WEBVTT 00:00:00.090 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:04.030 Commons license. 00:00:04.030 --> 00:00:06.330 Your support will help MIT OpenCourseWare 00:00:06.330 --> 00:00:10.690 continue to offer high-quality educational resources for free. 00:00:10.690 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:16.920 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.920 --> 00:00:18.220 at ocw.mit.edu. 00:00:31.136 --> 00:00:32.030 HERBERT GROSS: Hi. 00:00:32.030 --> 00:00:33.447 Well, I guess I really should have 00:00:33.447 --> 00:00:37.130 said "goodbye," because this is the last lecture 00:00:37.130 --> 00:00:38.330 in our course-- 00:00:38.330 --> 00:00:42.030 not the last assignment, but the last lecture. 00:00:42.030 --> 00:00:45.720 The reason I said "hi" was, why quit after all this time 00:00:45.720 --> 00:00:46.880 with saying that? 00:00:46.880 --> 00:00:49.820 And we've reached the stage now where 00:00:49.820 --> 00:00:52.820 we should clean up the vector spaces 00:00:52.820 --> 00:00:55.610 to the best of our ability and recognize 00:00:55.610 --> 00:00:58.820 that, from this point on, much of the treatment of vector 00:00:58.820 --> 00:01:02.360 spaces requires specialized concentration. 00:01:02.360 --> 00:01:04.519 In fact, I envy the real people who 00:01:04.519 --> 00:01:06.140 make regular movies where they have 00:01:06.140 --> 00:01:08.060 stuntmen when things get tough. 00:01:08.060 --> 00:01:09.860 I would continue on with this course, 00:01:09.860 --> 00:01:11.870 except that I don't have a stuntman 00:01:11.870 --> 00:01:14.270 to do these hard lectures for me. 00:01:14.270 --> 00:01:17.467 And also, the particular topic, as I told you last time, 00:01:17.467 --> 00:01:19.550 that I have in mind for today-- the subject called 00:01:19.550 --> 00:01:21.020 eigenvectors-- 00:01:21.020 --> 00:01:23.400 has two approaches to it. 00:01:23.400 --> 00:01:28.070 One is that it does have some very elaborate practical 00:01:28.070 --> 00:01:30.620 applications, many of which occur 00:01:30.620 --> 00:01:32.600 in more advanced subjects. 00:01:32.600 --> 00:01:35.270 It also has a very nice framework 00:01:35.270 --> 00:01:38.570 within the game of mathematics idea. 00:01:38.570 --> 00:01:42.350 And my own feeling was that, since we started this course 00:01:42.350 --> 00:01:45.350 with the concept of the game of mathematics, 00:01:45.350 --> 00:01:47.420 mathematical structures, I thought 00:01:47.420 --> 00:01:50.210 that, rather than go into complicated applications, 00:01:50.210 --> 00:01:53.240 I would treat eigenvectors in terms 00:01:53.240 --> 00:01:56.880 of the structure of mathematics as a game. 00:01:56.880 --> 00:01:58.910 In fact, as an interesting aside, 00:01:58.910 --> 00:02:00.890 I'd like to share with you a very famous story 00:02:00.890 --> 00:02:05.300 in mathematics, that, when the first book on matrix algebra 00:02:05.300 --> 00:02:09.949 was written by Hamilton, he inscribed the book with, 00:02:09.949 --> 00:02:12.890 "Here at last is a branch of mathematics 00:02:12.890 --> 00:02:16.190 for which there will never be found practical application." 00:02:16.190 --> 00:02:19.760 He did not invent the subject to solve difficult physics 00:02:19.760 --> 00:02:20.510 problems. 00:02:20.510 --> 00:02:22.940 He invented the subject because it was 00:02:22.940 --> 00:02:25.280 an elegant mathematical device. 00:02:25.280 --> 00:02:27.900 And so I thought that maybe, for our last lecture, 00:02:27.900 --> 00:02:29.960 we should end on that vein. 00:02:29.960 --> 00:02:34.760 At any rate, the subject for today is called eigenvectors. 00:02:34.760 --> 00:02:39.600 And from a purely game point of view, the idea is this. 00:02:39.600 --> 00:02:42.650 Let's suppose that we have a vector space V 00:02:42.650 --> 00:02:45.710 and a linear mapping, a linear transformation, f, 00:02:45.710 --> 00:02:48.690 that maps V onto itself, say. 00:02:48.690 --> 00:02:52.430 And the question is, are there any vectors in V other 00:02:52.430 --> 00:02:55.510 than the zero vector, such that f of v 00:02:55.510 --> 00:02:57.620 is some scalar multiple of v? 00:02:57.620 --> 00:03:00.440 You see, the reason I exclude the zero vector, first of all, 00:03:00.440 --> 00:03:03.230 is that we already know that, for a linear transformation, 00:03:03.230 --> 00:03:05.720 f of 0 is 0. 00:03:05.720 --> 00:03:08.810 And c times 0 is 0 for all c. 00:03:08.810 --> 00:03:13.250 So this would be trivially true if v were the zero vector. 00:03:13.250 --> 00:03:15.530 So what we're really interested in-- 00:03:15.530 --> 00:03:18.290 given a particular linear transformation, 00:03:18.290 --> 00:03:20.480 are there any vectors such that, relative 00:03:20.480 --> 00:03:23.960 to that linear transformation, the mapping of that vector 00:03:23.960 --> 00:03:27.420 is just a scalar multiple of that vector itself? 00:03:27.420 --> 00:03:30.830 In other words, does f preserve the direction 00:03:30.830 --> 00:03:33.320 of any vectors in V? 00:03:33.320 --> 00:03:36.650 By the way, don't confuse this with conformal mapping 00:03:36.650 --> 00:03:39.770 that we talked about in complex variables. 00:03:39.770 --> 00:03:43.700 In conformal mapping, we didn't preserve any-- in general, 00:03:43.700 --> 00:03:46.370 we did not preserve directions of lines. 00:03:46.370 --> 00:03:47.630 We preserved angles. 00:03:47.630 --> 00:03:49.670 In other words, an angle might have 00:03:49.670 --> 00:03:52.160 been rotated so that the direction of the two sides 00:03:52.160 --> 00:03:53.180 may have changed. 00:03:53.180 --> 00:03:54.770 It was the angle that was preserved 00:03:54.770 --> 00:03:56.240 in the conformal mapping. 00:03:56.240 --> 00:03:59.210 What we're asking now is, given a linear transformation, 00:03:59.210 --> 00:04:01.370 does it preserve any directions? 00:04:01.370 --> 00:04:04.230 And to illustrate this in terms of an example, 00:04:04.230 --> 00:04:06.140 let's suppose we think of mapping 00:04:06.140 --> 00:04:10.190 the xy-plane into the uv-plane under the linear mapping f 00:04:10.190 --> 00:04:17.839 bar, where f bar maps x, y into the 2-tuple x plus 4y comma 00:04:17.839 --> 00:04:19.130 x plus y. 00:04:19.130 --> 00:04:22.700 In other words, in terms of mapping the xy-plane 00:04:22.700 --> 00:04:25.640 into the uv-plane, this is the mapping-- 00:04:25.640 --> 00:04:29.150 u equals x plus 4y, v equals x plus y. 00:04:29.150 --> 00:04:31.190 It's understood here, when I'm referring 00:04:31.190 --> 00:04:35.890 to the typical xy-plane, that my basis vectors are i and j. 00:04:35.890 --> 00:04:39.370 Notice, by the way, that if I look at the vector i, 00:04:39.370 --> 00:04:42.380 i is the 2-tuple 1 comma 0. 00:04:42.380 --> 00:04:48.170 Notice that when x is 1 and y is 0, u is 1 and v is one. 00:04:48.170 --> 00:04:52.850 So f bar maps 1 comma 0 into 1 comma 1. 00:04:52.850 --> 00:04:55.280 Again, in terms of i and j vectors, 00:04:55.280 --> 00:04:58.820 f bar maps i into i plus j. 00:04:58.820 --> 00:05:02.930 And i plus j is certainly not a scalar multiple of i. 00:05:02.930 --> 00:05:05.990 Similarly, what does f bar do to j? 00:05:05.990 --> 00:05:10.190 j, relative to the basis i and j, is written as 0 comma 1. 00:05:10.190 --> 00:05:15.890 When x is 0 and y is 1, we obtain that u is 4 and v is 1. 00:05:15.890 --> 00:05:20.720 So under f bar, 0 comma 1 is mapped into 4 comma 1. 00:05:20.720 --> 00:05:23.540 Or in the language of i and j components, f 00:05:23.540 --> 00:05:27.440 bar maps j into 4i plus j. 00:05:27.440 --> 00:05:31.550 And that certainly is not a scalar multiple of j. 00:05:31.550 --> 00:05:36.050 In other words, 4i plus j is not parallel to j. 00:05:36.050 --> 00:05:39.090 On the other hand, let me pull this one out of the hat. 00:05:39.090 --> 00:05:41.510 Let's take the vector 2i plus j-- 00:05:41.510 --> 00:05:44.930 in other words, the 2-tuple 2 comma 1. 00:05:44.930 --> 00:05:50.200 When x is 2 and y is 1, we see that x plus 4y is 6 00:05:50.200 --> 00:05:52.220 and x plus y is 3. 00:05:52.220 --> 00:05:57.170 So f bar maps 2 comma 1 into 6 comma 3. 00:05:57.170 --> 00:06:00.290 And that certainly is 3 times 2 comma 00:06:00.290 --> 00:06:02.390 1-- see, by our rule of scalar multiplication. 00:06:02.390 --> 00:06:06.410 In other words, what this says is that the vector 2i plus j 00:06:06.410 --> 00:06:11.240 is mapped into the vector which has the same sense 00:06:11.240 --> 00:06:17.480 and direction as 2i plus j, but is 3 times as long. 00:06:17.480 --> 00:06:21.140 So you see, sometimes the linear transformation 00:06:21.140 --> 00:06:24.020 will map a vector into a scalar multiple of itself. 00:06:24.020 --> 00:06:25.340 Sometimes it won't. 00:06:25.340 --> 00:06:26.750 Sometimes there'll be no vectors, 00:06:26.750 --> 00:06:30.680 other than the zero vector, that they're mapped into-- 00:06:30.680 --> 00:06:34.150 scalar multiples of themselves, and things of this type. 00:06:34.150 --> 00:06:35.630 But that's not important right now. 00:06:35.630 --> 00:06:38.300 In terms of a game, what we're trying to do is what? 00:06:38.300 --> 00:06:40.430 Solve the equation [INAUDIBLE]---- well, 00:06:40.430 --> 00:06:42.140 let's give it in terms of a definition. 00:06:42.140 --> 00:06:45.470 First of all, if we have a vector space V, 00:06:45.470 --> 00:06:51.350 and f is a linear transformation mapping V into itself, 00:06:51.350 --> 00:06:55.760 if little v is any non-zero element of the vector space V, 00:06:55.760 --> 00:07:00.590 and if f of v equals c times v for some scalar-- 00:07:00.590 --> 00:07:02.150 for some number c-- 00:07:02.150 --> 00:07:08.750 then v is called an eigenvector and c is called an eigenvalue. 00:07:08.750 --> 00:07:12.320 In other words, if a vector has its direction preserved, 00:07:12.320 --> 00:07:14.150 geometrically speaking, all we're saying 00:07:14.150 --> 00:07:16.610 is that if the direction doesn't change, 00:07:16.610 --> 00:07:18.980 the vector is called an eigenvector. 00:07:18.980 --> 00:07:20.132 And the scaling factor-- 00:07:20.132 --> 00:07:20.840 which means what? 00:07:20.840 --> 00:07:23.210 Even though the direction doesn't change, 00:07:23.210 --> 00:07:25.880 the image may have a different magnitude, 00:07:25.880 --> 00:07:28.820 because the scalar c here doesn't have to be 1. 00:07:28.820 --> 00:07:31.820 That scalar is called an eigenvalue. 00:07:31.820 --> 00:07:35.240 And I'll give you more on this in the exercises, 00:07:35.240 --> 00:07:37.130 and perhaps even in supplementary notes 00:07:37.130 --> 00:07:39.380 if the exercises seem to get too sticky. 00:07:39.380 --> 00:07:41.300 But we'll see how things work out. 00:07:41.300 --> 00:07:43.610 For the time being, all I care about 00:07:43.610 --> 00:07:46.130 is that you understand what an eigenvector means 00:07:46.130 --> 00:07:48.260 and what an eigenvalue is. 00:07:48.260 --> 00:07:52.910 Quickly summarized, if f maps a vector space into itself, 00:07:52.910 --> 00:07:56.060 an eigenvector is any non-zero vector 00:07:56.060 --> 00:08:00.140 which has its direction preserved under the mapping f-- 00:08:00.140 --> 00:08:04.130 that f maps it into a scalar multiple of itself. 00:08:04.130 --> 00:08:07.910 There is a matrix approach for finding eigenvectors, 00:08:07.910 --> 00:08:09.980 and the matrix approach also gives us 00:08:09.980 --> 00:08:13.040 a very nice review of many of the techniques that we've 00:08:13.040 --> 00:08:15.350 used previously in our course. 00:08:15.350 --> 00:08:17.330 For the sake of argument, let's suppose 00:08:17.330 --> 00:08:19.730 that V is an n-dimensional vector space, 00:08:19.730 --> 00:08:23.660 and that we've again chosen a particular basis, u1 up to un, 00:08:23.660 --> 00:08:26.810 to represent V. Suppose, also, that f 00:08:26.810 --> 00:08:30.470 is a linear mapping carrying V into V. 00:08:30.470 --> 00:08:33.049 Remember how we used the matrix approach here? 00:08:33.049 --> 00:08:35.030 What we said was, look at. 00:08:35.030 --> 00:08:39.530 The vectors u1 up to un are carried into f of u1 00:08:39.530 --> 00:08:41.179 up to f of un. 00:08:41.179 --> 00:08:43.549 And that determines the linear transformation 00:08:43.549 --> 00:08:48.170 f because of the fact of the linearity properties. 00:08:48.170 --> 00:08:53.240 In other words, when you take f of a1 u1 plus a2 u2, 00:08:53.240 --> 00:08:56.840 it's just a1 f of u1 plus a2 f of u2. 00:08:56.840 --> 00:08:59.720 So once you know what happens to the basis vectors, 00:08:59.720 --> 00:09:02.150 you know what happens to everything. 00:09:02.150 --> 00:09:06.870 But since we're expressing V in terms of the basis u1 up to un, 00:09:06.870 --> 00:09:10.340 that means that f of u1, et cetera, f of un 00:09:10.340 --> 00:09:14.220 may all be expressed as linear combinations of u1 up to un. 00:09:14.220 --> 00:09:18.540 And that's precisely what I've written over here. 00:09:18.540 --> 00:09:20.420 We also know that the vector v that we're 00:09:20.420 --> 00:09:21.920 trying to find-- see, remember we're 00:09:21.920 --> 00:09:23.540 trying to find eigenvectors. 00:09:23.540 --> 00:09:27.476 The vector v, relative to the basis u1 up to un, 00:09:27.476 --> 00:09:31.723 can be written as the n-tuple x1 up to xn. 00:09:31.723 --> 00:09:33.890 And now I won't bother writing this, because I hope, 00:09:33.890 --> 00:09:35.570 by this time, you understand this. 00:09:35.570 --> 00:09:37.430 This is an abbreviation for saying what? 00:09:37.430 --> 00:09:41.930 The vector v is x1 u1 plus et cetera xn un, 00:09:41.930 --> 00:09:45.590 because I am always referring to the specific basis 00:09:45.590 --> 00:09:49.670 when I write n-tuples without any other qualifications. 00:09:49.670 --> 00:09:56.930 Now, the question was, how did the statement f of v equal cv 00:09:56.930 --> 00:10:00.260 translate in the language of matrices? 00:10:00.260 --> 00:10:02.870 Remember that we took this particular matrix 00:10:02.870 --> 00:10:05.420 of coefficients and wrote-- 00:10:05.420 --> 00:10:07.490 well, we transposed it. 00:10:07.490 --> 00:10:08.540 Remember what we said? 00:10:08.540 --> 00:10:10.610 We said that the matrix A would be 00:10:10.610 --> 00:10:14.630 the matrix whose first column would be the components of f 00:10:14.630 --> 00:10:19.640 of u1 and whose nth column would be the components of f of un. 00:10:19.640 --> 00:10:24.620 In other words, what we did was, is we said that to take f of v, 00:10:24.620 --> 00:10:26.990 we would just take the matrix-- 00:10:26.990 --> 00:10:33.870 notice how I've written it, now-- not a1,1, a1,2, a1,1, 00:10:33.870 --> 00:10:34.710 you see? 00:10:34.710 --> 00:10:37.380 a2,1, a3,1, et cetera-- 00:10:37.380 --> 00:10:41.250 you see, these make up the components of f of u1. 00:10:41.250 --> 00:10:43.950 These make up the components of f of un. 00:10:43.950 --> 00:10:47.082 That's what was called the matrix A. 00:10:47.082 --> 00:10:53.040 v itself was the n-tuple x1 up to xn, which became the column 00:10:53.040 --> 00:10:58.210 matrix X when we wrote it as a column vector. 00:10:58.210 --> 00:10:59.410 Remember that. 00:10:59.410 --> 00:11:02.070 And then what we said was the transpose 00:11:02.070 --> 00:11:05.340 of that would be c times this n-tuple. 00:11:05.340 --> 00:11:08.310 But if I write this n-tuple as a column vector, 00:11:08.310 --> 00:11:10.530 I don't need the transpose in here. 00:11:10.530 --> 00:11:13.110 In other words, in matrix language, 00:11:13.110 --> 00:11:16.080 with A being this matrix and X being 00:11:16.080 --> 00:11:19.530 this column matrix, this translates into the matrix 00:11:19.530 --> 00:11:24.000 equation A times X equals c times X, 00:11:24.000 --> 00:11:27.810 where we recall that the matrix A is what's given, 00:11:27.810 --> 00:11:30.930 and what we're trying to find is, first of all, 00:11:30.930 --> 00:11:35.900 are there any vectors X that are mapped into a scalar times X? 00:11:35.900 --> 00:11:38.010 In other words, are any column matrices 00:11:38.010 --> 00:11:41.100 that are mapped into a scalar times that column matrix 00:11:41.100 --> 00:11:42.525 with respect to A? 00:11:42.525 --> 00:11:46.950 And secondly, if there are such column matrices, what 00:11:46.950 --> 00:11:49.380 values of c correspond to that? 00:11:49.380 --> 00:11:52.410 Well, notice, by ordinary algebraic techniques-- 00:11:52.410 --> 00:11:57.180 because matrices do obey many of the ordinary rules of algebra-- 00:11:57.180 --> 00:12:04.530 AX equals cX is the same as saying AX minus cX is 0. 00:12:04.530 --> 00:12:06.960 Notice that we already know that matrices 00:12:06.960 --> 00:12:09.720 obey the distributive rule. 00:12:09.720 --> 00:12:12.780 In other words, I could factor out the matrix X from here. 00:12:12.780 --> 00:12:14.700 Of course, I have to be very, very careful. 00:12:14.700 --> 00:12:17.130 Notice that capital A is a matrix. 00:12:17.130 --> 00:12:18.990 Little c is a scalar. 00:12:18.990 --> 00:12:22.510 And to have A minus c wouldn't make much sense. 00:12:22.510 --> 00:12:25.110 In other words, since A is an n-by-n matrix, 00:12:25.110 --> 00:12:27.060 I want whatever I'm subtracting from it 00:12:27.060 --> 00:12:29.620 to also be an n-by-n matrix. 00:12:29.620 --> 00:12:33.360 So what I do is the little cute device of remembering 00:12:33.360 --> 00:12:36.390 the property of the identity matrix I sub n. 00:12:36.390 --> 00:12:41.350 I simply replace X by I sub n times X-- in other words, 00:12:41.350 --> 00:12:44.900 the identity matrix times X. That now says what? 00:12:44.900 --> 00:12:49.350 AX minus c, identity matrix n-by-n-- 00:12:49.350 --> 00:12:51.780 identity matrix times X equals 0. 00:12:51.780 --> 00:12:55.260 Now I can factor out the X. And I have what? 00:12:55.260 --> 00:12:59.220 The matrix A minus c times the identity matrix 00:12:59.220 --> 00:13:03.030 times the column matrix X equals 0. 00:13:03.030 --> 00:13:04.920 Now, remember, back when we were first 00:13:04.920 --> 00:13:08.160 talking about matrix algebra, we pointed out 00:13:08.160 --> 00:13:11.730 that matrices obey the same structure. 00:13:11.730 --> 00:13:16.200 Matrices obey the same structure with certain small reservations 00:13:16.200 --> 00:13:17.460 that numbers obey. 00:13:17.460 --> 00:13:22.890 For example, we saw that if A was not the zero matrix, 00:13:22.890 --> 00:13:27.690 AX equals zero did not imply that X equals zero like it 00:13:27.690 --> 00:13:29.220 did in ordinary arithmetic. 00:13:29.220 --> 00:13:32.590 But it did if A happened to be a non-singular matrix. 00:13:32.590 --> 00:13:34.410 In other words, what we did show was 00:13:34.410 --> 00:13:38.730 that if this particular matrix had an inverse, then, 00:13:38.730 --> 00:13:41.310 multiplying both sides of this equation 00:13:41.310 --> 00:13:43.470 by the inverse of this, the inverse 00:13:43.470 --> 00:13:47.640 would cancel this factor, and we'd be left with X equals 0. 00:13:47.640 --> 00:13:52.980 In other words, if A minus cI inverse exists, 00:13:52.980 --> 00:13:56.260 then X must be the zero column matrix. 00:13:56.260 --> 00:13:57.600 Now, remember what X is. 00:13:57.600 --> 00:14:01.515 X is the column matrix whose entries 00:14:01.515 --> 00:14:05.610 are the components of the v that we're looking for over here. 00:14:05.610 --> 00:14:08.700 Keep in mind that we were looking for a v which 00:14:08.700 --> 00:14:10.830 was unequal to zero. 00:14:10.830 --> 00:14:13.650 If v is unequal to 0, in particular, at least one 00:14:13.650 --> 00:14:16.560 of its components must be different from 0. 00:14:16.560 --> 00:14:19.320 So what we're saying is, if A-- 00:14:19.320 --> 00:14:23.250 if this matrix here, with its inverse, exists, 00:14:23.250 --> 00:14:26.190 then X must be the zero column matrix, 00:14:26.190 --> 00:14:29.050 which is the solution that we don't want. 00:14:29.050 --> 00:14:32.340 In other words, we won't get non-zero solutions. 00:14:32.340 --> 00:14:34.260 Or from a different point of view, 00:14:34.260 --> 00:14:37.800 what this says is, if we want to be able to find a column 00:14:37.800 --> 00:14:42.780 vector X which is not the zero column vector, in particular, 00:14:42.780 --> 00:14:46.750 A minus cI had better be a singular matrix. 00:14:46.750 --> 00:14:48.960 In other words, it's inverse doesn't exist. 00:14:48.960 --> 00:14:53.400 And as we saw back in block 4, when a matrix is singular, 00:14:53.400 --> 00:14:55.680 it means that its determinant is 0. 00:14:55.680 --> 00:14:58.890 Consequently, in order for there to be any chance 00:14:58.890 --> 00:15:03.210 that we can find non-zero solutions of this equation, 00:15:03.210 --> 00:15:08.190 it must be that the determinant of A minus cI must be 0. 00:15:08.190 --> 00:15:11.560 And by the way, what does A minus cI look like? 00:15:11.560 --> 00:15:15.430 Notice that I is the n-by-n identity matrix. 00:15:15.430 --> 00:15:19.860 When you multiply a matrix by a scalar, 00:15:19.860 --> 00:15:24.240 you multiply each entry of that matrix by that scalar. 00:15:24.240 --> 00:15:29.250 Since all of the entries off the diagonal are 0, c times 0 00:15:29.250 --> 00:15:30.520 will still be 0. 00:15:30.520 --> 00:15:35.140 So notice that c times the n-by-n identity matrix is just 00:15:35.140 --> 00:15:41.050 the n-by-n diagonal matrix, each of whose diagonal elements 00:15:41.050 --> 00:15:43.330 is c. 00:15:43.330 --> 00:15:46.870 And now, remembering what A is, and remembering 00:15:46.870 --> 00:15:50.170 how we subtract two matrices, we subtract 00:15:50.170 --> 00:15:52.840 them component-by-component. 00:15:52.840 --> 00:15:57.640 Notice that the only non-zero components of cI are the c's 00:15:57.640 --> 00:15:58.990 down the diagonal. 00:15:58.990 --> 00:16:04.480 What this says is, if we take our matrix A, A minus c In is 00:16:04.480 --> 00:16:07.210 simply, from a manipulative point of view, 00:16:07.210 --> 00:16:11.590 obtained by subtracting c from each of the diagonal elements. 00:16:11.590 --> 00:16:17.230 You see, it's a1,1 minus c, a2,2 minus c. 00:16:17.230 --> 00:16:19.600 But every place else, you're subtracting 0, 00:16:19.600 --> 00:16:21.940 because the entry here is 0. 00:16:21.940 --> 00:16:24.130 So this is what this matrix looks like. 00:16:24.130 --> 00:16:26.530 I want its determinant to be 0. 00:16:26.530 --> 00:16:30.490 Last time, we showed how we computed a determinant. 00:16:30.490 --> 00:16:33.400 Notice that the a's are given numbers. 00:16:33.400 --> 00:16:35.500 c is the only unknown. 00:16:35.500 --> 00:16:39.130 If we expand this determinant and equate it to 0, 00:16:39.130 --> 00:16:42.700 we get an nth degree polynomial in c. 00:16:42.700 --> 00:16:46.130 I'm not going to go into much detail about that now. 00:16:46.130 --> 00:16:49.090 In fact, I'm not going to go into any detail about this now. 00:16:49.090 --> 00:16:51.250 I will save that for the exercises. 00:16:51.250 --> 00:16:54.490 But what I will do is take the very simple case, where 00:16:54.490 --> 00:16:56.500 we have a two-dimensional vector space, 00:16:56.500 --> 00:16:59.710 and apply this theory to the two-by-two case. 00:16:59.710 --> 00:17:04.000 And I think that the easiest example to pick, in this case, 00:17:04.000 --> 00:17:08.290 is the same example as we started with-- example 1-- 00:17:08.290 --> 00:17:09.819 and revisit it. 00:17:09.819 --> 00:17:12.250 In other words, this is called "example 1 revisited." 00:17:12.250 --> 00:17:13.089 That doesn't sound-- 00:17:13.089 --> 00:17:14.920 Let's just call it example 2. 00:17:14.920 --> 00:17:18.460 In example 2, you were thinking of a two-dimensional space 00:17:18.460 --> 00:17:20.290 relative to a particular basis. 00:17:20.290 --> 00:17:26.470 The 2-tuple x comma y got mapped into x plus 4y comma x plus y. 00:17:26.470 --> 00:17:30.780 In particular, 1 comma 0 got mapped into 1 comma 1. 00:17:30.780 --> 00:17:33.640 0 comma 1 got mapped into 4 comma 1. 00:17:33.640 --> 00:17:35.410 The only reason I've left the bar off 00:17:35.410 --> 00:17:37.300 here is so that you don't get the feeling 00:17:37.300 --> 00:17:39.610 that this has to be interpreted geometrically. 00:17:39.610 --> 00:17:41.740 This could be any two-dimensional space 00:17:41.740 --> 00:17:43.960 relative to any basis. 00:17:43.960 --> 00:17:46.510 But at any rate, using this example, 00:17:46.510 --> 00:17:50.470 remembering how the matrix A is obtained, the first column of A 00:17:50.470 --> 00:17:51.850 are these components. 00:17:51.850 --> 00:17:52.990 That's 1, 1. 00:17:52.990 --> 00:17:56.110 The second column of A are these components-- 00:17:56.110 --> 00:17:57.040 4, 1. 00:17:57.040 --> 00:18:00.760 So the matrix A associated with f 00:18:00.760 --> 00:18:04.900 relative to our given basis that represents the 2-tuples here 00:18:04.900 --> 00:18:07.390 is 1, 4, 1, 1. 00:18:07.390 --> 00:18:11.180 If I want to now look at A minus c I2-- 00:18:11.180 --> 00:18:13.390 see, n is 2, in this case-- the two-by-two identity 00:18:13.390 --> 00:18:14.750 matrix-- what do I do? 00:18:14.750 --> 00:18:20.470 I just subtract c from each diagonal element this way 00:18:20.470 --> 00:18:23.620 so the determinant of that is this determinant. 00:18:23.620 --> 00:18:26.120 We already know how to expand a two-by-two determinant. 00:18:26.120 --> 00:18:29.320 It's this times this minus this times this. 00:18:29.320 --> 00:18:32.500 This is what? c squared minus 2c. 00:18:32.500 --> 00:18:35.050 Plus 1 minus 4 is minus 3. 00:18:35.050 --> 00:18:37.780 This is c squared minus 2c minus 3. 00:18:37.780 --> 00:18:41.140 And therefore, the only way this determinant can be 0 00:18:41.140 --> 00:18:42.580 is if this is 0. 00:18:42.580 --> 00:18:46.040 This factors into c minus 3 times c plus 1. 00:18:46.040 --> 00:18:50.240 Therefore, it must be that c is 3 or c is minus 1. 00:18:50.240 --> 00:18:52.630 In other words, the only possible characteristic-- or, 00:18:52.630 --> 00:18:57.970 eigenvalues, in this problem, are 3 and minus 1. 00:18:57.970 --> 00:19:00.700 And we'll see what that means in a moment in terms of looking 00:19:00.700 --> 00:19:02.000 at these cases separately. 00:19:02.000 --> 00:19:04.000 You may have noticed I made a slip of the tongue 00:19:04.000 --> 00:19:07.720 and said characteristic values instead of eigenvalues. 00:19:07.720 --> 00:19:12.130 You will find, in many textbooks, the word eigenvalue. 00:19:12.130 --> 00:19:14.980 In other books, you'll find characteristic value. 00:19:14.980 --> 00:19:17.650 These terms are used interchangeably. 00:19:17.650 --> 00:19:20.590 Eigenvalue was the German translation 00:19:20.590 --> 00:19:22.150 of characteristic value. 00:19:22.150 --> 00:19:25.120 So use these terms interchangeably, all right? 00:19:25.120 --> 00:19:27.010 But let's take a look at what happens 00:19:27.010 --> 00:19:30.640 in this particular example, in the case that c equals 3. 00:19:30.640 --> 00:19:34.810 If c equals 3, this tells me that my vector v 00:19:34.810 --> 00:19:38.050 is determined by f of v is 3 times v, where 00:19:38.050 --> 00:19:40.920 v is the 2-tuple x comma y. 00:19:40.920 --> 00:19:43.420 Writing out what this means in terms of my matrix now, 00:19:43.420 --> 00:19:45.985 my matrix A is 1, 4, 1, 1. 00:19:45.985 --> 00:19:48.340 v written as a column vector is this. 00:19:48.340 --> 00:19:51.640 And 3 times this column vector is this. 00:19:51.640 --> 00:19:53.980 Remembering that to compare two matrices to be equal, 00:19:53.980 --> 00:19:56.230 they must be equal entry by entry, 00:19:56.230 --> 00:19:58.870 the first entry of this product is what? 00:19:58.870 --> 00:20:01.450 It's x plus 4y. 00:20:01.450 --> 00:20:04.750 The second entry is x plus y. 00:20:04.750 --> 00:20:08.440 Therefore, it must be that x plus 4y equals 3x, 00:20:08.440 --> 00:20:11.020 and x plus y equals 3y. 00:20:11.020 --> 00:20:14.470 And both of these two conditions together say, quite simply, 00:20:14.470 --> 00:20:16.440 that x equals 2y. 00:20:16.440 --> 00:20:18.230 And what does that tell us? 00:20:18.230 --> 00:20:23.120 It says that if you take any 2-tuple of the form x comma y, 00:20:23.120 --> 00:20:24.820 where x is twice y-- 00:20:24.820 --> 00:20:30.230 in other words, if you take the set of all 2-tuples 2y comma y, 00:20:30.230 --> 00:20:32.570 these are eigenvectors. 00:20:32.570 --> 00:20:36.690 And they correspond to the eigenvalue 3. 00:20:36.690 --> 00:20:39.740 And I'm going to show you that pictorially in a few moments. 00:20:39.740 --> 00:20:41.690 I just want you to get used to the computation 00:20:41.690 --> 00:20:43.130 here for the time being. 00:20:43.130 --> 00:20:46.010 Secondly, if you want to think of this geometrically, 00:20:46.010 --> 00:20:50.840 x equals 2y doesn't have to be viewed as a set of vectors. 00:20:50.840 --> 00:20:54.500 It can be viewed as a line in the plane. 00:20:54.500 --> 00:20:58.190 And what this says is that f preserves the direction 00:20:58.190 --> 00:21:00.800 of the line x equals 2y. 00:21:00.800 --> 00:21:03.980 Oh, just as a quick check over here-- 00:21:03.980 --> 00:21:05.780 whichever interpretation you want-- 00:21:05.780 --> 00:21:09.020 notice that if you replace x by 2y-- 00:21:09.020 --> 00:21:11.930 remember what the definition of f was? 00:21:11.930 --> 00:21:13.190 f was what? 00:21:13.190 --> 00:21:17.330 f of x, y was x plus 4y comma x plus y. 00:21:17.330 --> 00:21:22.190 So if x is 2y, this becomes 6y comma 3y. 00:21:22.190 --> 00:21:26.660 In other words, f of 2y comma y is 6y comma 3y. 00:21:26.660 --> 00:21:30.200 That's the same as 3 times 2y comma y. 00:21:30.200 --> 00:21:32.640 And by the way, notice that the special case 00:21:32.640 --> 00:21:36.860 y equals 1 corresponded to part of our example number 00:21:36.860 --> 00:21:39.800 1, when we showed that f bar of 2 comma 00:21:39.800 --> 00:21:43.220 1 was three times 2 comma 1. 00:21:43.220 --> 00:21:45.980 In a similar way, c equals minus 1 00:21:45.980 --> 00:21:48.230 is the other characteristic value. 00:21:48.230 --> 00:21:52.520 Namely, if c equals minus 1, the equation f of v 00:21:52.520 --> 00:21:56.720 equals cv becomes f of v equals minus v. 00:21:56.720 --> 00:22:00.710 And in matrix language, that's AX equals minus X. 00:22:00.710 --> 00:22:03.710 Recalling what A and X are from before, 00:22:03.710 --> 00:22:08.390 remember, A times X will be the column matrix x plus 4y, 00:22:08.390 --> 00:22:10.070 x plus y. 00:22:10.070 --> 00:22:13.970 Minus X is the column matrix minus x minus y. 00:22:13.970 --> 00:22:18.020 Equating corresponding entries, we get this pair of equations. 00:22:18.020 --> 00:22:19.730 And notice that both of these equations 00:22:19.730 --> 00:22:22.670 say that x must equal minus 2y. 00:22:22.670 --> 00:22:26.180 In other words, if the x component is minus twice the y 00:22:26.180 --> 00:22:28.430 component, relative to the given basis 00:22:28.430 --> 00:22:30.530 that we're talking about here, notice 00:22:30.530 --> 00:22:35.090 that the set of all 2-tuples of the form minus 2y comma y 00:22:35.090 --> 00:22:37.400 are eigenvectors, in this case, and 00:22:37.400 --> 00:22:40.420 the corresponding eigenvalue is minus 1, 00:22:40.420 --> 00:22:43.490 and that f preserves the direction 00:22:43.490 --> 00:22:46.990 of the line x equals minus 2y. 00:22:46.990 --> 00:22:49.010 And I think, now, the time has come 00:22:49.010 --> 00:22:50.450 to show you what this thing means 00:22:50.450 --> 00:22:53.270 in terms of a simple geometric interpretation. 00:22:53.270 --> 00:22:55.400 Keep in mind, many of the applications 00:22:55.400 --> 00:22:58.070 of eigenvalues and eigenvectors come up 00:22:58.070 --> 00:23:02.010 in boundary value problems of partial differential equations. 00:23:02.010 --> 00:23:04.130 I will show you, in one of our exercises, 00:23:04.130 --> 00:23:07.730 that even our linear homogeneous differential equations may be 00:23:07.730 --> 00:23:09.710 viewed as eigenvector problems. 00:23:09.710 --> 00:23:11.330 They come up in many applications. 00:23:11.330 --> 00:23:13.730 But I'm saying, in terms of the spirit of a game, 00:23:13.730 --> 00:23:16.310 let's take the simplest physical interpretation. 00:23:16.310 --> 00:23:19.970 And that's simply the mapping of the xy-plane into the uv-plane. 00:23:19.970 --> 00:23:22.970 And all we're saying is that the mapping f bar 00:23:22.970 --> 00:23:25.550 that we're talking about-- what mapping are we talking about? 00:23:25.550 --> 00:23:28.130 The mapping that carries x, y into-- 00:23:28.130 --> 00:23:32.720 what was it that's written down here? x plus 4y comma x plus y. 00:23:32.720 --> 00:23:35.720 What that mapping does is it changes the direction 00:23:35.720 --> 00:23:37.430 of most lines in the plane. 00:23:37.430 --> 00:23:40.880 But there are two lines that it leaves alone in direction. 00:23:40.880 --> 00:23:44.870 Namely, the line x equal 2y gets mapped into the line u 00:23:44.870 --> 00:23:48.170 equals 2v, and the line x equals minus 2y 00:23:48.170 --> 00:23:51.200 gets mapped into the line u equals minus 2v. 00:23:51.200 --> 00:23:54.770 By the way, notice we are not saying that the points remain 00:23:54.770 --> 00:23:55.790 fixed here. 00:23:55.790 --> 00:23:59.630 Remember that the characteristic value corresponding 00:23:59.630 --> 00:24:01.955 to this eigenvector was 3. 00:24:01.955 --> 00:24:05.780 In other words, notice that 2 comma 1 doesn't get 00:24:05.780 --> 00:24:07.670 mapped into 2 comma 1 here. 00:24:07.670 --> 00:24:10.340 It got mapped into 6 comma 3-- 00:24:10.340 --> 00:24:12.950 that the characteristic value tells you, 00:24:12.950 --> 00:24:17.660 once you know what directions are preserved, how much 00:24:17.660 --> 00:24:19.230 the vector was stretched out. 00:24:19.230 --> 00:24:25.250 In other words, 2i plus j get stretched out into 6i plus 3j. 00:24:25.250 --> 00:24:27.500 Well, I'm not going to go into that in any more detail 00:24:27.500 --> 00:24:28.300 right now. 00:24:28.300 --> 00:24:31.220 All I do want to observe is that, if I 00:24:31.220 --> 00:24:34.230 was studying the particular mapping f bar, 00:24:34.230 --> 00:24:38.810 notice that the lines x equal 2y and x equal minus 2y are, 00:24:38.810 --> 00:24:44.000 in a way, a better coordinate system than the axes x and y, 00:24:44.000 --> 00:24:46.880 because notice that the x-axis and the y-axis 00:24:46.880 --> 00:24:49.880 have their directions changed under this mapping. 00:24:49.880 --> 00:24:53.420 But x equals 2y and x equals minus 2y don't 00:24:53.420 --> 00:24:55.260 have their directions changed. 00:24:55.260 --> 00:24:57.230 In fact, to look at this a different way, let's 00:24:57.230 --> 00:24:59.630 pick a representative vector from this line 00:24:59.630 --> 00:25:01.610 and a representative vector from this line. 00:25:01.610 --> 00:25:03.230 Let's take y to be 1. 00:25:03.230 --> 00:25:05.270 In this case, that would say x is 2. 00:25:05.270 --> 00:25:07.355 In this case, it says x is minus 2. 00:25:07.355 --> 00:25:12.010 Let's pick, as two new vectors, alpha 1 to be 2i plus j 00:25:12.010 --> 00:25:15.230 and alpha 2 to be minus 2i plus j. 00:25:15.230 --> 00:25:16.750 And my claim is-- 00:25:16.750 --> 00:25:18.380 I'll write them with arrows here, 00:25:18.380 --> 00:25:20.630 as long as we are going to think of this is a mapping. 00:25:20.630 --> 00:25:22.610 My claim is that alpha 1 and alpha 2 00:25:22.610 --> 00:25:26.000 is a very nice basis for E2 with respect 00:25:26.000 --> 00:25:28.560 to the linear transformation f. 00:25:28.560 --> 00:25:29.710 Well, y? 00:25:29.710 --> 00:25:32.110 Well, what do we already know about alpha 1? 00:25:32.110 --> 00:25:36.320 Alpha 1 is an eigenvector with characteristic value 3. 00:25:36.320 --> 00:25:40.720 In other words, f of alpha 1 is 3 alpha 1. 00:25:40.720 --> 00:25:43.870 Alpha 2 is also an eigenvector with characteristic value 00:25:43.870 --> 00:25:45.020 minus 1. 00:25:45.020 --> 00:25:48.360 So f of alpha 2 is minus alpha 2. 00:25:48.360 --> 00:25:51.520 Notice, then, therefore, from an algebraic point of view, 00:25:51.520 --> 00:25:55.270 if I pick alpha 1 and alpha 2 as my bases for-- well, 00:25:55.270 --> 00:25:56.770 I should be consistent here. 00:25:56.770 --> 00:26:01.600 I called this V. I suppose it should have been E2, 00:26:01.600 --> 00:26:03.520 simply to match the notation here. 00:26:03.520 --> 00:26:04.690 But that's not important. 00:26:04.690 --> 00:26:08.170 Suppose I pick alpha 1 and alpha 2 to be my new bases. 00:26:08.170 --> 00:26:15.316 Notice, you see, that f of alpha 1 is 3 alpha 1 plus 0 alpha 2. 00:26:15.316 --> 00:26:21.900 f of alpha 2 is 0 alpha 1 minus 1 alpha 2. 00:26:21.900 --> 00:26:27.590 So my matrix of f, relative to alpha 1 and alpha 2 as a basis, 00:26:27.590 --> 00:26:30.470 would have its first column being 3 and 0. 00:26:30.470 --> 00:26:34.230 It would have its second column being 0 and minus 1. 00:26:34.230 --> 00:26:39.020 In other words, the matrix now is a diagonal matrix, 3, 0, 0, 00:26:39.020 --> 00:26:40.340 minus 1. 00:26:40.340 --> 00:26:44.390 It's not only a diagonal matrix, but the diagonal elements 00:26:44.390 --> 00:26:47.660 themselves yield the eigenvalues. 00:26:47.660 --> 00:26:53.550 Notice how easy this matrix is to use for computing-- 00:26:53.550 --> 00:26:58.530 A times X-- if X happens to be written relative to the alphas, 00:26:58.530 --> 00:27:01.770 because the easiest type of matrix to multiply by 00:27:01.770 --> 00:27:03.900 is a diagonal matrix. 00:27:03.900 --> 00:27:08.520 And I'm not going to go through this here, but when you write-- 00:27:08.520 --> 00:27:12.260 when you pick the basis consisting of eigenvalues, 00:27:12.260 --> 00:27:15.960 eigenvectors, and write this diagonal matrix, 00:27:15.960 --> 00:27:18.600 the resulting diagonal matrix gives you 00:27:18.600 --> 00:27:21.240 a tremendous amount of insight as to what 00:27:21.240 --> 00:27:22.710 the space looks like. 00:27:22.710 --> 00:27:24.900 And I'll bring that out in the exercises. 00:27:24.900 --> 00:27:27.390 All I want you to get out of this overview 00:27:27.390 --> 00:27:31.510 is what eigenvectors are and how we compute them. 00:27:31.510 --> 00:27:33.510 And I thought that, to finish up with, 00:27:33.510 --> 00:27:37.980 I would like to give you a very, very profound result, which 00:27:37.980 --> 00:27:41.700 I won't prove for you, but which I will state-- 00:27:41.700 --> 00:27:43.500 has, also, a profound name. 00:27:43.500 --> 00:27:45.120 But I'll get to that in a moment. 00:27:45.120 --> 00:27:46.950 I call this an important aside. 00:27:46.950 --> 00:27:48.540 It really isn't an aside. 00:27:48.540 --> 00:27:53.200 It's the backbone of much of advanced matrix algebra. 00:27:53.200 --> 00:27:54.810 But the interesting thing is this. 00:27:54.810 --> 00:27:57.540 Remember, given an n-by-n matrix A, 00:27:57.540 --> 00:27:59.700 we were fooling around with looking 00:27:59.700 --> 00:28:05.040 at the determinant of A minus cI equaling 0 00:28:05.040 --> 00:28:08.037 and trying to find a scalar c that would do this for us. 00:28:08.037 --> 00:28:09.870 That's how we get the characteristic values. 00:28:09.870 --> 00:28:11.520 A was the given matrix. 00:28:11.520 --> 00:28:13.560 I was the given identity matrix. 00:28:13.560 --> 00:28:15.870 c was a scalar whose value we were 00:28:15.870 --> 00:28:21.270 trying to get the determinant of this matrix to be 0. 00:28:21.270 --> 00:28:25.980 The amazing point is that if you substitute the matrix 00:28:25.980 --> 00:28:30.885 A for c, in this equation, it will satisfy this equation. 00:28:30.885 --> 00:28:32.790 And what do I mean by that? 00:28:32.790 --> 00:28:38.790 Just replace c by A over here, and this equation is satisfied. 00:28:38.790 --> 00:28:40.470 By the way, that may look trivial. 00:28:40.470 --> 00:28:42.510 You may say to me, gee, whiz. 00:28:42.510 --> 00:28:43.440 What a big deal. 00:28:43.440 --> 00:28:45.720 If I take c and replace it by A, this 00:28:45.720 --> 00:28:47.760 is A times the identity matrix, which 00:28:47.760 --> 00:28:51.050 is still A. A minus A is the zero matrix. 00:28:51.050 --> 00:28:54.000 And the determinant of the zero matrix is clearly 0. 00:28:54.000 --> 00:28:59.930 The metaphysical thing here is, notice that c is a number. 00:28:59.930 --> 00:29:01.380 It's a scalar. 00:29:01.380 --> 00:29:04.650 And A is a matrix. 00:29:04.650 --> 00:29:07.620 Structurally, you cannot let c equal A. 00:29:07.620 --> 00:29:10.620 All we're saying is a remarkable result, 00:29:10.620 --> 00:29:13.830 that if you mechanically replace c by A, 00:29:13.830 --> 00:29:16.290 this equation is satisfied. 00:29:16.290 --> 00:29:19.680 And before I illustrate that for you, I've made a big decision. 00:29:19.680 --> 00:29:22.328 I'm going to tell you what this theorem is called. 00:29:22.328 --> 00:29:24.120 I wasn't originally going to tell you that. 00:29:24.120 --> 00:29:26.700 It's called the Cayley-Hamilton theorem. 00:29:26.700 --> 00:29:30.270 And by my telling you this name, you now know as much 00:29:30.270 --> 00:29:31.740 about the subject as I do. 00:29:31.740 --> 00:29:33.390 That's why I didn't want to tell you what the name was, 00:29:33.390 --> 00:29:35.910 so I'd still know something more than you did about it. 00:29:35.910 --> 00:29:37.770 But that's not too important. 00:29:37.770 --> 00:29:39.840 Let me illustrate how this thing works. 00:29:39.840 --> 00:29:45.120 Let's go back to our matrix 1, 4, 1, 1, all right? 00:29:45.120 --> 00:29:49.920 The determinant of A minus cI, we already saw, 00:29:49.920 --> 00:29:54.770 was c squared minus 2c minus 3. 00:29:54.770 --> 00:29:58.070 My claim is, if I replace c by A in here, 00:29:58.070 --> 00:30:01.410 this will still be obeyed, only with one slight modification. 00:30:01.410 --> 00:30:02.480 See, this becomes what? 00:30:02.480 --> 00:30:05.030 A squared minus 2A. 00:30:05.030 --> 00:30:08.900 And I can't write minus 3, because 3 is a number, not 00:30:08.900 --> 00:30:09.740 a matrix. 00:30:09.740 --> 00:30:11.390 It's always understood, when you're 00:30:11.390 --> 00:30:16.440 converting to matrix form, that the I is over here. 00:30:16.440 --> 00:30:17.900 And if you want to see why, you can 00:30:17.900 --> 00:30:23.060 think of this as being A to the 0, and think of the number 1 00:30:23.060 --> 00:30:25.070 as being c to the 0. 00:30:25.070 --> 00:30:26.690 In other words, structurally, this 00:30:26.690 --> 00:30:30.840 is A squared minus 2A minus 3A to the 0 power. 00:30:30.840 --> 00:30:35.000 This is c squared minus 2c minus 3c to the 0. 00:30:35.000 --> 00:30:37.790 And my claim is that this equation 00:30:37.790 --> 00:30:40.130 will be obeyed by the matrix A. 00:30:40.130 --> 00:30:42.240 Let's just check it out and see if it's true. 00:30:42.240 --> 00:30:45.180 Remember that A was the matrix 1, 4, 1, 1. 00:30:45.180 --> 00:30:47.660 To square it means multiply it by itself. 00:30:47.660 --> 00:30:50.960 If I go through the usual recipe for multiplying 00:30:50.960 --> 00:30:53.570 two two-by-two matrices, I very quickly 00:30:53.570 --> 00:30:56.460 see that the product is 5, 8, 2, 5. 00:30:56.460 --> 00:31:01.490 Notice that, since A is 1, 4, 1, 1, multiplying by minus 2 00:31:01.490 --> 00:31:03.840 multiplies each entry by minus 2. 00:31:03.840 --> 00:31:06.540 So minus 2A is this matrix. 00:31:06.540 --> 00:31:09.620 Notice that minus 3 times the identity matrix 00:31:09.620 --> 00:31:13.210 is a diagonal matrix that has minus 3 00:31:13.210 --> 00:31:15.580 as each main diagonal element-- 00:31:15.580 --> 00:31:17.040 in other words, this matrix here. 00:31:17.040 --> 00:31:19.550 And notice, now, just for the sake of argument, 00:31:19.550 --> 00:31:22.250 if I add these up, what do I get? 00:31:22.250 --> 00:31:26.720 5 minus 2 minus 3, which is 0, 8 minus 8 plus 0, 00:31:26.720 --> 00:31:31.820 which is 0, 2 minus 2 plus 0, which is 0, 5 minus 2 minus 3, 00:31:31.820 --> 00:31:32.720 which is 0. 00:31:32.720 --> 00:31:37.910 In other words, this sum is the zero matrix, not 00:31:37.910 --> 00:31:39.710 the zero number. 00:31:39.710 --> 00:31:43.190 You see, technically speaking here, in this equation here, 00:31:43.190 --> 00:31:46.190 the 0 refers to a number, because the determinant 00:31:46.190 --> 00:31:47.400 is a number. 00:31:47.400 --> 00:31:49.010 But here, we're talking about, it 00:31:49.010 --> 00:31:52.100 satisfied in matrix language. 00:31:52.100 --> 00:31:56.910 And what this means is that matrices can now 00:31:56.910 --> 00:31:59.035 be reduced by long division. 00:31:59.035 --> 00:32:00.660 So I'll give you a very simple example. 00:32:00.660 --> 00:32:02.880 But what the main impact of this is, 00:32:02.880 --> 00:32:06.710 I can now invent power series of matrices. 00:32:06.710 --> 00:32:10.410 In other words, I can define e to the x, where 00:32:10.410 --> 00:32:14.520 x is a matrix, to be 1 plus x plus x squared over 00:32:14.520 --> 00:32:18.600 2 factorial plus x cubed over 3 factorial, et cetera, the same 00:32:18.600 --> 00:32:20.220 as we did in scalar cases. 00:32:20.220 --> 00:32:24.940 And the main reason is that, once I'm given a matrix, 00:32:24.940 --> 00:32:29.520 and I find the basic polynomial equation that it satisfies, 00:32:29.520 --> 00:32:31.800 I can reduce every matrix to that. 00:32:31.800 --> 00:32:33.360 Let me give you an example. 00:32:33.360 --> 00:32:35.040 The key thing I want you to keep in mind 00:32:35.040 --> 00:32:38.400 here is that we already know, for this particular matrix A, 00:32:38.400 --> 00:32:43.140 that A squared minus 2A minus 3I is the zero matrix. 00:32:43.140 --> 00:32:45.747 Suppose, now, I wanted to compute A cubed. 00:32:45.747 --> 00:32:47.330 Now, in this assignment here I'm going 00:32:47.330 --> 00:32:51.480 to show you how I can reduce matrices by long division. 00:32:51.480 --> 00:32:53.820 And in the exercises, I'll actually 00:32:53.820 --> 00:32:55.740 do the long division for you. 00:32:55.740 --> 00:32:59.250 But what is long division in factoring form? 00:32:59.250 --> 00:33:03.180 What I'm saying is, I know that A squared minus 2A minus 3I 00:33:03.180 --> 00:33:04.380 is 0. 00:33:04.380 --> 00:33:06.930 So I would like to write A cubed in such a way 00:33:06.930 --> 00:33:10.350 that I can factor out an A squared minus 2A minus 3I. 00:33:10.350 --> 00:33:13.950 The way I do that is I notice that I must multiply A squared 00:33:13.950 --> 00:33:16.560 by A in order to get A cubed. 00:33:16.560 --> 00:33:19.920 The trouble is, when I multiply, I now have a minus 2A 00:33:19.920 --> 00:33:22.350 squared on the right-hand side that I don't want, 00:33:22.350 --> 00:33:23.580 because it's not here. 00:33:23.580 --> 00:33:26.100 And I have a minus 3A on the right-hand side that I don't 00:33:26.100 --> 00:33:27.970 want, because it's not here. 00:33:27.970 --> 00:33:31.740 So to compensate for that, I simply add on a 2A squared, 00:33:31.740 --> 00:33:33.030 and I add on a 3A. 00:33:33.030 --> 00:33:34.530 In other words, even though this may 00:33:34.530 --> 00:33:36.120 look like a funny way of doing it, 00:33:36.120 --> 00:33:41.140 a very funny way of writing A cubed is this expression here. 00:33:41.140 --> 00:33:43.290 And the reason that I choose this expression is, 00:33:43.290 --> 00:33:49.170 notice that this being 0 means that A cubed is just 2A squared 00:33:49.170 --> 00:33:51.060 plus 3A. 00:33:51.060 --> 00:33:54.690 Moreover, notice that I can still get the structural form A 00:33:54.690 --> 00:33:59.760 squared minus 2A minus 3I out of this thing by writing it. 00:33:59.760 --> 00:34:03.730 Seeing that my first term here is going to be 2A squared-- 00:34:03.730 --> 00:34:05.220 so I put a 2 over here-- 00:34:05.220 --> 00:34:07.860 this gives me my 2A squared, which I want. 00:34:07.860 --> 00:34:09.929 This gives me a minus 4A. 00:34:09.929 --> 00:34:12.659 But I want to have a plus 3A, so I add 00:34:12.659 --> 00:34:15.510 on 7A to compensate for that. 00:34:15.510 --> 00:34:19.120 This gives me a minus 6I, which I don't have up here. 00:34:19.120 --> 00:34:22.050 So to wipe that out, I add on a 6I. 00:34:22.050 --> 00:34:24.270 In other words, another way of writing I cubed, 00:34:24.270 --> 00:34:27.600 therefore, is this plus this. 00:34:27.600 --> 00:34:32.010 And since this is 0, this says that A cubed is nothing 00:34:32.010 --> 00:34:34.770 more than 7A plus 6I. 00:34:34.770 --> 00:34:36.969 In fact, in this particular case, 00:34:36.969 --> 00:34:40.710 notice that any power of A can be reduced 00:34:40.710 --> 00:34:44.340 to a linear combination of A and I, 00:34:44.340 --> 00:34:48.120 because as long as I have even a quadratic power in here, 00:34:48.120 --> 00:34:50.330 I can continue my long division. 00:34:50.330 --> 00:34:52.650 It's just like finding a remainder 00:34:52.650 --> 00:34:54.270 in ordinary long division. 00:34:54.270 --> 00:34:58.620 You keep on going until the remainder is of a lower 00:34:58.620 --> 00:35:00.835 degree than the divisor. 00:35:00.835 --> 00:35:02.460 In this particular case, I've shown you 00:35:02.460 --> 00:35:05.070 that A cubed is 7A plus 6I. 00:35:05.070 --> 00:35:07.440 And I picked an easy case just so we could check it. 00:35:07.440 --> 00:35:12.060 Notice that we already know that A squared is 5, 8, 2, 5. 00:35:12.060 --> 00:35:13.920 A is 1, 4, 1, 1. 00:35:13.920 --> 00:35:16.200 So A cubed is this times this. 00:35:16.200 --> 00:35:18.450 Multiplying these two-by-two's together, 00:35:18.450 --> 00:35:20.640 I get this particular matrix. 00:35:20.640 --> 00:35:25.890 On the other hand, knowing that A is 1, 4, 1, 1, 7A 00:35:25.890 --> 00:35:31.230 is this matrix, 6I is this matrix-- 00:35:31.230 --> 00:35:33.600 and if I now add 7A and 6I-- 00:35:33.600 --> 00:35:34.620 how do I add? 00:35:34.620 --> 00:35:36.030 Component-by-component. 00:35:36.030 --> 00:35:43.940 I get 7 plus 6, which is 13, 28 plus 0, which is 28, 7 plus 0, 00:35:43.940 --> 00:35:47.480 which is 7, 7 plus 6, which is 13. 00:35:47.480 --> 00:35:53.130 In other words, 7A plus 6I is, indeed, A cubed, as asserted. 00:35:53.130 --> 00:35:56.570 And I think you can see now why I wanted to end here. 00:35:56.570 --> 00:36:00.950 From here on in, the course becomes a very, very technical 00:36:00.950 --> 00:36:03.980 subject and one that's used best in conjunction 00:36:03.980 --> 00:36:07.370 with advanced math courses that are using these techniques. 00:36:07.370 --> 00:36:09.710 So we come to the end of part 2. 00:36:09.710 --> 00:36:11.780 I want to tell you what an enjoyable experience 00:36:11.780 --> 00:36:13.550 it was teaching you all. 00:36:13.550 --> 00:36:15.950 If nothing else, as I told you after part 1, 00:36:15.950 --> 00:36:19.670 I emerge smarter, because it takes a lot of preparation 00:36:19.670 --> 00:36:21.680 to get these boards pre-written. 00:36:21.680 --> 00:36:23.720 I couldn't do it alone, and I would like-- 00:36:23.720 --> 00:36:27.050 in addition to the camera people, the floor people, 00:36:27.050 --> 00:36:30.540 there are three people who work very closely with this project 00:36:30.540 --> 00:36:32.090 that I would like to single out. 00:36:32.090 --> 00:36:34.490 I would like to thank, especially, John Fitch, who 00:36:34.490 --> 00:36:38.270 is the manager of our self-study project, who also doubles 00:36:38.270 --> 00:36:42.970 in as director and producer of the tape, the film series, 00:36:42.970 --> 00:36:45.320 and is also my advisor for the study guide 00:36:45.320 --> 00:36:46.640 and things of the like. 00:36:46.640 --> 00:36:49.340 I would like to thank Charles Patton, who 00:36:49.340 --> 00:36:53.960 is the one responsible the most for the clear pictures 00:36:53.960 --> 00:36:57.320 and the excellent photogenic features that you notice of me, 00:36:57.320 --> 00:36:59.240 the sharpness of the camera. 00:36:59.240 --> 00:37:03.080 I would also like to thank Elise Pelletier, who, 00:37:03.080 --> 00:37:05.660 in addition to being a very able secretary, 00:37:05.660 --> 00:37:08.930 doubles in in the master control room as the master everything, 00:37:08.930 --> 00:37:12.620 from running the video tape recorder to making hasty phone 00:37:12.620 --> 00:37:14.600 calls and things of this type. 00:37:14.600 --> 00:37:17.660 I would also like to thank two other colleagues, Arthur 00:37:17.660 --> 00:37:20.330 [INAUDIBLE] and Paul Brown, administrative offices 00:37:20.330 --> 00:37:23.540 at the center, who have provided me most excellent working 00:37:23.540 --> 00:37:26.750 conditions, and finally, Harold [INAUDIBLE],, who 00:37:26.750 --> 00:37:28.760 was the first director of the center 00:37:28.760 --> 00:37:32.540 and whose idea it was to produce Calculus Revisited Part 00:37:32.540 --> 00:37:34.430 1 and Part 2. 00:37:34.430 --> 00:37:35.910 It has been my pleasure. 00:37:35.910 --> 00:37:38.560 I hope that our paths cross again. 00:37:38.560 --> 00:37:40.880 But until such a time, God bless you all. 00:37:43.530 --> 00:37:45.930 Funding for the publication of this video 00:37:45.930 --> 00:37:50.820 was provided by the Gabriella and Paul Rosenbaum Foundation. 00:37:50.820 --> 00:37:54.960 Help OCW continue to provide free and open access to MIT 00:37:54.960 --> 00:38:00.246 courses by making a donation at ocw.mit.edu/donate.