1 00:00:07,380 --> 00:00:08,230 OK. 2 00:00:08,230 --> 00:00:16,010 So this is the first lecture on eigenvalues and eigenvectors, 3 00:00:16,010 --> 00:00:19,250 and that's a big subject that will take up 4 00:00:19,250 --> 00:00:23,400 most of the rest of the course. 5 00:00:23,400 --> 00:00:31,500 It's, again, matrices are square and we're looking now 6 00:00:31,500 --> 00:00:34,820 for some special numbers, the eigenvalues, 7 00:00:34,820 --> 00:00:38,620 and some special vectors, the eigenvectors. 8 00:00:38,620 --> 00:00:45,360 And so this lecture is mostly about what are these numbers, 9 00:00:45,360 --> 00:00:48,850 and then the other lectures are about how do we use them, 10 00:00:48,850 --> 00:00:50,440 why do we want them. 11 00:00:50,440 --> 00:00:54,200 OK, so what's an eigenvector? 12 00:00:54,200 --> 00:00:56,480 Maybe I'll start with eigenvector. 13 00:00:56,480 --> 00:00:57,720 What's an eigenvector? 14 00:00:57,720 --> 00:01:02,900 So I have a matrix A. 15 00:01:02,900 --> 00:01:05,570 OK. 16 00:01:05,570 --> 00:01:07,430 What does a matrix do? 17 00:01:07,430 --> 00:01:09,290 It acts on vectors. 18 00:01:09,290 --> 00:01:12,250 It multiplies vectors x. 19 00:01:12,250 --> 00:01:21,110 So the way that matrix acts is in goes a vector x and out 20 00:01:21,110 --> 00:01:22,690 comes a vector Ax. 21 00:01:22,690 --> 00:01:23,930 It's like a function. 22 00:01:23,930 --> 00:01:27,870 With a function in calculus, in goes 23 00:01:27,870 --> 00:01:31,360 a number x, out comes f(x). 24 00:01:31,360 --> 00:01:34,960 Here in linear algebra we're up in more dimensions. 25 00:01:34,960 --> 00:01:39,350 In goes a vector x, out comes a vector Ax. 26 00:01:39,350 --> 00:01:43,060 And the vectors I'm specially interested in 27 00:01:43,060 --> 00:01:47,630 are the ones the come out in the same direction 28 00:01:47,630 --> 00:01:50,050 that they went in. 29 00:01:50,050 --> 00:01:52,050 That won't be typical. 30 00:01:52,050 --> 00:01:59,270 Most vectors, Ax is in -- points in some different direction. 31 00:01:59,270 --> 00:02:04,456 But there are certain vectors where Ax comes out parallel 32 00:02:04,456 --> 00:02:04,955 to x. 33 00:02:07,460 --> 00:02:09,660 And those are the eigenvectors. 34 00:02:09,660 --> 00:02:12,400 So Ax parallel to x. 35 00:02:18,130 --> 00:02:19,465 Those are the eigenvectors. 36 00:02:23,800 --> 00:02:25,990 And what do I mean by parallel? 37 00:02:25,990 --> 00:02:29,620 Oh, much easier to just state it in an equation. 38 00:02:29,620 --> 00:02:36,340 Ax is some multiple -- and everybody calls that multiple 39 00:02:36,340 --> 00:02:38,900 lambda -- of x. 40 00:02:38,900 --> 00:02:40,170 That's our big equation. 41 00:02:43,890 --> 00:02:46,780 We look for special vectors -- and remember most vectors 42 00:02:46,780 --> 00:02:49,780 won't be eigenvectors -- 43 00:02:49,780 --> 00:02:53,740 that -- for which Ax is in the same direction as x, 44 00:02:53,740 --> 00:02:57,380 and by same direction I allow it to be the very opposite 45 00:02:57,380 --> 00:03:03,830 direction, I allow lambda to be negative or zero. 46 00:03:03,830 --> 00:03:07,950 Well, I guess we've met the eigenvectors that 47 00:03:07,950 --> 00:03:10,600 have eigenvalue zero. 48 00:03:10,600 --> 00:03:13,600 Those are in the same direction, but they're -- 49 00:03:13,600 --> 00:03:18,020 in a kind of very special way. 50 00:03:18,020 --> 00:03:22,050 So this -- the eigenvector x. 51 00:03:22,050 --> 00:03:24,830 Lambda, whatever this multiplying factor 52 00:03:24,830 --> 00:03:31,750 is, whether it's six or minus six or zero or even 53 00:03:31,750 --> 00:03:34,800 some imaginary number, that's the eigenvalue. 54 00:03:34,800 --> 00:03:38,350 So there's the eigenvalue, there's the eigenvector. 55 00:03:38,350 --> 00:03:42,960 Let's just take a second on eigenvalue zero. 56 00:03:42,960 --> 00:03:45,630 From the point of view of eigenvalues, that's no special 57 00:03:45,630 --> 00:03:46,140 deal. 58 00:03:46,140 --> 00:03:48,910 That's, we have an eigenvector. 59 00:03:48,910 --> 00:03:51,240 If the eigenvalue happened to be zero, 60 00:03:51,240 --> 00:03:56,380 that would mean that Ax was zero x, in other words zero. 61 00:03:56,380 --> 00:04:00,280 So what would x, where would we look for -- what are the x-s? 62 00:04:00,280 --> 00:04:04,460 What are the eigenvectors with eigenvalue zero? 63 00:04:04,460 --> 00:04:08,200 They're the guys in the null space, Ax equals zero. 64 00:04:08,200 --> 00:04:12,540 So if our matrix is singular, let me write this down. 65 00:04:12,540 --> 00:04:23,210 If, if A is singular, then that -- what does singular mean? 66 00:04:23,210 --> 00:04:26,015 It means that it takes some vector x into zero. 67 00:04:29,190 --> 00:04:31,570 Some non-zero vector, that's why -- 68 00:04:31,570 --> 00:04:33,930 will be the eigenvector into zero. 69 00:04:33,930 --> 00:04:38,945 Then lambda equals zero is an eigenvalue. 70 00:04:42,530 --> 00:04:46,860 But we're interested in all eigenvalues now, 71 00:04:46,860 --> 00:04:50,480 lambda equals zero is not, like, so special anymore. 72 00:04:50,480 --> 00:04:51,140 OK. 73 00:04:51,140 --> 00:04:57,530 So the question is, how do we find these x-s and lambdas? 74 00:04:57,530 --> 00:05:01,830 And notice -- we don't have an equation Ax equal B anymore. 75 00:05:01,830 --> 00:05:02,830 I can't use elimination. 76 00:05:05,780 --> 00:05:07,670 I've got two unknowns, and in fact they're 77 00:05:07,670 --> 00:05:08,860 multiplied together. 78 00:05:08,860 --> 00:05:12,390 Lambda and x are both unknowns here. 79 00:05:12,390 --> 00:05:18,680 So, we need to, we need a good idea of how to find them. 80 00:05:18,680 --> 00:05:23,930 But before I, before I do that, and that's 81 00:05:23,930 --> 00:05:26,590 where determinant will come in, can I just 82 00:05:26,590 --> 00:05:29,330 give you some matrices? 83 00:05:29,330 --> 00:05:31,720 Like here you go. 84 00:05:31,720 --> 00:05:35,261 Take the matrix, a projection matrix. 85 00:05:35,261 --> 00:05:35,760 OK. 86 00:05:35,760 --> 00:05:42,950 So suppose we have a plane and our matrix P is -- 87 00:05:42,950 --> 00:05:45,450 what I've called A, now I'm going to call it P 88 00:05:45,450 --> 00:05:47,620 for the moment, because it's -- 89 00:05:47,620 --> 00:05:51,800 I'm thinking OK, let's look at a then this, 90 00:05:51,800 --> 00:05:57,270 this other new matrix, I just have an Ax, projection matrix. 91 00:05:57,270 --> 00:06:01,250 What are the eigenvalues of a projection matrix? 92 00:06:01,250 --> 00:06:03,240 So that's my question. 93 00:06:03,240 --> 00:06:07,720 What are the x-s, the eigenvectors, and the lambdas, 94 00:06:07,720 --> 00:06:13,190 the eigenvalues, thing,4 but the roots of that quadratic for -- 95 00:06:13,190 --> 00:06:17,170 and now let me say a projection matrix. 96 00:06:17,170 --> 00:06:19,940 My, my point is that we -- 97 00:06:19,940 --> 00:06:23,300 before we get into determinants and, and formulas 98 00:06:23,300 --> 00:06:26,280 and all that stuff, let's take some matrices 99 00:06:26,280 --> 00:06:28,690 where we know what they do. 100 00:06:28,690 --> 00:06:34,290 We know that if we take a vector b, what this matrix does 101 00:06:34,290 --> 00:06:38,480 is it projects it down to Pb. 102 00:06:38,480 --> 00:06:43,340 So is b an eigenvector in, in that picture? 103 00:06:43,340 --> 00:06:45,010 Is that vector b an eigenvector? 104 00:06:48,810 --> 00:06:50,270 No. 105 00:06:50,270 --> 00:06:53,850 Not so, so b is not an eigenvector 106 00:06:53,850 --> 00:06:59,090 c- because Pb, its projection, is in a different direction. 107 00:06:59,090 --> 00:07:04,030 So now tell me what vectors are eigenvectors of P? 108 00:07:04,030 --> 00:07:09,080 What vectors do get projected in the same direction that they 109 00:07:09,080 --> 00:07:09,580 start? 110 00:07:09,580 --> 00:07:12,310 So, so answer, tell me some x-s. 111 00:07:12,310 --> 00:07:16,590 Do you see what3 so it's if Ax equals lambda x, 112 00:07:16,590 --> 00:07:21,230 In this picture, where could I start with a vector b or x, 113 00:07:21,230 --> 00:07:27,070 do its projection, and end up in the same direction? 114 00:07:27,070 --> 00:07:33,360 Well, that would happen if the vector was right in that plane 115 00:07:33,360 --> 00:07:34,420 already. 116 00:07:34,420 --> 00:07:41,510 If the vector x was -- so let the vector x -- 117 00:07:41,510 --> 00:07:49,810 so any vector, any x in the plane will be an eigenvector. 118 00:07:49,810 --> 00:07:53,140 And what will happen when I multiply by P, 119 00:07:53,140 --> 00:07:57,110 when I project a vector x -- 120 00:07:57,110 --> 00:08:00,190 I called it b here, because this is our familiar picture, 121 00:08:00,190 --> 00:08:03,650 but now I'm going to say that b was no good for, for the, 122 00:08:03,650 --> 00:08:05,140 for our purposes. 123 00:08:05,140 --> 00:08:09,850 I'm interested in a vector x that's actually in the plane, 124 00:08:09,850 --> 00:08:14,100 and I project it, and what do I get back? 125 00:08:14,100 --> 00:08:15,370 x, of course. 126 00:08:15,370 --> 00:08:17,920 Doesn't move. can be complex numbers. 127 00:08:17,920 --> 00:08:24,190 So any x in the plane is unchanged by P, 128 00:08:24,190 --> 00:08:26,070 and what's that telling me? 129 00:08:26,070 --> 00:08:28,490 That's telling me that x is an eigenvector, 130 00:08:28,490 --> 00:08:32,909 and it's also telling me what's the eigenvalue, which is -- 131 00:08:32,909 --> 00:08:34,470 just compare it with that. 132 00:08:34,470 --> 00:08:39,549 The eigenvalue, the multiplier, is just one. 133 00:08:39,549 --> 00:08:41,510 Good. 134 00:08:41,510 --> 00:08:45,290 So we have actually a whole plane of eigenvectors. 135 00:08:45,290 --> 00:08:49,050 Now I ask, are there any other eigenvectors? 136 00:08:49,050 --> 00:08:51,020 And I expect the answer to be yes, 137 00:08:51,020 --> 00:08:53,290 because I would like to get three, 138 00:08:53,290 --> 00:08:55,470 if I'm in three dimensions, I would 139 00:08:55,470 --> 00:08:59,490 like to hope for three independent eigenvectors, two 140 00:08:59,490 --> 00:09:02,610 of them in the plane and one not in the plane. 141 00:09:02,610 --> 00:09:03,300 OK. 142 00:09:03,300 --> 00:09:06,940 So this guy b that I drew there was not any good. 143 00:09:06,940 --> 00:09:11,330 What's the right eigenvector that's not in the plane? 144 00:09:11,330 --> 00:09:16,910 The, the good one is the one that's perpendicular to the 145 00:09:16,910 --> 00:09:17,490 plane. 146 00:09:17,490 --> 00:09:22,170 There's an, another good x, because what's the projection? 147 00:09:22,170 --> 00:09:24,300 So these are eigenvectors. 148 00:09:24,300 --> 00:09:27,000 Another guy here would be another eigenvector. 149 00:09:27,000 --> 00:09:29,950 But now here is another one. two. 150 00:09:29,950 --> 00:09:37,710 Any x that's perpendicular to the plane, 151 00:09:37,710 --> 00:09:41,390 what's Px for that, for that, vector? 152 00:09:43,950 --> 00:09:47,070 What's the projection of this guy perpendicular to the plane? 153 00:09:47,070 --> 00:09:50,570 It is zero, of course. 154 00:09:50,570 --> 00:09:53,640 So -- there's the null space. 155 00:09:53,640 --> 00:09:58,800 Px and n- for those guys are zero, or zero x if we like, 156 00:09:58,800 --> 00:10:02,480 and the eigenvalue is zero. 157 00:10:02,480 --> 00:10:05,697 So my answer to the question is, what are the eigenvalues for 158 00:10:05,697 --> 00:10:07,280 In our example, the one we worked out, 159 00:10:07,280 --> 00:10:08,610 a projection matrix? 160 00:10:08,610 --> 00:10:09,660 There they are. 161 00:10:09,660 --> 00:10:11,880 One and zero. 162 00:10:11,880 --> 00:10:13,060 OK. 163 00:10:13,060 --> 00:10:17,440 We know projection matrices. 164 00:10:17,440 --> 00:10:22,660 We can write them down as that A, A transpose, A inverse, 165 00:10:22,660 --> 00:10:28,400 A transpose thing, but without doing that from the picture 166 00:10:28,400 --> 00:10:31,420 we could see what are the eigenvectors. 167 00:10:31,420 --> 00:10:32,280 OK. 168 00:10:32,280 --> 00:10:34,580 Are there other matrices? 169 00:10:34,580 --> 00:10:36,790 Let me take a second example. 170 00:10:36,790 --> 00:10:39,650 How about a permutation matrix? 171 00:10:39,650 --> 00:10:41,619 What about the matrix, I'll call it A now. 172 00:10:41,619 --> 00:10:42,410 Zero one, one zero. 173 00:10:42,410 --> 00:10:49,120 A equals zero one one zero, that had eigenvalue one and 174 00:10:49,120 --> 00:10:53,740 Can you tell me a vector x -- 175 00:10:53,740 --> 00:10:55,680 see, we'll have a system soon enough, 176 00:10:55,680 --> 00:10:58,040 so I, I would like to just do these e- 177 00:10:58,040 --> 00:11:03,950 these couple of examples, just to see the picture before we, 178 00:11:03,950 --> 00:11:11,300 before we let it all, go into a system 179 00:11:11,300 --> 00:11:13,850 where that, matrix isn't anything special. 180 00:11:13,850 --> 00:11:15,590 Because it is special. 181 00:11:15,590 --> 00:11:20,260 And what, so what vector could I multiply by and end up 182 00:11:20,260 --> 00:11:21,430 in the same direction? 183 00:11:21,430 --> 00:11:25,800 Can you spot an eigenvector for this guy? 184 00:11:25,800 --> 00:11:32,380 That's a matrix that permutes x1 and x2, right? 185 00:11:32,380 --> 00:11:37,190 It switches the two components of x. 186 00:11:37,190 --> 00:11:43,710 How could the vector with its x2 x1, with -- 187 00:11:43,710 --> 00:11:49,630 permuted turn out to be a multiple of x1 x2, 188 00:11:49,630 --> 00:11:51,060 the vector we start with? 189 00:11:51,060 --> 00:11:53,300 Can you tell me an eigenvector here for this guy? 190 00:11:53,300 --> 00:11:56,450 x equal -- what is -- actually, can you tell me one vector that 191 00:11:56,450 --> 00:11:58,690 which is lambda x, and I have a three x, 192 00:11:58,690 --> 00:12:01,160 And of course you -- everybody knows that they're -- what, 193 00:12:01,160 --> 00:12:01,830 has eigenvalue one? 194 00:12:01,830 --> 00:12:03,670 So what, what vector would have eigenvalue one, 195 00:12:03,670 --> 00:12:05,544 just above what we2 found here. so that if I, 196 00:12:05,544 --> 00:12:09,530 if I permute it it doesn't change? right? 197 00:12:09,530 --> 00:12:13,450 There, that could be one one, thanks. 198 00:12:13,450 --> 00:12:14,460 One one. 199 00:12:14,460 --> 00:12:16,780 OK, take that vector one one. 200 00:12:16,780 --> 00:12:20,470 That will be an eigenvector, because if I do Ax 201 00:12:20,470 --> 00:12:23,120 I get one one. 202 00:12:23,120 --> 00:12:27,230 So that's the eigenvalue is one. 203 00:12:27,230 --> 00:12:28,710 Great. 204 00:12:28,710 --> 00:12:30,600 That's one eigenvalue. 205 00:12:30,600 --> 00:12:33,420 But I have here a two by two matrix, 206 00:12:33,420 --> 00:12:37,620 and I figure there's going to be a second eigenvalue. 207 00:12:40,180 --> 00:12:42,740 And eigenvector. 208 00:12:42,740 --> 00:12:44,410 Now, what about that? 209 00:12:44,410 --> 00:12:50,280 What's a vector, OK, maybe we can just, like, guess it. 210 00:12:50,280 --> 00:12:55,280 A vector that the other -- actually, 211 00:12:55,280 --> 00:12:58,940 this one that I'm thinking of is going to be a vector that has 212 00:12:58,940 --> 00:13:00,630 eigenvalue minus one. 213 00:13:03,240 --> 00:13:06,670 That's going to be my other eigenvalue for this matrix. 214 00:13:06,670 --> 00:13:09,430 It's a -- notice the nice positive or not negative 215 00:13:09,430 --> 00:13:13,240 matrix, but an eigenvalue is going to come out negative. 216 00:13:13,240 --> 00:13:15,520 And can you guess, spot the x that will work for 217 00:13:15,520 --> 00:13:18,910 Times x is supposed to give me zero, right? that? 218 00:13:18,910 --> 00:13:20,960 So I want a, a vector. 219 00:13:20,960 --> 00:13:27,140 When I multiply by A, which reverses the two components, 220 00:13:27,140 --> 00:13:31,120 I want the thing to come out minus the original. 221 00:13:31,120 --> 00:13:34,480 So what shall I send in in that case? 222 00:13:34,480 --> 00:13:37,650 If I send in negative one one. 223 00:13:40,550 --> 00:13:47,150 Then when I apply A, I get I do that multiplication, 224 00:13:47,150 --> 00:13:52,440 and I get one negative one, so it reversed sign. 225 00:13:52,440 --> 00:13:55,840 So Ax is -x. 226 00:13:55,840 --> 00:13:57,710 Lambda is minus one. 227 00:13:57,710 --> 00:14:04,130 Ax -- so Ax was x there and Ax is minus x here. 228 00:14:04,130 --> 00:14:08,080 Can I just mention, like, jump ahead, have, 229 00:14:08,080 --> 00:14:10,550 give a perfectly innocent-looking quadratic 230 00:14:10,550 --> 00:14:14,990 and point out a special little fact about eigenvalues. 231 00:14:14,990 --> 00:14:18,940 n by n matrices will have n eigenvalues. 232 00:14:18,940 --> 00:14:23,390 And I get this matrix4 zero zero zero one, 233 00:14:23,390 --> 00:14:28,390 And it's not like -- suppose n is three or four or more. 234 00:14:28,390 --> 00:14:32,120 It's not so easy to find them. 235 00:14:32,120 --> 00:14:35,570 We'd have a third degree or a fourth degree or an n-th degree 236 00:14:35,570 --> 00:14:36,330 equation. 237 00:14:36,330 --> 00:14:37,960 But here's one nice fact. 238 00:14:37,960 --> 00:14:40,270 There, there's one pleasant fact. we -- 239 00:14:40,270 --> 00:14:42,570 the eigenvalues came out four and two. 240 00:14:42,570 --> 00:14:45,540 That the sum of the eigenvalues equals the sum 241 00:14:45,540 --> 00:14:46,530 down the diagonal. 242 00:14:46,530 --> 00:14:50,150 That's called the trace, and I put that in the lecture 243 00:14:50,150 --> 00:14:54,590 Now I add three I to that matrix. content specifically. 244 00:14:54,590 --> 00:15:03,390 So this is a neat fact, the fact that sthe sum of the lambdas, 245 00:15:03,390 --> 00:15:08,390 add up the lambdas, equals the sum -- 246 00:15:08,390 --> 00:15:11,220 what would you like me to, shall I write that down? 247 00:15:11,220 --> 00:15:17,330 What I'm want to say in words is the sum down the diagonal of A. 248 00:15:17,330 --> 00:15:18,640 Shall I write a11+a22+...+ ann. 249 00:15:18,640 --> 00:15:22,570 That's add up the diagonal entries. 250 00:15:27,470 --> 00:15:31,700 In this example, it's zero. 251 00:15:31,700 --> 00:15:36,560 In other words, once I found this eigenvalue of one, 252 00:15:36,560 --> 00:15:39,110 I knew the other one had to be minus one 253 00:15:39,110 --> 00:15:43,710 in this two by two case, because in the two by two case, which 254 00:15:43,710 --> 00:15:51,970 is a good one to, to, play with, the trace tells you 255 00:15:51,970 --> 00:15:54,804 right away what the other eigenvalue is. 256 00:15:54,804 --> 00:15:56,970 So if I tell you one eigenvalue, you can tell me the 257 00:15:56,970 --> 00:15:57,780 other one. 258 00:15:57,780 --> 00:16:02,270 We'll, we'll have that -- we'll, minus one and eigenvectors one 259 00:16:02,270 --> 00:16:05,850 one and eigenvector minus one we'll see that again. 260 00:16:05,850 --> 00:16:06,350 OK. 261 00:16:06,350 --> 00:16:07,980 Now can I -- 262 00:16:07,980 --> 00:16:11,240 I could give more examples, but maybe it's 263 00:16:11,240 --> 00:16:16,140 time to face the, the equation, Ax equal lambda x, and figure 264 00:16:16,140 --> 00:16:19,810 how are we going to find x and lambda. 265 00:16:19,810 --> 00:16:22,571 And that is lambda one times lambda3 266 00:16:22,571 --> 00:16:23,070 OK. 267 00:16:23,070 --> 00:16:25,590 So this, so the question now is how 268 00:16:25,590 --> 00:16:30,630 to find eigenvalues and eigenvectors. 269 00:16:30,630 --> 00:16:35,430 How to solve, how to solve Ax equal lambda x from the three 270 00:16:35,430 --> 00:16:43,670 x, so it's just I mean, when we've got two unknowns 271 00:16:43,670 --> 00:16:47,500 both in the equation. 272 00:16:47,500 --> 00:16:48,460 OK. 273 00:16:48,460 --> 00:16:51,330 Here's the trick. 274 00:16:51,330 --> 00:16:53,240 Simple idea. 275 00:16:53,240 --> 00:16:58,980 Bring this onto the same side. 276 00:16:58,980 --> 00:16:59,950 Rewrite. 277 00:16:59,950 --> 00:17:05,589 Bring this over as A minus lambda times the identity x 278 00:17:05,589 --> 00:17:09,770 One. equals zero. 279 00:17:09,770 --> 00:17:10,740 Right? 280 00:17:10,740 --> 00:17:14,150 I have Ax minus lambda x, so I brought that over 281 00:17:14,150 --> 00:17:14,260 and I've got zero left on the, on the right-hand side. 282 00:17:14,260 --> 00:17:16,740 What's the relation between that problem and -- 283 00:17:16,740 --> 00:17:19,170 let me write 284 00:17:19,170 --> 00:17:19,980 OK. 285 00:17:19,980 --> 00:17:23,200 I don't know lambda and I don't know x, but I do know something 286 00:17:23,200 --> 00:17:23,700 here. 287 00:17:26,490 --> 00:17:28,460 What I know is if I, if I'm going 288 00:17:28,460 --> 00:17:31,870 to be able to solve this thing, for some x that's not the zero 289 00:17:31,870 --> 00:17:35,760 vector, that's not, that's a useless eigenvector, 290 00:17:35,760 --> 00:17:37,410 doesn't count. 291 00:17:37,410 --> 00:17:44,860 What I know now is that this matrix must be what? 292 00:17:44,860 --> 00:17:48,350 If I'm going to be -- if there is an x -- 293 00:17:48,350 --> 00:17:51,520 I don't -- right now I don't know what it is. 294 00:17:51,520 --> 00:17:54,490 I'm going to find lambda first, actually. 295 00:17:54,490 --> 00:18:00,300 And -- but if there is an x, it tells me that this matrix, 296 00:18:00,300 --> 00:18:05,160 this special combination, which is like the matrix A with 297 00:18:05,160 --> 00:18:10,430 lambda -- shifted by lambda, shifted by lambda I, 298 00:18:10,430 --> 00:18:14,550 that it has to be singular. 299 00:18:14,550 --> 00:18:17,030 This matrix must be singular, otherwise 300 00:18:17,030 --> 00:18:21,510 the only x would be the zero x, and zero matrix.OK. 301 00:18:21,510 --> 00:18:22,700 So this is singular. 302 00:18:22,700 --> 00:18:30,320 And what do I now know about singular matrices? 303 00:18:30,320 --> 00:18:31,520 So, so take three away. 304 00:18:31,520 --> 00:18:35,140 Their determinant is zero. 305 00:18:35,140 --> 00:18:38,790 So I've -- so from the fact that that has to be singular, 306 00:18:38,790 --> 00:18:44,750 I know that the determinant of A minus lambda I has to be zero. 307 00:18:48,590 --> 00:18:52,430 And that, now I've got x out of it. 308 00:18:52,430 --> 00:18:55,680 I've got an equation for lambda, that the key equation -- 309 00:18:55,680 --> 00:19:01,020 it's called the characteristic equation or the eigenvalue 310 00:19:01,020 --> 00:19:01,520 equation. 311 00:19:04,180 --> 00:19:08,370 And that -- in other words, I'm now in a position to find 312 00:19:08,370 --> 00:19:10,480 lambda first. 313 00:19:10,480 --> 00:19:15,940 So -- the idea will be to find lambda first. 314 00:19:18,890 --> 00:19:21,660 And actually, I won't find one lambda, 315 00:19:21,660 --> 00:19:24,120 I'll find N different lambdas. 316 00:19:24,120 --> 00:19:28,440 Well, n lambdas, maybe not n different ones. 317 00:19:28,440 --> 00:19:31,430 A lambda could be repeated. 318 00:19:31,430 --> 00:19:38,040 A repeated lambda is the source of all trouble in 18.06. 319 00:19:38,040 --> 00:19:43,620 So, let's hope for the moment that they're not repeated. 320 00:19:43,620 --> 00:19:48,310 There, there they were different, right? 321 00:19:48,310 --> 00:19:51,920 One and minus one in that, in that, for that permutation. 322 00:19:51,920 --> 00:19:52,800 OK. 323 00:19:52,800 --> 00:19:59,460 So and after I found this lambda, can I just look ahead? 324 00:19:59,460 --> 00:20:02,200 How I going to find x? 325 00:20:02,200 --> 00:20:06,670 After I have found this lambda, the lambda being this -- 326 00:20:06,670 --> 00:20:10,551 one of the numbers that makes this matrix singular. 327 00:20:10,551 --> 00:20:11,550 Their product was eight. 328 00:20:11,550 --> 00:20:15,110 Then of course finding x is just by elimination. 329 00:20:15,110 --> 00:20:15,610 Right? 330 00:20:15,610 --> 00:20:18,630 It's just -- now I've got a singular matrix, 331 00:20:18,630 --> 00:20:20,640 I'm looking for the null space. 332 00:20:20,640 --> 00:20:24,020 We're experts at finding the null space. 333 00:20:24,020 --> 00:20:26,570 You know, you do elimination, you identify 334 00:20:26,570 --> 00:20:31,780 the, the, the pivot columns and so on, you're -- 335 00:20:31,780 --> 00:20:36,900 and, give values to the free variables. 336 00:20:36,900 --> 00:20:39,300 Probably there'll only be one free variable. 337 00:20:39,300 --> 00:20:42,480 We'll give it the value one, like there. 338 00:20:42,480 --> 00:20:44,550 And we find the other variable. 339 00:20:44,550 --> 00:20:45,390 OK. 340 00:20:45,390 --> 00:20:52,610 So let's -- find the x second will be a doable job. 341 00:20:52,610 --> 00:20:54,540 That's my big equation for x. 342 00:20:54,540 --> 00:20:58,070 Let's go, let's look at the first job of finding lambda. 343 00:20:58,070 --> 00:20:59,500 Can I take another example? 344 00:20:59,500 --> 00:21:00,000 OK. 345 00:21:00,000 --> 00:21:02,130 And let's, let's work that one out. 346 00:21:02,130 --> 00:21:02,630 OK. 347 00:21:02,630 --> 00:21:07,590 So let me take the example, say, let me make it easy. 348 00:21:07,590 --> 00:21:09,240 it's just sitting there. 349 00:21:09,240 --> 00:21:12,550 Three three one and one. what do you 350 00:21:12,550 --> 00:21:14,610 know about the complex numbers? 351 00:21:14,610 --> 00:21:16,680 So I've made it easy. 352 00:21:16,680 --> 00:21:19,160 I've made it two by two. 353 00:21:19,160 --> 00:21:20,810 I've made it symmetric. 354 00:21:20,810 --> 00:21:24,530 And I even made it constant down the diagonal. 355 00:21:24,530 --> 00:21:27,830 That a matrix, a perfectly real matrix could 356 00:21:27,830 --> 00:21:32,380 So that -- so the more, like, special properties I stick 357 00:21:32,380 --> 00:21:36,270 into the matrix, the more special outcome I get 358 00:21:36,270 --> 00:21:38,370 for the eigenvalues. 359 00:21:38,370 --> 00:21:42,050 For example, this symmetric matrix, 360 00:21:42,050 --> 00:21:48,130 I know that it'll come out with real eigenvalues. one. 361 00:21:48,130 --> 00:21:52,530 The eigenvalues will turn out to be nice real numbers. 362 00:21:52,530 --> 00:21:57,760 And up in our previous example, that was a symmetric matrix. 363 00:21:57,760 --> 00:22:02,130 Actually, while we're at it, that was a symmetric matrix. 364 00:22:02,130 --> 00:22:05,500 Its eigenvalues were nice real numbers, one and minus one. 365 00:22:05,500 --> 00:22:07,630 And do you notice anything about its eigenvectors? 366 00:22:07,630 --> 00:22:08,970 And what do you notice? 367 00:22:08,970 --> 00:22:11,640 Anything particular about those two vectors, one one and minus 368 00:22:11,640 --> 00:22:14,840 And now comes that thing that I wanted to be reminded of. 369 00:22:14,840 --> 00:22:15,380 one one? 370 00:22:15,380 --> 00:22:18,850 They just happen to be -- no, I can't say they just happen 371 00:22:18,850 --> 00:22:20,610 to be, because that's the whole point, 372 00:22:20,610 --> 00:22:23,390 is that they had to be -- what? 373 00:22:23,390 --> 00:22:25,610 What are they? 374 00:22:25,610 --> 00:22:27,380 They're perpendicular. 375 00:22:27,380 --> 00:22:30,180 The vector, when I -- if I see a vector one one and a one -- 376 00:22:30,180 --> 00:22:33,121 and a minus one one, my mind immediately takes that 377 00:22:33,121 --> 00:22:33,620 dot product. 378 00:22:33,620 --> 00:22:36,570 It's zero. what's the determinant of that matrix? 379 00:22:36,570 --> 00:22:38,140 Those vectors are perpendicular. 380 00:22:38,140 --> 00:22:40,090 That'll happen here too. 381 00:22:40,090 --> 00:22:42,670 Well, let's find the eigenvalues. 382 00:22:42,670 --> 00:22:45,770 Actually, oh, my example's too easy. 383 00:22:45,770 --> 00:22:48,350 My example is too easy. 384 00:22:48,350 --> 00:22:53,550 Let me tell you in advance what's going to happen. 385 00:22:53,550 --> 00:22:56,210 May I? 386 00:22:56,210 --> 00:22:58,510 Or shall I do the determinant of A minus lambda, 387 00:22:58,510 --> 00:23:00,720 and then point out at the end? 388 00:23:00,720 --> 00:23:03,180 Will you remind me at the -- after I've found 389 00:23:03,180 --> 00:23:10,830 the eigenvalues to say why they were -- why they were easy from 390 00:23:10,830 --> 00:23:17,050 That -- it had to be eight, because we factored into lambda 391 00:23:17,050 --> 00:23:20,160 the, from the example we did? 392 00:23:20,160 --> 00:23:23,270 OK, let's do the job here. 393 00:23:23,270 --> 00:23:27,410 Let's compute determinant of A minus lambda I. 394 00:23:27,410 --> 00:23:29,490 So that's a determinant. 395 00:23:29,490 --> 00:23:32,600 And what's, what is this thing? 396 00:23:32,600 --> 00:23:36,090 It's the matrix A with lambda removed from the diagonal. 397 00:23:36,090 --> 00:23:38,570 for this matrix? 398 00:23:38,570 --> 00:23:40,770 So the diagonal matrix is shifted, 399 00:23:40,770 --> 00:23:43,800 and then I'm taking the determinant. 400 00:23:43,800 --> 00:23:44,370 OK. 401 00:23:44,370 --> 00:23:47,390 So I multiply this out. 402 00:23:47,390 --> 00:23:49,340 So what is that determinant? 403 00:23:49,340 --> 00:23:52,520 Do you notice, I didn't take lambda away 404 00:23:52,520 --> 00:23:55,090 from all the entries. 405 00:23:55,090 --> 00:23:56,790 It's lambda I, so it's lambda along the 406 00:23:56,790 --> 00:23:58,040 Lambda plus three x. diagonal. 407 00:23:58,040 --> 00:24:04,040 So I get three minus lambda squared and then minus one, 408 00:24:04,040 --> 00:24:06,380 right? 409 00:24:06,380 --> 00:24:08,000 And I want that to be zero. 410 00:24:08,000 --> 00:24:09,709 And what is A minus lambda I x? 411 00:24:09,709 --> 00:24:11,000 Well, I'm going to simplify it. 412 00:24:11,000 --> 00:24:11,050 And what will I get? 413 00:24:11,050 --> 00:24:11,190 So if I multiply this out, I get lambda squared minus six 414 00:24:11,190 --> 00:24:11,290 What's -- how is this matrix related to that matrix? 415 00:24:11,290 --> 00:24:11,330 lambda plus what? 416 00:24:11,330 --> 00:24:11,350 Plus eight. 417 00:24:11,350 --> 00:24:11,400 But it's out there. 418 00:24:11,400 --> 00:24:11,490 And that I'm going to set to zero. 419 00:24:11,490 --> 00:24:11,560 And I'm going to solve it. 420 00:24:11,560 --> 00:24:11,640 So and it's, it's a quadratic equation. 421 00:24:11,640 --> 00:24:11,750 I can use factorization, I can use the quadratic formula. 422 00:24:11,750 --> 00:24:12,625 I'll get two lambdas. 423 00:24:12,625 --> 00:24:30,690 Before I do it, tell me what's that number six that's 424 00:24:30,690 --> 00:24:41,560 showing up in this equation? 425 00:24:41,560 --> 00:24:48,080 It's the trace. 426 00:24:48,080 --> 00:25:03,300 That number six is three plus three. 427 00:25:03,300 --> 00:25:05,820 And while we're at it, what's the number eight that's 428 00:25:05,820 --> 00:25:08,590 showing up in this equation? 429 00:25:08,590 --> 00:25:10,340 It's the determinant. 430 00:25:10,340 --> 00:25:13,290 That our matrix has determinant eight. 431 00:25:13,290 --> 00:25:16,820 So in a two by two case, it's really nice. 432 00:25:16,820 --> 00:25:21,630 It's lambda squared minus the trace times lambda -- 433 00:25:21,630 --> 00:25:24,430 the trace is the linear coefficient -- 434 00:25:24,430 --> 00:25:27,880 and plus the determinant, the constant term. 435 00:25:27,880 --> 00:25:28,540 OK. 436 00:25:28,540 --> 00:25:32,730 So let's -- can, can we find the roots? 437 00:25:32,730 --> 00:25:36,310 I guess the easy way is to factor that as something times 438 00:25:36,310 --> 00:25:37,980 something. 439 00:25:37,980 --> 00:25:41,280 If we couldn't factor it, then we'd have to use the old 440 00:25:41,280 --> 00:25:46,590 b^2-4ac formula, but I, I think we can factor that into lambda 441 00:25:46,590 --> 00:25:50,870 minus what times lambda minus what? 442 00:25:50,870 --> 00:25:54,850 Can you do that factorization? 443 00:25:54,850 --> 00:25:57,240 Four and two? 444 00:25:57,240 --> 00:25:59,172 Lambda minus four times lambda minus two. 445 00:25:59,172 --> 00:26:00,880 So the, the eigenvalues are four and two. 446 00:26:00,880 --> 00:26:01,210 So the eigenvalues are -- one eigenvalue, lambda one, 447 00:26:01,210 --> 00:26:01,300 Now I'm looking for x, the eigenvector. let's say, 448 00:26:01,300 --> 00:26:01,320 is four. 449 00:26:01,320 --> 00:26:01,490 Lambda two, the other eigenvalue, is two. 450 00:26:01,490 --> 00:26:03,570 The eigenvalues are four and two. 451 00:26:03,570 --> 00:26:08,540 And then I can go for the eigenvectors. 452 00:26:08,540 --> 00:26:15,380 Suppose I have a matrix A, and Ax equal lambda x. 453 00:26:15,380 --> 00:26:16,620 equals zero. 454 00:26:16,620 --> 00:26:20,980 You see I got the eigenvalues first. 455 00:26:20,980 --> 00:26:25,950 So if they, if this had eigenvalue lambda, 456 00:26:25,950 --> 00:26:26,860 Four and two. 457 00:26:26,860 --> 00:26:29,030 Now for the eigenvectors. 458 00:26:29,030 --> 00:26:31,300 So what are the eigenvectors? 459 00:26:31,300 --> 00:26:34,880 They're these guys in the null space when 460 00:26:34,880 --> 00:26:40,090 I take away, when I make the matrix singular by taking 461 00:26:40,090 --> 00:26:42,740 four I or two I away. 462 00:26:42,740 --> 00:26:46,470 So we're -- we got to do those separately. 463 00:26:46,470 --> 00:26:50,100 I'll -- let me find the eigenvector for four first. 464 00:26:50,100 --> 00:26:57,420 So I'll subtract four, so A minus four I is -- 465 00:26:57,420 --> 00:27:00,590 so taking four away will put minus ones there. 466 00:27:04,070 --> 00:27:07,000 And what's the point about that matrix? 467 00:27:07,000 --> 00:27:10,260 If four is an eigenvalue, then A minus four 468 00:27:10,260 --> 00:27:13,220 I had better be a what kind of matrix? 469 00:27:13,220 --> 00:27:14,800 Singular. 470 00:27:14,800 --> 00:27:17,800 If that matrix isn't singular, the four wasn't correct. 471 00:27:17,800 --> 00:27:21,280 But we're OK, that matrix is singular. 472 00:27:21,280 --> 00:27:23,110 And what's the x now? 473 00:27:23,110 --> 00:27:25,290 The x is in the null space. 474 00:27:25,290 --> 00:27:28,910 So what's the x1 that goes with, with the lambda one? 475 00:27:28,910 --> 00:27:32,580 eigenvalue, eigenvector, eigenvalue for this, 476 00:27:32,580 --> 00:27:38,294 So that A -- so this is -- now I'm doing A x1 is lambda one 477 00:27:38,294 --> 00:27:38,794 x1. 478 00:27:41,640 --> 00:27:45,160 So I took A minus lambda one I, that's this matrix, 479 00:27:45,160 --> 00:27:48,630 and now I'm looking for the x1 in its null space, 480 00:27:48,630 --> 00:27:49,510 and who is he? 481 00:27:49,510 --> 00:27:51,760 What's the vector x in the null space? 482 00:27:51,760 --> 00:27:53,380 Of course it's one one. 483 00:27:53,380 --> 00:27:56,310 So that's the eigenvector that goes with that eigenvalue. 484 00:27:56,310 --> 00:27:57,610 So, so now -- 485 00:27:57,610 --> 00:28:00,540 Let's just spend one more minute on this bad 486 00:28:00,540 --> 00:28:02,490 Now how about the eigenvector that 487 00:28:02,490 --> 00:28:04,110 goes with the other eigenvalue? 488 00:28:04,110 --> 00:28:06,040 Can I do that with, with erasing? 489 00:28:06,040 --> 00:28:08,940 I take A minus two I. 490 00:28:08,940 --> 00:28:11,380 So now I take two away from the diagonal, 491 00:28:11,380 --> 00:28:15,670 and that leaves me with a one and a one. 492 00:28:15,670 --> 00:28:19,490 So A minus two I has again produced a singular matrix, 493 00:28:19,490 --> 00:28:21,710 as it had to. 494 00:28:21,710 --> 00:28:24,990 I'm looking for the null space of that guy. 495 00:28:24,990 --> 00:28:27,210 What vector is in its null space? 496 00:28:27,210 --> 00:28:29,320 Well, of course, a whole line of vectors. 497 00:28:32,040 --> 00:28:35,990 So when I say the eigenvector, I'm not speaking correctly. 498 00:28:35,990 --> 00:28:38,490 There's a whole line of eigenvectors, and you just -- 499 00:28:38,490 --> 00:28:40,760 I just want a basis. 500 00:28:40,760 --> 00:28:43,360 And for a line I just want one vector. 501 00:28:43,360 --> 00:28:48,130 But -- You could, you're -- there's some freedom 502 00:28:48,130 --> 00:28:50,860 in choosing that one, but choose a reasonable one. 503 00:28:50,860 --> 00:28:54,200 What's a vector in the null space of that? 504 00:28:54,200 --> 00:28:58,800 Well, the natural vector to pick as the eigenvector 505 00:28:58,800 --> 00:29:01,540 with, with lambda two is minus one one. 506 00:29:05,130 --> 00:29:07,760 If I did elimination on that vector 507 00:29:07,760 --> 00:29:10,520 and set that, the free variable to be one, 508 00:29:10,520 --> 00:29:15,010 I would get minus one and get that eigenvector. 509 00:29:15,010 --> 00:29:22,250 So you see then that I've got eigenvector, 510 00:29:22,250 --> 00:29:32,890 Now the other neat fact is that the determinant, 511 00:29:32,890 --> 00:29:36,906 How are those two matrices related? 512 00:29:40,530 --> 00:29:46,710 Well, one is just three I more than the other one, right? two. 513 00:29:46,710 --> 00:29:52,410 I just took that matrix and I -- 514 00:29:52,410 --> 00:30:29,276 I took this matrix and I added three I. 515 00:30:29,276 --> 00:30:31,900 So my question is, what happened to the minus four times lambda 516 00:30:31,900 --> 00:30:34,720 minus two. eigenvalues and what happened to the eigenvectors? 517 00:30:34,720 --> 00:30:37,530 That's the, that's like the question we keep asking now 518 00:30:37,530 --> 00:30:39,500 in this chapter. 519 00:30:39,500 --> 00:30:42,690 If I, if I do something to the matrix, what happens if I -- 520 00:30:42,690 --> 00:30:44,730 or I know something about the matrix, 521 00:30:44,730 --> 00:30:48,550 what's the what's the conclusion for its eigenvectors 522 00:30:48,550 --> 00:30:49,280 and eigenvalues? 523 00:30:49,280 --> 00:30:54,180 Because -- those eigenvalues and eigenvectors are going to tell 524 00:30:54,180 --> 00:30:57,280 us important information about the matrix. 525 00:30:57,280 --> 00:31:00,970 And here what are we seeing? 526 00:31:00,970 --> 00:31:04,330 What's happening to these eigenvalues, one and minus 527 00:31:04,330 --> 00:31:06,460 one, when I add three I? 528 00:31:10,040 --> 00:31:13,160 It just added three to the eigenvalues. 529 00:31:13,160 --> 00:31:16,640 I got four and two, three more than one and minus 530 00:31:16,640 --> 00:31:18,220 one. 531 00:31:18,220 --> 00:31:19,960 What happened to the eigenvectors? 532 00:31:19,960 --> 00:31:21,490 Nothing at all. 533 00:31:21,490 --> 00:31:25,170 One one is -- and minus -- and one -- and minus one one are -- 534 00:31:25,170 --> 00:31:27,050 is still the eigenvectors. 535 00:31:27,050 --> 00:31:34,200 In other words, simple but useful observation. 536 00:31:34,200 --> 00:31:40,320 If I add three I to a matrix, its eigenvectors don't change 537 00:31:40,320 --> 00:31:42,870 and its eigenvalues are three bigger. 538 00:31:42,870 --> 00:31:44,130 Let's, let's just see why. 539 00:31:44,130 --> 00:32:02,150 Let me keep all this on the same board. but just so you see -- 540 00:32:02,150 --> 00:32:08,410 so I'll try to do that. 541 00:32:08,410 --> 00:32:39,970 this has eigenvalue lambda plus three. 542 00:32:39,970 --> 00:32:46,590 And x, the eigenvector, is the same x for both matrices. 543 00:32:46,590 --> 00:32:48,790 OK. 544 00:32:48,790 --> 00:32:50,195 So that's, great. 545 00:32:53,440 --> 00:32:54,670 Of course, it's special. 546 00:32:54,670 --> 00:32:57,570 We got the new matrix by adding three I. 547 00:32:57,570 --> 00:32:59,500 Suppose I had added another matrix. 548 00:32:59,500 --> 00:33:02,410 Suppose I know the eigenvalues and eigenvectors of A. 549 00:33:02,410 --> 00:33:07,250 So I took A minus lambda I x, and what kind of a matrix I 550 00:33:07,250 --> 00:33:09,510 So this is, this, this little board 551 00:33:09,510 --> 00:33:11,875 here is going to be not so great. 552 00:33:17,730 --> 00:33:21,270 Suppose I have a matrix A and it has an eigenvector x with 553 00:33:21,270 --> 00:33:23,260 an eigenvalue lambda. 554 00:33:23,260 --> 00:33:28,730 You remember, I solve A minus lambda I x 555 00:33:28,730 --> 00:33:31,020 And now I add on some other matrix. 556 00:33:31,020 --> 00:33:34,000 So, so what I'm asking you is, if you know the eigenvalues 557 00:33:34,000 --> 00:33:38,430 of A and you know the eigenvalues of B, 558 00:33:38,430 --> 00:33:43,990 let me say suppose B -- so this is if -- let me put an if here. 559 00:33:43,990 --> 00:33:49,230 If Ax equals lambda x, fine, and B has, eigenvalues, 560 00:33:49,230 --> 00:33:52,240 has eigenvalues -- 561 00:33:52,240 --> 00:33:56,800 what shall we call them? 562 00:33:56,800 --> 00:34:02,270 Alpha, alpha one and alpha -- 563 00:34:02,270 --> 00:34:05,020 let's say -- 564 00:34:05,020 --> 00:34:07,270 I'll use alpha for the eigenvalues of B 565 00:34:07,270 --> 00:34:08,199 for no good reason. 566 00:34:11,489 --> 00:34:16,389 What a- you see what I'm going to ask is, how about A plus B? 567 00:34:20,949 --> 00:34:24,070 Let me, let me give you the, let me give you, 568 00:34:24,070 --> 00:34:27,770 what you might think first. 569 00:34:27,770 --> 00:34:28,730 OK. 570 00:34:28,730 --> 00:34:35,460 If Ax equals lambda x and if B has an eigenvalue alpha, 571 00:34:35,460 --> 00:34:40,630 then I allowed to say -- what's the matter with this argument? 572 00:34:40,630 --> 00:34:43,300 That gave us the constant term eight. 573 00:34:43,300 --> 00:34:44,070 It's wrong. 574 00:34:44,070 --> 00:34:47,130 What I'm going to write up is wrong. 575 00:34:47,130 --> 00:34:50,179 I'm going to say Bx is alpha x. 576 00:34:50,179 --> 00:34:55,909 Add those up, and you get A plus B x equals lambda plus alpha x. 577 00:34:55,909 --> 00:35:00,500 So you would think that if you know the eigenvalues of A 578 00:35:00,500 --> 00:35:03,490 and you knew the eigenvalues of B, 579 00:35:03,490 --> 00:35:07,900 then if you added you would know the eigenvalues of A plus B. 580 00:35:07,900 --> 00:35:11,310 But that's false. 581 00:35:11,310 --> 00:35:17,840 A plus B -- well, when B was three I, that worked great. 582 00:35:17,840 --> 00:35:21,290 But this is not so great. 583 00:35:21,290 --> 00:35:25,720 And what's the matter with that argument there? 584 00:35:25,720 --> 00:35:32,610 We have no reason to believe that x is also 585 00:35:32,610 --> 00:35:33,500 an eigenvector of 586 00:35:33,500 --> 00:35:38,930 B has some eigenvalues, B. but it's 587 00:35:38,930 --> 00:35:43,600 got some different eigenvectors normally. 588 00:35:43,600 --> 00:35:45,520 It's a different matrix. 589 00:35:45,520 --> 00:35:47,250 I don't know anything special. 590 00:35:47,250 --> 00:35:50,150 If I don't know anything special, then as far as I know, 591 00:35:50,150 --> 00:35:52,670 it's got some different eigenvector y, 592 00:35:52,670 --> 00:35:55,790 and when I add I get just rubbish. 593 00:35:55,790 --> 00:35:57,480 I mean, I get -- 594 00:35:57,480 --> 00:35:59,970 I can add, but I don't learn anything. 595 00:35:59,970 --> 00:36:05,380 So not so great is A plus B. 596 00:36:05,380 --> 00:36:09,460 Or A times B. 597 00:36:09,460 --> 00:36:12,670 Normally the eigenvalues of A plus B 598 00:36:12,670 --> 00:36:18,630 or A times B are not eigenvalues of A plus eigenvalues of B. 599 00:36:18,630 --> 00:36:22,500 Ei- eigenvalues are not, like, linear. 600 00:36:22,500 --> 00:36:24,610 Or -- and they don't multiply. 601 00:36:24,610 --> 00:36:27,580 Because, eigenvectors are usually different 602 00:36:27,580 --> 00:36:31,600 and, and there's just no way to find out 603 00:36:31,600 --> 00:36:33,520 what A plus B does to affect 604 00:36:33,520 --> 00:36:34,960 What do I do now? it. 605 00:36:34,960 --> 00:36:35,470 OK. 606 00:36:35,470 --> 00:36:41,660 So that's, like, a caution. 607 00:36:41,660 --> 00:36:44,610 Don't, if B is a multiple of the identity, great, 608 00:36:44,610 --> 00:36:50,340 but if B is some general matrix, then for A plus B you've got 609 00:36:50,340 --> 00:36:54,580 to find -- you've got to solve the eigenvalue problem. 610 00:36:54,580 --> 00:36:59,720 Now I want to do another example that brings out a, 611 00:36:59,720 --> 00:37:02,200 OK. another point about eigenvalues. 612 00:37:02,200 --> 00:37:06,280 Let me make this example a rotation matrix. 613 00:37:06,280 --> 00:37:09,390 possibility of complex numbers. 614 00:37:09,390 --> 00:37:10,170 OK. 615 00:37:10,170 --> 00:37:13,280 So here's another example. 616 00:37:13,280 --> 00:37:16,390 So a rotate -- 617 00:37:16,390 --> 00:37:21,060 oh, I'd better call it Q. 618 00:37:21,060 --> 00:37:26,500 I often use Q for, for, rotations 619 00:37:26,500 --> 00:37:32,600 because those are the, like, very important examples 620 00:37:32,600 --> 00:37:34,580 of orthogonal matrices. 621 00:37:34,580 --> 00:37:38,290 Let me make it a ninety degree rotation. 622 00:37:38,290 --> 00:37:41,709 So -- my matrix is going to be the one that rotates every 623 00:37:41,709 --> 00:37:43,750 And that's the sum, that's lambda one plus lambda 624 00:37:43,750 --> 00:37:46,640 vector by ninety degrees. 625 00:37:46,640 --> 00:37:49,480 So do you remember that matrix? 626 00:37:49,480 --> 00:37:51,950 It's the cosine of ninety degrees, which 627 00:37:51,950 --> 00:37:54,790 is zero, the sine of ninety degrees, 628 00:37:54,790 --> 00:38:03,990 which is one, minus the sine of ninety, the cosine of ninety. 629 00:38:03,990 --> 00:38:09,360 So that matrix deserves the letter Q. 630 00:38:09,360 --> 00:38:15,500 It's an orthogonal matrix, very, very orthogonal matrix. 631 00:38:15,500 --> 00:38:21,420 Now I'm interested in its eigenvalues and eigenvectors. 632 00:38:21,420 --> 00:38:24,500 Two by two, it can't be that tough. 633 00:38:24,500 --> 00:38:27,200 We know that the eigenvalues add to zero. 634 00:38:30,780 --> 00:38:33,310 Actually, we know something already here. 635 00:38:33,310 --> 00:38:35,750 The eigen- what's the sum of the two eigenvalues? 636 00:38:35,750 --> 00:38:38,880 Just tell me what I just said. 637 00:38:38,880 --> 00:38:40,430 Zero, right. 638 00:38:40,430 --> 00:38:42,310 From that trace business. 639 00:38:42,310 --> 00:38:46,550 The sum of the eigenvalues is, is going to come out zero. 640 00:38:46,550 --> 00:38:48,440 And the product of the eigenvalues, 641 00:38:48,440 --> 00:38:50,190 did I tell you about the determinant being 642 00:38:50,190 --> 00:38:50,870 the product of the eigenvalues? 643 00:38:50,870 --> 00:38:50,880 No. 644 00:38:50,880 --> 00:38:50,950 But that's a good thing to know. 645 00:38:50,950 --> 00:38:51,030 We pointed out how that eight appeared in 646 00:38:51,030 --> 00:38:52,863 the, in the quadratic equation. eigenvalues, 647 00:38:52,863 --> 00:39:07,700 we can postpone that evil day, 648 00:39:07,700 --> 00:39:23,700 So let me just say this. 649 00:39:23,700 --> 00:39:49,930 The trace is zero plus zero, obviously. 650 00:39:49,930 --> 00:39:52,675 And that was the determinant. 651 00:39:52,675 --> 00:39:53,175 OK. 652 00:39:56,230 --> 00:39:58,390 What I'm leading up to with this example 653 00:39:58,390 --> 00:40:03,610 is that something's going to go wrong. 654 00:40:03,610 --> 00:40:09,080 Something goes wrong for rotation 655 00:40:09,080 --> 00:40:16,410 because what vector can come out parallel to itself 656 00:40:16,410 --> 00:40:18,820 after a rotation? 657 00:40:18,820 --> 00:40:23,790 If this matrix rotates every vector by ninety degrees, 658 00:40:23,790 --> 00:40:26,880 what could be an eigenvector? 659 00:40:26,880 --> 00:40:31,190 Do you see we're, we're, we're going to have trouble. 660 00:40:31,190 --> 00:40:33,780 eigenvectors are -- 661 00:40:36,060 --> 00:40:36,560 Well. 662 00:40:36,560 --> 00:40:39,220 Our, our picture of eigenvectors, 663 00:40:39,220 --> 00:40:42,410 of, of coming out in the same direction that they went in, 664 00:40:42,410 --> 00:40:45,140 there won't be it. 665 00:40:45,140 --> 00:40:48,820 And with, and with eigenvalues we're going to have trouble. 666 00:40:48,820 --> 00:40:50,440 From these equations. 667 00:40:50,440 --> 00:40:51,710 Let's see. 668 00:40:51,710 --> 00:40:53,780 Why I expecting trouble? 669 00:40:53,780 --> 00:40:56,640 The, the first equation says that the eigenvalues 670 00:40:56,640 --> 00:40:57,320 add to zero. 671 00:40:59,920 --> 00:41:01,170 So there's a plus and a minus. 672 00:41:01,170 --> 00:41:02,211 So I take the eigenvalue. 673 00:41:04,360 --> 00:41:06,490 But then the second equation says 674 00:41:06,490 --> 00:41:09,450 that the product is plus one. 675 00:41:09,450 --> 00:41:10,880 We're in trouble. 676 00:41:10,880 --> 00:41:14,810 But there's a way out. 677 00:41:14,810 --> 00:41:17,860 So how -- let's do the usual stuff. 678 00:41:17,860 --> 00:41:21,700 Look at determinant of Q minus lambda I. 679 00:41:21,700 --> 00:41:27,230 So I'll just follow the rules, take the determinant, 680 00:41:27,230 --> 00:41:31,940 subtract lambda from the diagonal, where I had zeros, 681 00:41:31,940 --> 00:41:34,400 the rest is the same. 682 00:41:34,400 --> 00:41:37,180 Rest of Q is just copied. 683 00:41:37,180 --> 00:41:38,540 Compute that determinant. 684 00:41:38,540 --> 00:41:42,720 OK, so what does that determinant equal? 685 00:41:42,720 --> 00:41:47,740 Lambda squared minus minus one plus what? 686 00:41:51,930 --> 00:41:54,060 What's up? 687 00:41:54,060 --> 00:41:56,020 There's my equation. 688 00:41:56,020 --> 00:41:59,050 My equation for the eigenvalues is lambda 689 00:41:59,050 --> 00:42:00,950 squared plus one equals zero. 690 00:42:00,950 --> 00:42:04,620 What are the eigenvalues lambda one and lambda two? 691 00:42:04,620 --> 00:42:27,340 They're I, whatever that is, and minus it, right. 692 00:42:27,340 --> 00:42:30,240 Those are the right numbers. 693 00:42:30,240 --> 00:42:36,270 To be real numbers even though the matrix was perfectly real. 694 00:42:36,270 --> 00:42:38,190 So this can happen. 695 00:42:41,060 --> 00:42:44,950 Complex numbers are going to -- have to enter eighteen oh six 696 00:42:44,950 --> 00:42:51,500 at this moment. 697 00:42:51,500 --> 00:42:54,600 Boo, right. 698 00:42:54,600 --> 00:42:56,950 All right. 699 00:42:56,950 --> 00:43:02,970 If I just choose good matrices that have real 700 00:43:02,970 --> 00:43:08,500 supposed to have here? 701 00:43:24,360 --> 00:43:35,590 We do know a little information about the, 702 00:43:35,590 --> 00:43:38,672 the two complex numbers. 703 00:43:38,672 --> 00:43:40,380 They're complex conjugates of each other. 704 00:43:45,850 --> 00:43:51,590 If, if lambda is an eigenvalue, then when I change, 705 00:43:51,590 --> 00:43:54,360 when I go -- you remember what complex conjugates are? 706 00:43:54,360 --> 00:43:57,170 You switch the sign of the imaginary part. 707 00:43:57,170 --> 00:43:59,910 Well, this was only imaginary, had no real part, 708 00:43:59,910 --> 00:44:02,690 so we just switched its sign. 709 00:44:02,690 --> 00:44:06,720 So that eigenvalues come in pairs like that, 710 00:44:06,720 --> 00:44:08,440 but they're complex. 711 00:44:08,440 --> 00:44:11,040 A complex conjugate pair. 712 00:44:11,040 --> 00:44:14,530 And that can happen with a perfectly real matrix. 713 00:44:14,530 --> 00:44:17,060 And as a matter of fact -- 714 00:44:17,060 --> 00:44:18,930 so that was my, my point earlier, 715 00:44:18,930 --> 00:44:23,240 that if a matrix was symmetric, it wouldn't happen. 716 00:44:23,240 --> 00:44:27,090 So if we stick to matrices that are symmetric or, like, close 717 00:44:27,090 --> 00:44:32,610 to symmetric, then the eigenvalues will stay real. 718 00:44:32,610 --> 00:44:35,400 But if we move far away from symmetric -- 719 00:44:35,400 --> 00:44:39,520 and that's as far as you can move, because that matrix is -- 720 00:44:39,520 --> 00:44:44,470 how is Q transpose related to Q for that matrix? 721 00:44:44,470 --> 00:44:46,760 That matrix is anti-symmetric. 722 00:44:46,760 --> 00:44:49,520 Q transpose is minus Q. 723 00:44:49,520 --> 00:44:52,070 That's the very opposite of symmetry. 724 00:44:52,070 --> 00:44:54,570 When I flip across the diagonal I get -- 725 00:44:54,570 --> 00:44:56,350 I reverse all the signs. 726 00:44:56,350 --> 00:45:00,840 Those are the guys that have pure imaginary eigenvalues. 727 00:45:00,840 --> 00:45:03,320 So they're the extreme case. 728 00:45:03,320 --> 00:45:07,380 And in between are, are matrices that 729 00:45:07,380 --> 00:45:10,990 are not symmetric or anti-symmetric but, 730 00:45:10,990 --> 00:45:13,710 but they have partly a symmetric part 731 00:45:13,710 --> 00:45:15,110 and an anti-symmetric part. 732 00:45:15,110 --> 00:45:16,660 OK. 733 00:45:16,660 --> 00:45:23,220 So I'm doing a bunch of examples here to show the possibilities. 734 00:45:23,220 --> 00:45:28,320 The good possibilities being perpendicular eigenvectors, 735 00:45:28,320 --> 00:45:30,360 real eigenvalues. 736 00:45:30,360 --> 00:45:33,910 The bad possibilities being complex eigenvalues. 737 00:45:33,910 --> 00:45:37,360 We could say that's bad. 738 00:45:37,360 --> 00:45:39,455 There's another even worse. 739 00:45:42,380 --> 00:45:45,740 I'm getting through the, the bad things here today. 740 00:45:45,740 --> 00:45:51,750 Then, then the next lecture can, can, 741 00:45:51,750 --> 00:45:56,750 can be like pure happiness. 742 00:45:56,750 --> 00:45:57,720 OK. 743 00:45:57,720 --> 00:46:03,260 Here's one more bad thing that could happen. 744 00:46:03,260 --> 00:46:05,940 So I, again, I'll do it with an example. 745 00:46:05,940 --> 00:46:10,740 Suppose my matrix is, suppose I take this three three one 746 00:46:10,740 --> 00:46:13,140 and I change that guy to zero. 747 00:46:18,150 --> 00:46:21,280 What are the eigenvalues of that matrix? 748 00:46:21,280 --> 00:46:22,740 What are the eigenvectors? 749 00:46:22,740 --> 00:46:25,380 This is always our question. 750 00:46:25,380 --> 00:46:26,860 Of course, the next section we're 751 00:46:26,860 --> 00:46:29,580 going to show why are, why do we care. 752 00:46:29,580 --> 00:46:33,121 But for the moment, this lecture is introducing 753 00:46:33,121 --> 00:46:33,620 them. 754 00:46:33,620 --> 00:46:35,990 And let's just find them. 755 00:46:35,990 --> 00:46:36,640 OK. 756 00:46:36,640 --> 00:46:40,530 What are the eigenvalues of that matrix? 757 00:46:40,530 --> 00:46:45,910 Let me tell you -- at a glance we could answer that question. 758 00:46:45,910 --> 00:46:49,490 Because the matrix is triangular. 759 00:46:49,490 --> 00:46:53,540 It's really useful to know -- if you've got properties like 760 00:46:53,540 --> 00:46:55,320 a triangular matrix. 761 00:46:55,320 --> 00:46:57,980 It's very useful to know you can read the eigenvalues 762 00:46:57,980 --> 00:46:58,610 off. 763 00:46:58,610 --> 00:47:01,920 They're right on the diagonal. 764 00:47:01,920 --> 00:47:05,470 So the eigenvalue is three and also three. 765 00:47:05,470 --> 00:47:07,290 Three is a repeated eigenvalue. 766 00:47:07,290 --> 00:47:09,330 But let's see that happen. 767 00:47:09,330 --> 00:47:10,660 Let me do it right. 768 00:47:10,660 --> 00:47:15,370 The determinant of A minus lambda I, what I always 769 00:47:15,370 --> 00:47:17,119 have to do is this determinant. 770 00:47:17,119 --> 00:47:18,660 I take away lambda from the diagonal. 771 00:47:21,600 --> 00:47:24,070 I leave the rest. 772 00:47:24,070 --> 00:47:28,450 I compute the determinant, so I get a three minus lambda 773 00:47:28,450 --> 00:47:32,090 times a three minus lambda. 774 00:47:32,090 --> 00:47:35,680 And nothing. 775 00:47:35,680 --> 00:47:38,860 So that's where the triangular part came in. 776 00:47:38,860 --> 00:47:41,260 Triangular part, the one thing we know about triangular 777 00:47:41,260 --> 00:47:44,660 matrices is the determinant is just the product down 778 00:47:44,660 --> 00:47:45,890 the diagonal. 779 00:47:45,890 --> 00:47:48,990 And in this case, it's this same, repeated -- 780 00:47:48,990 --> 00:47:51,775 so lambda one is one -- 781 00:47:51,775 --> 00:47:53,900 sorry, lambda one is three and lambda two is three. 782 00:47:53,900 --> 00:47:58,540 That was easy. 783 00:47:58,540 --> 00:48:05,900 I mean, no -- why should I be pessimistic about a matrix 784 00:48:05,900 --> 00:48:10,650 whose eigenvalues can be read off right away? 785 00:48:10,650 --> 00:48:14,820 The problem with this matrix is in the eigenvectors. 786 00:48:14,820 --> 00:48:16,450 So let's go to the eigenvectors. 787 00:48:16,450 --> 00:48:18,960 So how do I find the eigenvectors? 788 00:48:18,960 --> 00:48:22,010 I'm looking for a couple of eigenvectors. 789 00:48:22,010 --> 00:48:22,800 Singular, right? 790 00:48:22,800 --> 00:48:31,590 It's supposed to be singular. 791 00:48:31,590 --> 00:48:40,530 And then it's got some vectors -- which it is. 792 00:48:40,530 --> 00:48:59,410 So it's got some vector x in the null space. 793 00:49:12,890 --> 00:49:15,670 And what, what's the, what's -- give me a basis for the null 794 00:49:15,670 --> 00:49:18,360 space for that guy. 795 00:49:18,360 --> 00:49:21,860 Tell me, what's a vector x in the null space, so that'll 796 00:49:21,860 --> 00:49:25,710 be the, the eigenvector that goes with lambda one 797 00:49:25,710 --> 00:49:27,030 equals three. 798 00:49:27,030 --> 00:49:31,030 The eigenvector is -- so what's in the null space? 799 00:49:31,030 --> 00:49:32,180 One zero, is it? 800 00:49:34,700 --> 00:49:35,200 Great. 801 00:49:38,730 --> 00:49:40,600 Now, what's the other eigenvector? 802 00:49:40,600 --> 00:49:47,960 What's, what's the eigenvector that goes with lambda two? 803 00:49:47,960 --> 00:49:51,620 Well, lambda two is three again. 804 00:49:51,620 --> 00:49:53,150 So I get the same thing again. 805 00:49:53,150 --> 00:49:55,500 Give me another vector -- 806 00:49:55,500 --> 00:49:57,750 I want it to be independent. 807 00:49:57,750 --> 00:49:59,340 If I'm going to write down an x2, 808 00:49:59,340 --> 00:50:02,440 I'm never going to let it be dependent on x1. 809 00:50:02,440 --> 00:50:05,240 I'm looking for independent eigenvectors, 810 00:50:05,240 --> 00:50:08,720 and what's the conclusion? 811 00:50:08,720 --> 00:50:11,010 There isn't one. 812 00:50:11,010 --> 00:50:17,050 This is a degenerate matrix. 813 00:50:17,050 --> 00:50:22,960 It's only got one line of eigenvectors instead of two. 814 00:50:22,960 --> 00:50:27,250 It's this possibility of a repeated eigenvalue 815 00:50:27,250 --> 00:50:34,390 opens this further possibility of a shortage of eigenvectors. 816 00:50:34,390 --> 00:50:43,310 And so there's no second independent eigenvector x2. 817 00:50:43,310 --> 00:50:48,010 So it's a matrix, it's a two by two matrix, 818 00:50:48,010 --> 00:50:51,850 but with only one independent eigenvector. 819 00:50:51,850 --> 00:50:56,910 So that will be -- the matrices that -- 820 00:50:56,910 --> 00:51:01,830 where eigenvectors are -- don't give the complete story. 821 00:51:01,830 --> 00:51:02,330 OK. 822 00:51:02,330 --> 00:51:05,930 My lecture on Monday will give the complete story 823 00:51:05,930 --> 00:51:11,290 for all the other matrices. 824 00:51:11,290 --> 00:51:12,360 Thanks. 825 00:51:12,360 --> 00:51:16,640 Have a good weekend. 826 00:51:16,640 --> 00:51:22,480 A real New England weekend.