:41:12.880 --> 00:41:14.530 Displacement, I'm keeping. 00:41:14.530 --> 00:41:16.770 But I'm also going to keep what I 00:41:16.770 --> 00:41:20.120 think of as the other important unknown, w. 00:41:20.120 --> 00:41:22.500 So the other important unknown is w. 00:41:22.500 --> 00:41:25.810 So now I have two equations and one of them 00:41:25.810 --> 00:41:28.700 is just that, is just the current, 00:41:28.700 --> 00:41:32.110 is just the current law, the balance law A transpose w equal 00:41:32.110 --> 00:41:35.610 f. 00:41:35.610 --> 00:41:39.480 The only guy I'm eliminating is e. 00:41:39.480 --> 00:41:45.910 Initially you could say I've a three field system. u, e and w. 00:41:45.910 --> 00:41:50.090 Now, e and w are so easily connected 00:41:50.090 --> 00:41:53.850 that I'm going to eliminate e. e is c inverse w. 00:41:53.850 --> 00:41:56.240 I did it actually here. 00:41:56.240 --> 00:42:00.610 This w is C(b-Au), that's we know. 00:42:00.610 --> 00:42:06.680 If I multiply by the C inverse, then I have C inverse w is, 00:42:06.680 --> 00:42:11.900 and I bring the Au over to the far left and I have only the b 00:42:11.900 --> 00:42:12.670 left. 00:42:12.670 --> 00:42:19.200 Everybody saw that, I did a C inverse there 00:42:19.200 --> 00:42:26.330 to get e off by itself and then I substituted for e, 00:42:26.330 --> 00:42:29.420 I put in the u part so e's now gone, 00:42:29.420 --> 00:42:40.540 and the equation is C inverse w plus Au equals b. 00:42:40.540 --> 00:42:46.090 That's my two field system. 00:42:46.090 --> 00:42:50.920 Now, there's a matrix here. 00:42:50.920 --> 00:42:53.187 This is really nice. 00:42:53.187 --> 00:42:54.270 I just want to write that. 00:42:54.270 --> 00:42:57.520 I've got to what I want but now I want to look at it. 00:42:57.520 --> 00:43:04.190 So I think of a two by two block matrix multiplying [w, u] 00:43:04.190 --> 00:43:07.590 and giving me [b, f]. 00:43:07.590 --> 00:43:10.460 And I want to ask you about that matrix. 00:43:10.460 --> 00:43:17.290 So this is the matrix for when I've only eliminated e 00:43:17.290 --> 00:43:21.340 and I've still got w as well as u. 00:43:21.340 --> 00:43:26.850 OK, you can read off what's in that matrix. 00:43:26.850 --> 00:43:28.870 What goes up here? 00:43:28.870 --> 00:43:30.650 C inverse, of course. 00:43:30.650 --> 00:43:33.210 Positive diagonal, usually. 00:43:33.210 --> 00:43:34.060 Easy. 00:43:34.060 --> 00:43:37.980 What goes here is a rectangular, that guy is rectangular. 00:43:37.980 --> 00:43:41.830 A transpose is multiplying w, so it goes down here, 00:43:41.830 --> 00:43:47.750 and this equation has no u in it. 00:43:47.750 --> 00:43:55.190 That matrix is worth noticing. 00:43:55.190 --> 00:43:58.410 And let's spend the rest of this, the remaining minutes, 00:43:58.410 --> 00:44:00.320 just to think about that matrix. 00:44:00.320 --> 00:44:05.930 I just want to say, what if I keep w and u, 00:44:05.930 --> 00:44:08.710 this is an important possibility. 00:44:08.710 --> 00:44:12.820 And it's important in finite elements which as you know 00:44:12.820 --> 00:44:17.010 is just a terrific way to solve a whole lot of continuum 00:44:17.010 --> 00:44:18.100 problems. 00:44:18.100 --> 00:44:23.470 And what's it called when I have w and u together, 00:44:23.470 --> 00:44:27.040 both unknowns, not eliminating w now, 00:44:27.040 --> 00:44:29.020 it's called the mixed method. 00:44:29.020 --> 00:44:36.580 So this corresponds to the mixed method in finite elements. 00:44:36.580 --> 00:44:45.420 It corresponds to the possibility of keeping w and u. 00:44:45.420 --> 00:44:47.490 Well, and you might say, wait, isn't there 00:44:47.490 --> 00:44:50.690 a third possibility? 00:44:50.690 --> 00:44:52.670 And what would that be? 00:44:52.670 --> 00:44:56.430 Keep only w. 00:44:56.430 --> 00:44:59.670 Here we kept only u, here we've got them both, 00:44:59.670 --> 00:45:07.570 this is kind of the mother of all equilibrium equations. 00:45:07.570 --> 00:45:10.570 And another possibly would be to keep 00:45:10.570 --> 00:45:14.670 only the w's, to make the currents the primary unknowns. 00:45:14.670 --> 00:45:19.620 And that, in the finite element structural context, 00:45:19.620 --> 00:45:23.590 that would be saying make the stresses. 00:45:23.590 --> 00:45:25.885 So of course it'd be called the stress method. 00:45:25.885 --> 00:45:27.260 It'd be called the stress method, 00:45:27.260 --> 00:45:31.660 and Professor Pian in Course 16, now retired, 00:45:31.660 --> 00:45:35.000 was one of the major developers of the stress method. 00:45:35.000 --> 00:45:36.960 The difficulty with the stress method, 00:45:36.960 --> 00:45:41.330 the reason it didn't win big time, 00:45:41.330 --> 00:45:46.375 is that the w's, if you make them the unknowns you've got 00:45:46.375 --> 00:45:49.970 a constraint on them, Kirchhoff's-- Not all w's are 00:45:49.970 --> 00:45:52.230 allowed. 00:45:52.230 --> 00:45:55.510 Somehow over here all u's are allowed, 00:45:55.510 --> 00:46:00.830 and that made it much easier to set up the problem. 00:46:00.830 --> 00:46:05.320 So the displacement method is the 95 percent winner. 00:46:05.320 --> 00:46:10.400 But there are problems where maybe C inverse is complicated, 00:46:10.400 --> 00:46:12.530 or C is too complicated and you're 00:46:12.530 --> 00:46:18.540 better to-- We can see that. 00:46:18.540 --> 00:46:21.470 That's later in the book, but we want 00:46:21.470 --> 00:46:24.400 to see now about that matrix. 00:46:24.400 --> 00:46:27.970 Well, if I wrote that matrix down, 00:46:27.970 --> 00:46:32.080 and let me write just so you-- I want to ask you 00:46:32.080 --> 00:46:34.070 about that block matrix. 00:46:34.070 --> 00:46:36.840 What's its size? 00:46:36.840 --> 00:46:39.260 Now just focus entirely on that block matrix, 00:46:39.260 --> 00:46:41.460 because that's what I care about. 00:46:41.460 --> 00:46:46.300 What's the size of that matrix? 00:46:46.300 --> 00:46:50.790 Let's see, what's the size of C? m by m. 00:46:50.790 --> 00:46:53.740 What's the size of A? n by n. 00:46:53.740 --> 00:46:54.840 So what do I have here? 00:46:54.840 --> 00:46:58.470 I've got n rows and m plus n columns, 00:46:58.470 --> 00:46:59.680 and there's n more rows. 00:46:59.680 --> 00:47:01.310 It's of size m+n. 00:47:05.880 --> 00:47:08.760 It's got the n u's and the m w's. 00:47:08.760 --> 00:47:10.910 Of course, m+n. 00:47:10.910 --> 00:47:12.890 And the natural size, right. 00:47:12.890 --> 00:47:15.500 So it's got more unknowns but we'll 00:47:15.500 --> 00:47:17.880 see, oh in optimization you bring 00:47:17.880 --> 00:47:22.050 in Lagrange multipliers, that's just exactly parallel to what 00:47:22.050 --> 00:47:23.070 we're doing here. 00:47:23.070 --> 00:47:26.080 You have more, you have extra bunch of unknowns. 00:47:26.080 --> 00:47:27.460 That's what we have. 00:47:27.460 --> 00:47:32.190 Now what else about that matrix? 00:47:32.190 --> 00:47:35.880 I was going to write down a very, very tiny model 00:47:35.880 --> 00:47:37.800 for that matrix. 00:47:37.800 --> 00:47:39.220 I'll just make it two by two. 00:47:39.220 --> 00:47:44.830 Here's a model for that matrix where C is just a one 00:47:44.830 --> 00:47:48.560 and A is just a one. 00:47:48.560 --> 00:47:51.000 I mean, it's kind of laughable, right? 00:47:51.000 --> 00:47:53.190 That model, this is the real thing. 00:47:53.190 --> 00:48:00.330 But it gives you an example to check against. 00:48:00.330 --> 00:48:04.250 OK, what's a property of that matrix? 00:48:04.250 --> 00:48:06.070 It's, again? 00:48:06.070 --> 00:48:06.880 Symmetric. 00:48:06.880 --> 00:48:07.610 Good. 00:48:07.610 --> 00:48:09.820 That's a symmetric matrix. 00:48:09.820 --> 00:48:13.360 Because what happens if I transpose that block matrix? 00:48:13.360 --> 00:48:18.700 That A block will flip over here as A transpose, 00:48:18.700 --> 00:48:21.310 the A transpose block will flip up there 00:48:21.310 --> 00:48:26.050 as A, what happens to the C inverse block? 00:48:26.050 --> 00:48:28.520 C is a symmetric guy. 00:48:28.520 --> 00:48:32.900 In fact, it was just diagonal in our imagination. 00:48:32.900 --> 00:48:37.110 The key point is it's symmetric, its inverse is symmetric, 00:48:37.110 --> 00:48:39.320 its transpose is the same. 00:48:39.320 --> 00:48:42.200 So that's a symmetric matrix. 00:48:42.200 --> 00:48:43.450 That's a good thing, right? 00:48:43.450 --> 00:48:49.270 Now we've got a matrix that's symmetric, square symmetric. 00:48:49.270 --> 00:48:53.490 OK, what's my other question about that matrix? 00:48:53.490 --> 00:49:00.570 Is it or is it not positive definite, right? 00:49:00.570 --> 00:49:03.510 We've got to answer that question. 00:49:03.510 --> 00:49:06.040 Have we got a positive definite matrix? 00:49:06.040 --> 00:49:08.830 Would all the pivots be positive? 00:49:08.830 --> 00:49:11.990 Would the eigenvalues be positive? 00:49:11.990 --> 00:49:15.970 What's your guess? 00:49:15.970 --> 00:49:16.790 No. 00:49:16.790 --> 00:49:19.160 That matrix is not positive. 00:49:19.160 --> 00:49:22.440 No way that a matrix with a zero there, 00:49:22.440 --> 00:49:25.740 a zero block, or that matrix with a zero number 00:49:25.740 --> 00:49:27.060 could be positive definite. 00:49:27.060 --> 00:49:28.210 No, no way. 00:49:28.210 --> 00:49:39.770 The energy in this guy, this u transpose Au, you remember, 00:49:39.770 --> 00:49:46.280 would be u_1 squared and u_1*u_2, twice. 00:49:46.280 --> 00:49:50.040 And no u_2 squareds. 00:49:50.040 --> 00:49:55.300 And that thing is definitely indefinite. 00:49:55.300 --> 00:49:59.050 Right? 00:49:59.050 --> 00:50:01.840 In the u_1 direction it looks good, 00:50:01.840 --> 00:50:03.800 that's things positive there. 00:50:03.800 --> 00:50:08.070 But if I took u_1 and u_2 to have opposite signs, 00:50:08.070 --> 00:50:11.470 and made u_2 big enough then of course 00:50:11.470 --> 00:50:12.650 this just brings it down. 00:50:12.650 --> 00:50:16.030 So the graph of that would be a saddle point. 00:50:16.030 --> 00:50:18.170 The graph of that would be a saddle. 00:50:18.170 --> 00:50:26.040 OK, and now here I have the same thing on an m+n size. 00:50:26.040 --> 00:50:26.920 So what do I have? 00:50:26.920 --> 00:50:28.890 Actually, you could see. 00:50:28.890 --> 00:50:34.120 The last exercise is mentally do elimination on that matrix. 00:50:34.120 --> 00:50:37.160 Mentally do elimination on that matrix. 00:50:37.160 --> 00:50:42.910 So start with the first m rows. 00:50:42.910 --> 00:50:45.000 We'll work with those first. 00:50:45.000 --> 00:50:48.090 What will elimination do, what will the pivots be like, 00:50:48.090 --> 00:50:51.460 what will happen at the beginning of elimination? 00:50:51.460 --> 00:50:55.790 When I start with this matrix? 00:50:55.790 --> 00:50:59.430 Well it meets C inverse right away, that diagonal matrix, 00:50:59.430 --> 00:51:00.850 and it's extremely happy. 00:51:00.850 --> 00:51:02.300 Those will be the pivots, right? 00:51:02.300 --> 00:51:03.740 They're sitting on the diagonals, 00:51:03.740 --> 00:51:05.510 zero off the diagonals. 00:51:05.510 --> 00:51:09.460 They'll be positive pivots, I'll have m positive pivots here. 00:51:09.460 --> 00:51:15.740 And then I get down to where A comes in the picture. 00:51:15.740 --> 00:51:20.800 So on the last board here, let me just copy this matrix, 00:51:20.800 --> 00:51:25.320 [C inverse, A; A transpose, 0]. 00:51:25.320 --> 00:51:29.210 An elimination is going to, it'll be very happy with that. 00:51:29.210 --> 00:51:35.570 But it's going to put, so it's happy with that row. 00:51:35.570 --> 00:51:36.660 Block row. 00:51:36.660 --> 00:51:41.130 It's going to do an elimination to get a bunch of zeroes there 00:51:41.130 --> 00:51:45.740 and what did it do? 00:51:45.740 --> 00:51:48.310 This was elimination, this was subtracting, 00:51:48.310 --> 00:51:50.700 yeah what did it subtract here? 00:51:50.700 --> 00:51:55.080 It multiplied these pivot rows by something 00:51:55.080 --> 00:52:01.270 and subtracted from these lower rows and got the zero block. 00:52:01.270 --> 00:52:04.130 And what did it multiply by? 00:52:04.130 --> 00:52:07.480 What do I multiply that block row, and this 00:52:07.480 --> 00:52:11.730 is a perfect, perfect exercise to see how blocks are just 00:52:11.730 --> 00:52:12.880 like numbers. 00:52:12.880 --> 00:52:14.110 You can deal with them. 00:52:14.110 --> 00:52:17.230 What do I multiply that block row by 00:52:17.230 --> 00:52:22.880 and subtract from the row below to get a zero. 00:52:22.880 --> 00:52:27.090 You said C A transpose, but I don't think that's it. 00:52:27.090 --> 00:52:31.100 A transpose C. You've got to multiply by A transpose C. 00:52:31.100 --> 00:52:34.690 First of all, C A transpose wouldn't be a possibility. 00:52:34.690 --> 00:52:35.970 Wrong shapes. 00:52:35.970 --> 00:52:42.210 A transpose C is the four by five, five by five guy. 00:52:42.210 --> 00:52:45.470 So you multiply A transpose C, that cancels that, 00:52:45.470 --> 00:52:48.720 leaves the A transpose, when you subtract it gives you a zero, 00:52:48.720 --> 00:52:51.100 and what does it give you there? 00:52:51.100 --> 00:52:55.310 What shows up there? 00:52:55.310 --> 00:53:00.710 A transpose C multiplies at A, subtracts 00:53:00.710 --> 00:53:04.780 so it's actually what shows up there is minus A transpose C 00:53:04.780 --> 00:53:08.200 A. Let me write it in there. 00:53:08.200 --> 00:53:11.980 Minus A transpose C A. So that matrix 00:53:11.980 --> 00:53:13.880 is exactly what comes from this one. 00:53:13.880 --> 00:53:16.870 It's exactly what we do when we eliminate w. 00:53:16.870 --> 00:53:18.250 That's what elimination is. 00:53:18.250 --> 00:53:21.400 I just eliminated w by getting a zero there. 00:53:21.400 --> 00:53:25.500 And I got only an equation, but notice the minus. 00:53:25.500 --> 00:53:32.030 So, final question, what are the signs of the last n pivots? 00:53:32.030 --> 00:53:35.370 The first m were all positive, and they were sitting 00:53:35.370 --> 00:53:38.120 on the diagonal already. 00:53:38.120 --> 00:53:42.430 The last n are not so easy to see, 00:53:42.430 --> 00:53:44.860 but we can see what sign they have. 00:53:44.860 --> 00:53:48.390 And what sign do the last n pivots have? 00:53:48.390 --> 00:53:50.090 Minus. 00:53:50.090 --> 00:53:53.160 Because they come from a negative definite. 00:53:53.160 --> 00:53:55.850 Minus A transpose C A is shown up there. 00:53:55.850 --> 00:53:58.630 So that's the saddle point. 00:53:58.630 --> 00:54:03.230 Saddle points are, when you have two-field problems, 00:54:03.230 --> 00:54:05.680 you're talking about saddle points, 00:54:05.680 --> 00:54:09.590 and the mixed method in finite elements is exactly that. 00:54:09.590 --> 00:54:17.420 And the tricky part is then with the mixed method, 00:54:17.420 --> 00:54:19.800 you're sort of not so perfectly guaranteed 00:54:19.800 --> 00:54:21.980 that the matrix is invertible. 00:54:21.980 --> 00:54:26.210 Because we have plus stuff and minus stuff. 00:54:26.210 --> 00:54:28.620 OK, thank you, that's great. 00:54:28.620 --> 00:54:33.830 And I'll see you Monday all about the exam and review, 00:54:33.830 --> 00:54:36.310 it's a great chance to think back.