1 00:00:00,000 --> 00:00:11,200 -- one and -- 2 00:00:11,200 --> 00:00:14,470 the lecture on symmetric matrixes. 3 00:00:14,470 --> 00:00:17,270 So that's the most important class 4 00:00:17,270 --> 00:00:19,960 of matrixes, symmetric matrixes. 5 00:00:19,960 --> 00:00:22,710 A equals A transpose. 6 00:00:22,710 --> 00:00:26,930 So the first points, the main points of the lecture 7 00:00:26,930 --> 00:00:31,170 I'll tell you right away. 8 00:00:31,170 --> 00:00:33,740 What's special about the eigenvalues? 9 00:00:33,740 --> 00:00:35,900 What's special about the eigenvectors? 10 00:00:35,900 --> 00:00:42,000 This is -- the way we now look at a matrix. 11 00:00:42,000 --> 00:00:45,180 We want to know about its eigenvalues and eigenvectors 12 00:00:45,180 --> 00:00:48,580 and if we have a special type of matrix, 13 00:00:48,580 --> 00:00:51,490 that should tell us something about eigenvalues 14 00:00:51,490 --> 00:00:53,220 and eigenvectors. 15 00:00:53,220 --> 00:00:56,800 Like Markov matrixes, they have an eigenvalue equal 16 00:00:56,800 --> 00:00:57,440 one. 17 00:00:57,440 --> 00:01:03,270 Now symmetric matrixes, can I just tell you right off what 18 00:01:03,270 --> 00:01:06,250 the main facts -- the two main facts are? 19 00:01:06,250 --> 00:01:09,510 The eigenvalues of a symmetric matrix, real -- 20 00:01:09,510 --> 00:01:12,320 this is a real symmetric matrix, we -- 21 00:01:12,320 --> 00:01:15,400 talking mostly about real matrixes. 22 00:01:15,400 --> 00:01:17,850 The eigenvalues are also real. 23 00:01:21,440 --> 00:01:25,600 So our examples of rotation matrixes, where -- 24 00:01:25,600 --> 00:01:29,170 where we got E- eigenvalues that were complex, 25 00:01:29,170 --> 00:01:31,750 that won't happen now. 26 00:01:31,750 --> 00:01:34,610 For symmetric matrixes, the eigenvalues are real 27 00:01:34,610 --> 00:01:37,960 and the eigenvectors are also very special. 28 00:01:37,960 --> 00:01:41,330 The eigenvectors are perpendicular, orthogonal, 29 00:01:41,330 --> 00:01:42,520 so which do you prefer? 30 00:01:42,520 --> 00:01:44,300 I'll say perpendicular. 31 00:01:44,300 --> 00:01:46,830 Perp- well, they're both long words. 32 00:01:52,560 --> 00:01:55,290 Okay, right. 33 00:01:55,290 --> 00:02:01,090 So -- I have a -- you should say "why?" 34 00:02:01,090 --> 00:02:06,650 and I'll at least answer why for case one, maybe case two, 35 00:02:06,650 --> 00:02:09,070 the checking the Eigen -- that the eigenvectors are 36 00:02:09,070 --> 00:02:16,330 perpendicular, I'll leave to, the -- to the book. 37 00:02:16,330 --> 00:02:19,640 But let's just realize what -- 38 00:02:19,640 --> 00:02:24,520 well, first I have to say, it -- 39 00:02:24,520 --> 00:02:29,230 it could happen, like for the identity matrix -- 40 00:02:29,230 --> 00:02:31,170 there's a symmetric matrix. 41 00:02:31,170 --> 00:02:33,400 Its eigenvalues are certainly all real, 42 00:02:33,400 --> 00:02:36,460 they're all one for the identity matrix. 43 00:02:36,460 --> 00:02:38,480 What about the eigenvectors? 44 00:02:38,480 --> 00:02:44,020 Well, for the identity, every vector is an eigenvector. 45 00:02:44,020 --> 00:02:46,930 So how can I say they're perpendicular? 46 00:02:46,930 --> 00:02:49,400 What I really mean is the -- they -- 47 00:02:49,400 --> 00:02:57,520 this word are should really be written can be chosen 48 00:02:57,520 --> 00:02:59,230 perpendicular. 49 00:02:59,230 --> 00:03:02,650 That is, if we have -- it's the usual case. 50 00:03:02,650 --> 00:03:05,210 If the eigenvalues are all different, 51 00:03:05,210 --> 00:03:09,280 then each eigenvalue has one line of eigenvectors 52 00:03:09,280 --> 00:03:13,250 and those lines are perpendicular here. 53 00:03:13,250 --> 00:03:16,530 But if an eigenvalue's repeated, then there's 54 00:03:16,530 --> 00:03:20,730 a whole plane of eigenvectors and all I'm saying 55 00:03:20,730 --> 00:03:25,330 is that in that plain, we can choose perpendicular ones. 56 00:03:25,330 --> 00:03:28,520 So that's why it's a can be chosen part, is -- 57 00:03:28,520 --> 00:03:31,160 this is in the case of a repeated eigenvalue where 58 00:03:31,160 --> 00:03:34,160 there's some real, substantial freedom. 59 00:03:34,160 --> 00:03:39,110 But the typical case is different eigenvalues, 60 00:03:39,110 --> 00:03:43,670 all real, one dimensional eigenvector space, 61 00:03:43,670 --> 00:03:46,030 Eigen spaces, and all perpendicular. 62 00:03:46,030 --> 00:03:50,470 So, just -- let's just see the conclusion. 63 00:03:50,470 --> 00:03:56,270 If we accept those as correct, what happens -- 64 00:03:56,270 --> 00:03:59,332 and I also mean that there's a full set of them. 65 00:03:59,332 --> 00:04:01,790 so forgive me for doing such a thing, but, I'll look at the 66 00:04:01,790 --> 00:04:06,480 I -- so that's part of this picture here, that there -- 67 00:04:06,480 --> 00:04:08,880 there's a complete set of eigenvectors, 68 00:04:08,880 --> 00:04:10,160 perpendicular ones. 69 00:04:10,160 --> 00:04:14,260 So, having a complete set of eigenvectors means -- 70 00:04:14,260 --> 00:04:16,600 so normal -- 71 00:04:16,600 --> 00:04:20,649 so the usual -- maybe I put the -- usually -- usual -- 72 00:04:20,649 --> 00:04:26,060 usual case is that the matrix A we can write in terms of its 73 00:04:26,060 --> 00:04:31,610 eigenvalue matrix and its eigenvector matrix this way, 74 00:04:31,610 --> 00:04:33,350 right? 75 00:04:33,350 --> 00:04:36,430 We can do that in the usual case, 76 00:04:36,430 --> 00:04:42,460 but now what's special when the matrix is symmetric? 77 00:04:42,460 --> 00:04:45,740 So this is the usual case, and now 78 00:04:45,740 --> 00:04:48,960 let me go to the symmetric case. 79 00:04:52,260 --> 00:04:55,990 So in the symmetric case, A, this -- 80 00:04:55,990 --> 00:04:58,900 this should become somehow a little special. 81 00:04:58,900 --> 00:05:03,710 Well, the lambdas on the diagonal are still on the 82 00:05:03,710 --> 00:05:04,500 diagonal. 83 00:05:04,500 --> 00:05:08,370 They're -- they're real, but that's where they are. 84 00:05:08,370 --> 00:05:12,160 What about the eigenvector matrix? 85 00:05:12,160 --> 00:05:15,620 So what can I do now special about the eigenvector matrix 86 00:05:15,620 --> 00:05:18,850 when -- when the A itself is symmetric, 87 00:05:18,850 --> 00:05:22,670 that says something good about the eigenvector matrix, 88 00:05:22,670 --> 00:05:25,520 so what is this -- what does this lead to? 89 00:05:25,520 --> 00:05:31,120 This -- these perpendicular eigenvectors, I can not only -- 90 00:05:31,120 --> 00:05:33,590 I can not only guarantee they're perpendicular, 91 00:05:33,590 --> 00:05:36,680 I could also make them unit vectors, no problem, 92 00:05:36,680 --> 00:05:38,910 just s- scale their length to 93 00:05:38,910 --> 00:05:39,640 one. 94 00:05:39,640 --> 00:05:41,260 So what do I have? 95 00:05:41,260 --> 00:05:45,990 I have orthonormal eigenvectors. 96 00:05:49,950 --> 00:05:55,320 And what does that tell me about the eigenvector matrix? 97 00:05:55,320 --> 00:05:59,617 What -- what letter should I now use in place of S -- 98 00:05:59,617 --> 00:06:02,200 I've got -- those two equations are identical,1 remember S has 99 00:06:02,200 --> 00:06:06,080 the eigenvectors in its columns, but now those columns are 100 00:06:06,080 --> 00:06:11,810 orthonormal, so the right letter to use is Q. 101 00:06:11,810 --> 00:06:15,000 So that's where -- so we've got the letter all set up, book. 102 00:06:15,000 --> 00:06:19,730 so this should be Q lambda Q inverse. 103 00:06:19,730 --> 00:06:25,150 Q standing in our minds always for this matrix -- in this case 104 00:06:25,150 --> 00:06:27,550 it's square, it's -- 105 00:06:27,550 --> 00:06:29,915 so these are the Okay. columns of Q, of course. 106 00:06:35,190 --> 00:06:39,040 And one more thing. 107 00:06:39,040 --> 00:06:39,950 What's Q inverse? 108 00:06:43,070 --> 00:06:45,120 For a matrix that has these orthonormal columns, 109 00:06:45,120 --> 00:06:47,370 So I took the dot product -- ye, somehow, it didn't -- 110 00:06:47,370 --> 00:06:52,010 I we know that the inverse is the same as the transpose. 111 00:06:52,010 --> 00:06:56,710 So here is the beautiful -- 112 00:06:56,710 --> 00:06:59,150 there is the -- the great haven't learned anything. 113 00:06:59,150 --> 00:07:03,900 description, the factorization of a symmetric matrix. 114 00:07:03,900 --> 00:07:06,300 And this is, like, one of the famous theorems 115 00:07:06,300 --> 00:07:10,230 of linear algebra, that if I have a symmetric matrix, 116 00:07:10,230 --> 00:07:14,160 it can be factored in this form. 117 00:07:14,160 --> 00:07:17,230 An orthogonal matrix times diagonal times 118 00:07:17,230 --> 00:07:20,270 the transpose of that orthogonal matrix. 119 00:07:20,270 --> 00:07:24,160 And, of course, everybody immediately says yes, 120 00:07:24,160 --> 00:07:28,510 and if this is possible, then that's 121 00:07:28,510 --> 00:07:29,780 clearly symmetric, right? 122 00:07:29,780 --> 00:07:33,890 That -- take -- we've looked at products of three guys like 123 00:07:33,890 --> 00:07:38,550 that and taken their transpose and we got it back again. 124 00:07:38,550 --> 00:07:42,060 So do you -- do you see the beauty of this -- 125 00:07:42,060 --> 00:07:45,030 of this factorization, then? 126 00:07:45,030 --> 00:07:49,800 It -- it completely displays the eigenvalues and eigenvectors 127 00:07:49,800 --> 00:07:53,760 the symmetry of the -- of the whole thing, because -- 128 00:07:53,760 --> 00:07:56,990 that product, Q times lambda times Q transpose, 129 00:07:56,990 --> 00:08:02,200 if I transpose it, it -- this comes in this position and we 130 00:08:02,200 --> 00:08:04,260 get that matrix back again. 131 00:08:04,260 --> 00:08:08,450 So that's -- in mathematics, that's called the spectral 132 00:08:08,450 --> 00:08:12,260 Spectrum is the set of eigenvalues of a matrix. 133 00:08:12,260 --> 00:08:12,900 theorem. 134 00:08:12,900 --> 00:08:17,410 Spec- it somehow comes from the idea of the spectrum of light 135 00:08:17,410 --> 00:08:22,330 as a combination of pure things -- 136 00:08:22,330 --> 00:08:28,060 where our matrix is broken down into pure eigenvalues 137 00:08:28,060 --> 00:08:30,100 and eigenvectors -- 138 00:08:30,100 --> 00:08:35,940 in mechanics it's often called the principle axis theorem. 139 00:08:35,940 --> 00:08:37,100 It's very useful. 140 00:08:37,100 --> 00:08:40,080 It means that if you have -- 141 00:08:40,080 --> 00:08:42,730 we'll see it geometrically. 142 00:08:42,730 --> 00:08:45,020 It means that if I have some material -- 143 00:08:45,020 --> 00:08:49,960 if I look at the right axis, it becomes diagonal, it becomes -- 144 00:08:49,960 --> 00:08:50,460 the -- the 145 00:08:50,460 --> 00:08:51,530 I- I've done something dumb, because I've got the -- 146 00:08:51,530 --> 00:08:52,700 I should've taken the dot product of this guy here with 147 00:08:52,700 --> 00:08:55,180 -- that's directions don't couple together. 148 00:08:55,180 --> 00:08:55,920 Okay. 149 00:08:55,920 --> 00:08:59,420 So that's -- that -- that's what to remember from -- 150 00:08:59,420 --> 00:09:00,770 from this lecture. 151 00:09:00,770 --> 00:09:06,020 Now, I would like to say why are the eigenvalues real? 152 00:09:06,020 --> 00:09:07,080 Can I do that? 153 00:09:07,080 --> 00:09:10,200 So -- so -- because that -- something useful comes out. 154 00:09:10,200 --> 00:09:16,251 So I'll just come back -- come to that question why real 155 00:09:16,251 --> 00:09:16,750 eigenvalues? 156 00:09:22,910 --> 00:09:24,120 Okay. 157 00:09:24,120 --> 00:09:25,920 So I have to start from the only thing 158 00:09:25,920 --> 00:09:28,435 we know, Ax equal lambda x. 159 00:09:33,710 --> 00:09:34,560 Okay. 160 00:09:34,560 --> 00:09:37,090 But as far as I know at this moment, 161 00:09:37,090 --> 00:09:40,100 lambda could be complex. 162 00:09:40,100 --> 00:09:44,570 I'm going to prove it's not -- and x could be complex. 163 00:09:44,570 --> 00:09:48,370 In fact, for the moment, even A could be -- 164 00:09:48,370 --> 00:09:51,120 we could even think, well, what happens if A is complex? 165 00:09:51,120 --> 00:09:53,750 Well, one thing we can always do -- this is -- 166 00:09:53,750 --> 00:09:56,000 this is like always -- 167 00:09:56,000 --> 00:09:58,080 always okay -- 168 00:09:58,080 --> 00:10:03,030 I can -- if I have an equation, I can take the complex 169 00:10:03,030 --> 00:10:05,130 conjugate of everything. 170 00:10:05,130 --> 00:10:09,610 That's -- no -- no -- so A conjugate x conjugate equal 171 00:10:09,610 --> 00:10:16,036 lambda conjugate x conjugate, it just means that everywhere over 172 00:10:16,036 --> 00:10:18,410 here that there was a -- an equals x bar transpose lambda 173 00:10:18,410 --> 00:10:21,780 bar x bar. i, then here I changed it to a-i. 174 00:10:21,780 --> 00:10:24,500 That's -- that -- you know that that step -- 175 00:10:24,500 --> 00:10:28,100 that conjugate business, that a+ib, 176 00:10:28,100 --> 00:10:31,650 if I conjugate it it's a-ib. 177 00:10:31,650 --> 00:10:36,820 That's the meaning of conjugate -- and products behave right, 178 00:10:36,820 --> 00:10:39,390 I can conjugate every factor. 179 00:10:39,390 --> 00:10:43,640 So I haven't done anything yet except to say what would be 180 00:10:43,640 --> 00:10:53,200 true if, x -- in any case, even if x and lambda were complex. 181 00:10:53,200 --> 00:10:56,130 Of course, our -- we're speaking about real matrixes A, 182 00:10:56,130 --> 00:10:58,630 so I can take that out. 183 00:10:58,630 --> 00:11:04,050 Actually, this already tells me something about real matrixes. 184 00:11:04,050 --> 00:11:07,060 I haven't used any assumption of A -- 185 00:11:07,060 --> 00:11:08,950 A transpose yet. 186 00:11:08,950 --> 00:11:12,440 Symmetry is waiting in the wings to be used. 187 00:11:12,440 --> 00:11:18,580 This tells me that if a real matrix has an eigenvalue lambda 188 00:11:18,580 --> 00:11:22,040 what I was going to do. and an eigenvector x, it also has -- 189 00:11:22,040 --> 00:11:25,040 another of its eigenvalues is lambda bar 190 00:11:25,040 --> 00:11:27,800 with eigenvector x bar. 191 00:11:27,800 --> 00:11:32,500 Real matrixes, the eigenvalues come in lambda, lambda bar -- 192 00:11:32,500 --> 00:11:37,500 the complex eigenvalues come in lambda and lambda bar pairs. 193 00:11:37,500 --> 00:11:39,600 But, of course, I'm aiming to show 194 00:11:39,600 --> 00:11:44,190 that they're not complex at all, here, by getting symmetry in. 195 00:11:44,190 --> 00:11:46,100 So how I going to use symmetry? 196 00:11:46,100 --> 00:11:51,630 I'm going to transpose this equation to x bar 197 00:11:51,630 --> 00:11:59,040 transpose A transpose equals x bar transpose lambda bar. 198 00:11:59,040 --> 00:12:03,150 That's just a number, so I don't mind wear I put that number. 199 00:12:03,150 --> 00:12:05,750 This is -- this is -- 200 00:12:05,750 --> 00:12:06,480 this is a -- then 201 00:12:06,480 --> 00:12:08,250 again okay. 202 00:12:08,250 --> 00:12:10,000 Ax equals lambda x bar transpose x, right? 203 00:12:10,000 --> 00:12:14,880 But now I'm ready to use symmetry. 204 00:12:14,880 --> 00:12:18,890 I'm ready -- so this was all just mechanics. 205 00:12:18,890 --> 00:12:22,040 Now -- now comes the moment to say, okay, 206 00:12:22,040 --> 00:12:24,040 if the matrix is this from the right with x bar, 207 00:12:24,040 --> 00:12:25,665 I get x bar transpose Ax bar symmetric, 208 00:12:25,665 --> 00:12:29,030 then this A transpose is the same as A. 209 00:12:29,030 --> 00:12:31,910 You see, at that moment I used the assumption. 210 00:12:31,910 --> 00:12:34,380 Now let me finish the discussion. 211 00:12:34,380 --> 00:12:37,610 Here -- here's the way I finish. 212 00:12:37,610 --> 00:12:42,460 I look at this original equation and I take the inner product. 213 00:12:42,460 --> 00:12:45,170 I multiply both sides by -- 214 00:12:45,170 --> 00:12:46,680 oh, maybe I'll do it with this one. 215 00:12:49,260 --> 00:12:51,540 I take -- 216 00:12:51,540 --> 00:12:55,200 I multiply both sides by x bar transpose. 217 00:12:55,200 --> 00:12:58,670 x bar transpose Ax bar equals lambda 218 00:12:58,670 --> 00:13:03,620 bar x bar transpose x bar. 219 00:13:03,620 --> 00:13:04,470 Okay, fine. 220 00:13:08,330 --> 00:13:12,170 All right, now what's the other one? 221 00:13:12,170 --> 00:13:16,880 Oh, for the other one I'll probably use this guy. 222 00:13:16,880 --> 00:13:20,580 A- I happy about this? 223 00:13:20,580 --> 00:13:21,080 No. 224 00:13:21,080 --> 00:13:22,910 For some reason I'm not. 225 00:13:22,910 --> 00:13:26,830 I'm -- I want to -- 226 00:13:26,830 --> 00:14:04,850 if I take the inner product of Okay. 227 00:14:24,690 --> 00:14:26,660 So that -- that was -- that's fine. 228 00:14:26,660 --> 00:14:30,230 That comes directly from that, multiplying both sides by x bar 229 00:14:30,230 --> 00:14:33,860 transpose, but now I'd like to get -- 230 00:14:33,860 --> 00:14:38,320 why do I have x bars over there? 231 00:14:38,320 --> 00:14:40,860 Oh, yes. 232 00:14:40,860 --> 00:14:43,161 Forget this. 233 00:14:43,161 --> 00:14:43,660 Okay. 234 00:14:43,660 --> 00:14:45,820 On this one -- right. 235 00:14:45,820 --> 00:14:49,010 On this one, I took it like that, I multiply on the right 236 00:14:49,010 --> 00:14:50,400 by x. 237 00:14:50,400 --> 00:14:54,080 That's the idea. 238 00:14:54,080 --> 00:14:55,540 Okay. 239 00:14:55,540 --> 00:15:02,150 Now why I happier with this situation now? 240 00:15:02,150 --> 00:15:04,180 A proof is coming here. 241 00:15:04,180 --> 00:15:10,140 Because I compare this guy with this one. 242 00:15:10,140 --> 00:15:12,570 And they have the same left hand side. 243 00:15:12,570 --> 00:15:14,410 So they have the same right hand side. 244 00:15:14,410 --> 00:15:16,240 So comparing those two, can -- 245 00:15:16,240 --> 00:15:19,910 I'll raise the board to do this comparison -- 246 00:15:19,910 --> 00:15:25,150 this thing, lambda x bar transpose x 247 00:15:25,150 --> 00:15:32,880 is equal to lambda bar x bar transpose x. 248 00:15:32,880 --> 00:15:33,380 Okay. 249 00:15:35,890 --> 00:15:38,190 And the conclusion I'm going to reach -- 250 00:15:38,190 --> 00:15:43,280 I -- I on the right track here? 251 00:15:43,280 --> 00:15:44,830 The conclusion I'm going to reach 252 00:15:44,830 --> 00:15:47,060 is lambda equal lambda bar. 253 00:15:52,170 --> 00:15:55,330 I would have to track down the other possibility that this -- 254 00:15:55,330 --> 00:15:58,400 this thing is zero, but let me -- 255 00:15:58,400 --> 00:16:01,330 oh -- oh, yes, that's important. 256 00:16:01,330 --> 00:16:03,370 It's not zero. 257 00:16:03,370 --> 00:16:08,760 So once I know that this isn't zero, I just cancel it 258 00:16:08,760 --> 00:16:11,220 and I learn that lambda equals lambda bar. 259 00:16:11,220 --> 00:16:14,360 And so what can you -- do you -- 260 00:16:14,360 --> 00:16:17,823 have you got the reasoning altogether? 261 00:16:20,640 --> 00:16:24,130 What does this tell us? 262 00:16:24,130 --> 00:16:27,750 Lambda's an eigenvalue of this symmetric matrix. 263 00:16:27,750 --> 00:16:30,300 We've just proved that it equaled lambda bar, 264 00:16:30,300 --> 00:16:34,630 so we have just proved that lambda is real, 265 00:16:34,630 --> 00:16:35,710 right? 266 00:16:35,710 --> 00:16:39,980 If, if a number is equal to its own complex conjugate, 267 00:16:39,980 --> 00:16:42,290 then there's no imaginary part at all. 268 00:16:42,290 --> 00:16:43,280 The number is real. 269 00:16:43,280 --> 00:16:45,435 So lambda is real. 270 00:16:48,260 --> 00:16:49,380 Good. 271 00:16:49,380 --> 00:16:51,740 Good. 272 00:16:51,740 --> 00:16:56,310 Now, what -- but it depended on this little expression, 273 00:16:56,310 --> 00:16:59,040 on knowing that that wasn't zero, 274 00:16:59,040 --> 00:17:03,960 so that I could cancel it out -- so can we just take a second 275 00:17:03,960 --> 00:17:05,400 on that one? 276 00:17:05,400 --> 00:17:08,650 Because it's an important quantity. 277 00:17:08,650 --> 00:17:11,160 x bar transpose x. 278 00:17:11,160 --> 00:17:18,280 Okay, now remember, as far as we know, x is complex. 279 00:17:18,280 --> 00:17:20,650 So this is -- 280 00:17:20,650 --> 00:17:25,260 here -- x is complex, x has these components, x1, 281 00:17:25,260 --> 00:17:28,310 x2 down to xn. 282 00:17:28,310 --> 00:17:35,890 And x bar transpose, well, it's transposed and it's conjugated, 283 00:17:35,890 --> 00:17:42,570 so that's x1 conjugated x2 conjugated up to xn conjugated. 284 00:17:42,570 --> 00:17:43,520 I'm -- I'm -- 285 00:17:43,520 --> 00:17:46,390 I'm really reminding you of crucial facts 286 00:17:46,390 --> 00:17:48,440 about complex numbers that are going 287 00:17:48,440 --> 00:17:51,230 to come into the next lecture as well as this one. 288 00:17:51,230 --> 00:17:57,700 So w- what can you tell me about that product -- 289 00:17:57,700 --> 00:18:01,150 I -- I guess what I'm trying to say is, 290 00:18:01,150 --> 00:18:05,830 if I had a complex vector, this would be the quantity I would 291 00:18:05,830 --> 00:18:06,330 -- 292 00:18:06,330 --> 00:18:07,730 I would like. 293 00:18:07,730 --> 00:18:09,310 This is the quantity I like. 294 00:18:09,310 --> 00:18:13,390 I would take the vector times its transpose -- now what -- 295 00:18:13,390 --> 00:18:17,070 what happens usually if I take a vector -- a -- a -- x transpose 296 00:18:17,070 --> 00:18:18,070 x? 297 00:18:18,070 --> 00:18:22,330 I mean, that's a quantity we see all the time, x transpose x. 298 00:18:22,330 --> 00:18:25,200 That's the length of x squared, right? 299 00:18:25,200 --> 00:18:28,350 That's this positive length squared, it's Pythagoras, 300 00:18:28,350 --> 00:18:31,640 it's x1 squared plus x2 squared and so on. 301 00:18:31,640 --> 00:18:36,100 Now our vector's complex, and you see the effect? 302 00:18:36,100 --> 00:18:39,090 I'm conjugating one of these guys. 303 00:18:39,090 --> 00:18:41,180 So now when I do this multiplication, 304 00:18:41,180 --> 00:18:49,370 I have x1 bar times x1 and x2 bar times x2 and so on. 305 00:18:49,370 --> 00:18:53,640 So this is an -- this is sum a+ib. 306 00:18:53,640 --> 00:18:57,960 And this is sum a-ib. 307 00:18:57,960 --> 00:19:02,440 I mean, what's the point here? 308 00:19:02,440 --> 00:19:05,450 What's the point -- when I multiply a number by its 309 00:19:05,450 --> 00:19:11,480 conjugate, a complex number by its conjugate, what do I get? 310 00:19:11,480 --> 00:19:16,140 I get a n- the -- the imaginary part is gone. 311 00:19:16,140 --> 00:19:20,680 When I multiply a+ib by its conjugate, what's -- 312 00:19:20,680 --> 00:19:23,400 what's the result of that -- of each of those separate little 313 00:19:23,400 --> 00:19:25,210 multiplications? 314 00:19:25,210 --> 00:19:29,970 There's an a squared and -- and what -- how many -- what's -- 315 00:19:29,970 --> 00:19:34,010 b squared comes in with a plus or a minus? 316 00:19:34,010 --> 00:19:35,290 A plus. 317 00:19:35,290 --> 00:19:39,260 i times minus i is a plus b squared. 318 00:19:39,260 --> 00:19:41,240 And what about the imaginary part? 319 00:19:43,830 --> 00:19:46,770 Gone, right? 320 00:19:46,770 --> 00:19:49,050 An iab and a minus iab. 321 00:19:49,050 --> 00:19:52,870 So this -- this is the right thing to do. 322 00:19:52,870 --> 00:20:00,600 If you want a decent answer, then multiply numbers 323 00:20:00,600 --> 00:20:02,910 by their conjugates. 324 00:20:02,910 --> 00:20:08,900 Multiply vectors by the conjugates of x transpose. 325 00:20:08,900 --> 00:20:13,750 So this quantity is positive, this quantity is positive -- 326 00:20:13,750 --> 00:20:17,460 the whole thing is positive except for the zero vector 327 00:20:17,460 --> 00:20:23,010 and that allows me to know that this is a positive number, 328 00:20:23,010 --> 00:20:27,220 which I safely cancel out and I reach the conclusion. 329 00:20:27,220 --> 00:20:33,290 So actually, in this discussion here, I've done two things. 330 00:20:33,290 --> 00:20:35,690 If I reached the conclusion that lambda's 331 00:20:35,690 --> 00:20:38,510 real, which I wanted to do. 332 00:20:38,510 --> 00:20:41,600 But at the same time, we sort of saw what 333 00:20:41,600 --> 00:20:43,790 to do if things were complex. 334 00:20:43,790 --> 00:20:48,490 If a vector is complex, then it's x bar transpose x, 335 00:20:48,490 --> 00:20:55,350 this is its length squared. 336 00:20:55,350 --> 00:20:59,510 And as I said, the next lecture Monday, we'll -- 337 00:20:59,510 --> 00:21:03,220 we'll repeat that this is the right thing and then do 338 00:21:03,220 --> 00:21:06,280 the right thing for matrixes and all other -- 339 00:21:06,280 --> 00:21:10,950 all other, complex possibilities. 340 00:21:10,950 --> 00:21:12,070 Okay. 341 00:21:12,070 --> 00:21:17,070 But the main point, then, is that the eigenvalues 342 00:21:17,070 --> 00:21:20,720 of a symmetric matrix, it just -- do you -- do -- 343 00:21:20,720 --> 00:21:23,620 where did we use symmetry, by the way? 344 00:21:23,620 --> 00:21:24,860 We used it here, right? 345 00:21:24,860 --> 00:21:27,870 Let -- can I just -- 346 00:21:27,870 --> 00:21:31,980 let -- suppose A was a complex. 347 00:21:31,980 --> 00:21:34,800 Suppose A had been a complex number. 348 00:21:34,800 --> 00:21:37,230 Could -- could I have made all this work? 349 00:21:37,230 --> 00:21:40,780 If A was a complex number -- complex matrix, 350 00:21:40,780 --> 00:21:45,070 then here I should have written A bar. 351 00:21:45,070 --> 00:21:47,600 I erased the bar because I assumed A was real. 352 00:21:47,600 --> 00:21:50,560 But now let's suppose for a moment it's not. 353 00:21:50,560 --> 00:21:55,260 Then when I took this step, what should I have? 354 00:21:55,260 --> 00:21:56,710 What did I do on that step? 355 00:21:56,710 --> 00:21:57,820 I transposed. 356 00:21:57,820 --> 00:22:00,215 So I should have A bar transpose. 357 00:22:03,560 --> 00:22:05,990 In the symmetric case, that was A, 358 00:22:05,990 --> 00:22:08,590 and that's what made everything work, right? 359 00:22:08,590 --> 00:22:12,720 This -- this led immediately to that. 360 00:22:12,720 --> 00:22:17,840 This one led immediately to this when the matrix was real, 361 00:22:17,840 --> 00:22:20,300 so that didn't matter, and it was symmetric, 362 00:22:20,300 --> 00:22:21,790 so that didn't matter. 363 00:22:21,790 --> 00:22:23,600 Then I got A. 364 00:22:23,600 --> 00:22:26,760 But -- so now I just get to ask you. 365 00:22:26,760 --> 00:22:30,900 Suppose the matrix had been complex. 366 00:22:30,900 --> 00:22:35,620 What's the right equivalent of sym- symmetry? 367 00:22:38,320 --> 00:22:41,480 So the good matrix -- so here, let me say -- 368 00:22:41,480 --> 00:22:52,776 good matrixes -- by good I mean real lambdas and perpendicular 369 00:22:52,776 --> 00:22:53,276 x-s. 370 00:22:57,310 --> 00:23:02,860 And tell me now, which matrixes are good? 371 00:23:02,860 --> 00:23:04,800 If they're -- 372 00:23:04,800 --> 00:23:08,070 If they're real matrixes, the good ones 373 00:23:08,070 --> 00:23:10,790 are symmetric, because then everything went through. 374 00:23:10,790 --> 00:23:13,010 The -- so the good -- 375 00:23:13,010 --> 00:23:15,050 I'm saying now what's good. 376 00:23:15,050 --> 00:23:17,430 This is -- this is -- these are the good matrixes. 377 00:23:17,430 --> 00:23:21,030 They have real eigenvalues, perpendicular eigenvectors -- 378 00:23:21,030 --> 00:23:27,910 good means A equal A transpose if real. 379 00:23:27,910 --> 00:23:30,560 Then -- then that was what -- our proof worked. 380 00:23:30,560 --> 00:23:35,690 But if A is complex, all -- our proof will still work provided 381 00:23:35,690 --> 00:23:38,060 A bar transpose is A. 382 00:23:38,060 --> 00:23:41,740 Do you see what I'm saying? 383 00:23:41,740 --> 00:23:47,700 I'm saying if we have complex matrixes and we want to say are 384 00:23:47,700 --> 00:23:51,330 they -- are they as good as symmetric matrixes, 385 00:23:51,330 --> 00:23:56,750 then we should not only transpose the thing, 386 00:23:56,750 --> 00:23:58,760 but conjugate it. 387 00:23:58,760 --> 00:24:00,800 Those are good matrixes. 388 00:24:00,800 --> 00:24:03,090 And of course, the most important 389 00:24:03,090 --> 00:24:06,980 s- the most important case is when they're real, 390 00:24:06,980 --> 00:24:09,410 this part doesn't matter and I just have 391 00:24:09,410 --> 00:24:11,330 A equal A transpose symmetric. 392 00:24:11,330 --> 00:24:12,070 Do you -- I -- 393 00:24:12,070 --> 00:24:15,260 I'll just repeat that. 394 00:24:15,260 --> 00:24:20,290 The good matrixes, if complex, are these. 395 00:24:20,290 --> 00:24:23,590 If real, that doesn't make any difference 396 00:24:23,590 --> 00:24:25,900 so I'm just saying symmetric. 397 00:24:25,900 --> 00:24:30,530 And of course, 99% of examples and applications 398 00:24:30,530 --> 00:24:34,740 to the matrixes are real and we don't have that 399 00:24:34,740 --> 00:24:38,230 and then symmetric is the key property. 400 00:24:38,230 --> 00:24:40,950 Okay. 401 00:24:40,950 --> 00:24:48,170 So that -- that's, these main facts and now let me just -- 402 00:24:48,170 --> 00:24:53,690 let me just -- so that's this x bar transpose x, okay. 403 00:24:53,690 --> 00:24:59,210 So I'll just, write it once more in this form. 404 00:24:59,210 --> 00:25:05,370 So perpendicular orthonormal eigenvectors, real eigenvalues, 405 00:25:05,370 --> 00:25:08,410 transposes of orthonormal eigenvectors. 406 00:25:08,410 --> 00:25:13,690 That's the symmetric case, A equal A transpose. 407 00:25:13,690 --> 00:25:15,320 Okay. 408 00:25:15,320 --> 00:25:18,030 Good. 409 00:25:18,030 --> 00:25:23,340 Actually, I'll even take one more step here. 410 00:25:23,340 --> 00:25:25,860 Suppose -- I -- 411 00:25:25,860 --> 00:25:29,350 I can break this down to show you really 412 00:25:29,350 --> 00:25:34,000 what that says about a symmetric matrix. 413 00:25:34,000 --> 00:25:35,180 I can break that down. 414 00:25:35,180 --> 00:25:40,110 Let me here -- here go these eigenvectors. 415 00:25:40,110 --> 00:25:46,540 I -- here go these eigenvalues, lambda one, lambda two and so 416 00:25:46,540 --> 00:25:47,130 on. 417 00:25:47,130 --> 00:25:50,355 Here go these eigenvectors transposed. 418 00:25:54,870 --> 00:25:59,640 And what happens if I actually do out that multiplication? 419 00:25:59,640 --> 00:26:03,630 Do you see what will happen? 420 00:26:03,630 --> 00:26:07,700 There's lambda one times q1 transpose. 421 00:26:07,700 --> 00:26:11,850 So the first row here is just lambda one q1 transpose. 422 00:26:11,850 --> 00:26:15,650 If I multiply column times row -- 423 00:26:15,650 --> 00:26:17,850 you remember I could do that? 424 00:26:17,850 --> 00:26:24,180 When I multiply matrixes, I can multiply columns times rows? 425 00:26:24,180 --> 00:26:27,130 So when I do that, I get lambda one and then 426 00:26:27,130 --> 00:26:31,900 the column and then the row and then 427 00:26:31,900 --> 00:26:34,885 lambda two and then the column and the row. 428 00:26:41,770 --> 00:26:46,980 Every symmetric matrix breaks up into these pieces. 429 00:26:46,980 --> 00:26:54,950 So these pieces have real lambdas and they have these 430 00:26:54,950 --> 00:26:56,833 Eigen -- these orthonormal eigenvectors. 431 00:27:00,490 --> 00:27:04,560 And, maybe you even could tell me what kind of a matrix 432 00:27:04,560 --> 00:27:07,370 have I got there? 433 00:27:07,370 --> 00:27:12,750 Suppose I take a unit vector times its transpose? 434 00:27:12,750 --> 00:27:17,620 So column times row, I'm getting a matrix. 435 00:27:17,620 --> 00:27:22,040 That's a matrix with a special name. 436 00:27:22,040 --> 00:27:24,290 What's it's -- what kind of a matrix is it? 437 00:27:24,290 --> 00:27:27,860 We've seen those matrixes, now, in chapter four. 438 00:27:27,860 --> 00:27:32,440 It's -- is A A transpose with a unit vector, 439 00:27:32,440 --> 00:27:36,210 so I don't have to divide by A transpose A. 440 00:27:36,210 --> 00:27:40,580 That matrix is a projection matrix. 441 00:27:40,580 --> 00:27:41,920 That's a projection matrix. 442 00:27:41,920 --> 00:27:46,550 It's symmetric and if I square it there'll be another -- 443 00:27:46,550 --> 00:27:50,040 there'll be a q1 transpose q1, which is one. 444 00:27:50,040 --> 00:27:53,820 So I'll get that matrix back again. 445 00:27:53,820 --> 00:27:57,070 Every -- so every symmetric matrix -- 446 00:27:57,070 --> 00:28:06,740 every symmetric matrix is a combination of -- 447 00:28:06,740 --> 00:28:14,230 of mutually perpendicular -- so perpendicular projection 448 00:28:14,230 --> 00:28:15,500 matrixes. 449 00:28:15,500 --> 00:28:17,442 Projection matrixes. 450 00:28:20,610 --> 00:28:21,530 Okay. 451 00:28:21,530 --> 00:28:23,940 That's another way that people like 452 00:28:23,940 --> 00:28:27,770 to think of the spectral theorem, 453 00:28:27,770 --> 00:28:31,790 that every symmetric matrix can be broken up that way. 454 00:28:31,790 --> 00:28:35,220 That -- I guess at this moment -- 455 00:28:35,220 --> 00:28:36,800 first I haven't done an example. 456 00:28:36,800 --> 00:28:41,700 I could create a symmetric matrix, check that it's -- 457 00:28:41,700 --> 00:28:44,430 find its eigenvalues, they would come out real, 458 00:28:44,430 --> 00:28:47,710 find its eigenvectors, they would come out perpendicular 459 00:28:47,710 --> 00:28:51,230 and you would see it in numbers, but maybe I'll leave it here 460 00:28:51,230 --> 00:28:54,650 for the moment in letters. 461 00:28:54,650 --> 00:28:58,520 Oh, I -- maybe I will do it with numbers, for this reason. 462 00:28:58,520 --> 00:29:02,720 Because there's one more remarkable fact. 463 00:29:02,720 --> 00:29:05,730 Can I just put this further great fact 464 00:29:05,730 --> 00:29:07,890 about symmetric matrixes on the board? 465 00:29:11,720 --> 00:29:13,560 When I have symmetric matrixes, I 466 00:29:13,560 --> 00:29:18,140 know their eigenvalues are So then I can get interested 467 00:29:18,140 --> 00:29:22,040 in the question are they positive real. or negative? 468 00:29:22,040 --> 00:29:23,980 And you remember why that's important. 469 00:29:23,980 --> 00:29:27,740 For differential equations, that decides between instability 470 00:29:27,740 --> 00:29:29,920 and stability. 471 00:29:29,920 --> 00:29:32,620 So I'm -- after I know they're real, 472 00:29:32,620 --> 00:29:34,970 then the next question is are they positive, 473 00:29:34,970 --> 00:29:37,480 are they negative? 474 00:29:37,480 --> 00:29:44,140 And I hate to have to compute those eigenvalues to answer 475 00:29:44,140 --> 00:29:46,120 that question, right? 476 00:29:46,120 --> 00:29:49,120 Because computing the eigenvalues of a symmetric 477 00:29:49,120 --> 00:29:51,230 matrix of order let's say 50 -- 478 00:29:51,230 --> 00:29:53,930 compute its 50 eigenvalues -- 479 00:29:53,930 --> 00:29:58,630 is a job. 480 00:29:58,630 --> 00:30:04,050 I mean, by pencil and paper it's a lifetime's job. 481 00:30:04,050 --> 00:30:11,480 I mean, which -- and in fact, a few years ago -- well, say, 482 00:30:11,480 --> 00:30:17,660 20 years ago, or 30, nobody really knew how to do it. 483 00:30:17,660 --> 00:30:22,140 I mean, so, like, science was stuck on this problem. 484 00:30:22,140 --> 00:30:24,810 If you have a matrix of order 50 or 100, 485 00:30:24,810 --> 00:30:27,320 how do you find its eigenvalues? 486 00:30:27,320 --> 00:30:29,530 Numerically, now, I'm just saying, 487 00:30:29,530 --> 00:30:34,330 because pencil and paper is -- we're going to run out of time 488 00:30:34,330 --> 00:30:36,380 or paper or something before we get it. 489 00:30:38,970 --> 00:30:41,840 Well -- and you might think, okay, 490 00:30:41,840 --> 00:30:47,850 get Matlab to compute the determinant of lambda minus A, 491 00:30:47,850 --> 00:30:52,330 A minus lambda I, this polynomial of 50th degree, 492 00:30:52,330 --> 00:30:53,465 and then find the roots. 493 00:30:56,230 --> 00:30:59,340 Matlab will do it, but it will complain, 494 00:30:59,340 --> 00:31:04,920 because it's a very bad way to find the eigenvalues. 495 00:31:04,920 --> 00:31:08,480 I'm sorry to be saying this, because it's the way I 496 00:31:08,480 --> 00:31:10,220 taught you to do it, right? 497 00:31:10,220 --> 00:31:12,000 I taught you to find the eigenvalues 498 00:31:12,000 --> 00:31:14,830 by doing that determinant and taking 499 00:31:14,830 --> 00:31:16,790 the roots of that polynomial. 500 00:31:16,790 --> 00:31:20,030 But now I'm saying, okay, I really meant that for two 501 00:31:20,030 --> 00:31:21,820 by twos and three by threes but I 502 00:31:21,820 --> 00:31:24,680 didn't mean you to do it on a 50 by 50 503 00:31:24,680 --> 00:31:27,980 and you're not too unhappy, probably, 504 00:31:27,980 --> 00:31:29,620 because you didn't want to do it. 505 00:31:29,620 --> 00:31:36,650 But -- good, because it would be a very unstable way -- 506 00:31:36,650 --> 00:31:40,590 the 50 answers that would come out would be highly unreliable. 507 00:31:40,590 --> 00:31:45,590 So, new ways are -- are much better to find those 50 508 00:31:45,590 --> 00:31:46,310 eigenvalues. 509 00:31:46,310 --> 00:31:50,270 That's a -- that's a part of numerical linear algebra. 510 00:31:50,270 --> 00:31:54,850 But here's the remarkable fact -- 511 00:31:54,850 --> 00:32:00,170 that Matlab would quite happily find the 50 pivots, right? 512 00:32:00,170 --> 00:32:03,720 Now the pivots are not the same as the eigenvalues. 513 00:32:03,720 --> 00:32:06,440 But here's the great thing. 514 00:32:06,440 --> 00:32:11,070 If I had a real matrix, I could find those 50 pivots 515 00:32:11,070 --> 00:32:14,340 and I could see maybe 28 of them are positive 516 00:32:14,340 --> 00:32:15,680 and 22 are negative 517 00:32:15,680 --> 00:32:16,660 pivots. 518 00:32:16,660 --> 00:32:20,150 And I can compute those safely and quickly. 519 00:32:20,150 --> 00:32:23,900 And the great fact is that 28 of the eigenvalues would be 520 00:32:23,900 --> 00:32:27,410 positive and 22 would be negative -- 521 00:32:27,410 --> 00:32:31,740 that the sines of the pivots -- so this is, like -- 522 00:32:31,740 --> 00:32:34,780 I hope you think this -- this is kind of a nice thing, 523 00:32:34,780 --> 00:32:39,970 that the sines of the pivots -- 524 00:32:39,970 --> 00:32:42,650 for symmetric, I'm always talking about symmetric 525 00:32:42,650 --> 00:32:43,700 matrixes -- 526 00:32:43,700 --> 00:32:45,890 so I'm really, like, trying to convince you 527 00:32:45,890 --> 00:32:50,100 that symmetric matrixes are better than the rest. 528 00:32:50,100 --> 00:32:58,700 So the sines of the pivots are same as the sines 529 00:32:58,700 --> 00:33:02,070 of the eigenvalues. 530 00:33:02,070 --> 00:33:04,780 The same number. 531 00:33:04,780 --> 00:33:10,490 The number of pivots greater than zero, 532 00:33:10,490 --> 00:33:16,120 the number of positive pivots is equal to the number 533 00:33:16,120 --> 00:33:21,330 of positive eigenvalues. 534 00:33:21,330 --> 00:33:25,750 So that, actually, is a very useful -- that gives you a g- 535 00:33:25,750 --> 00:33:31,340 a good start on a decent way to compute eigenvalues, 536 00:33:31,340 --> 00:33:33,470 because you can narrow them down, 537 00:33:33,470 --> 00:33:35,410 you can find out how many are positive, 538 00:33:35,410 --> 00:33:37,490 how many are negative. 539 00:33:37,490 --> 00:33:42,680 Then you could shift the matrix by seven times the identity. 540 00:33:42,680 --> 00:33:46,420 That would shift all the eigenvalues by seven. 541 00:33:46,420 --> 00:33:48,500 Then you could take the pivots of that matrix 542 00:33:48,500 --> 00:33:52,700 and you would know how many eigenvalues of the original 543 00:33:52,700 --> 00:33:53,760 were above seven and 544 00:33:53,760 --> 00:33:54,870 below seven. 545 00:33:54,870 --> 00:33:59,070 So this -- this neat little theorem, that, 546 00:33:59,070 --> 00:34:05,490 symmetric matrixes have this connection between the -- 547 00:34:05,490 --> 00:34:09,150 nobody's mixing up and thinking the pivots are the eigenvalues 548 00:34:09,150 --> 00:34:10,820 -- 549 00:34:10,820 --> 00:34:13,050 I mean, the only thing I can think of 550 00:34:13,050 --> 00:34:15,880 is the product of the pivots equals 551 00:34:15,880 --> 00:34:19,210 the product of the eigenvalues, why is that? 552 00:34:19,210 --> 00:34:21,139 So if I asked you for the reason on that, 553 00:34:21,139 --> 00:34:24,679 why is the product of the pivots for a symmetric matrix 554 00:34:24,679 --> 00:34:27,639 the same as the product of the eigenvalues? 555 00:34:27,639 --> 00:34:34,500 Because they both equal the determinant. 556 00:34:34,500 --> 00:34:35,000 Right. 557 00:34:35,000 --> 00:34:37,060 The product of the pivots gives the determinant 558 00:34:37,060 --> 00:34:40,710 if no row exchanges, the product of the eigenvalues 559 00:34:40,710 --> 00:34:42,340 always gives the determinant. 560 00:34:42,340 --> 00:34:47,100 So -- so the products -- but that doesn't tell you anything 561 00:34:47,100 --> 00:34:51,020 about the 50 individual ones, which this does. 562 00:34:51,020 --> 00:34:51,810 Okay. 563 00:34:51,810 --> 00:34:57,566 So that's -- those are essential facts about symmetric matrixes. 564 00:34:57,566 --> 00:34:58,066 Okay. 565 00:35:00,930 --> 00:35:06,410 Now I -- I said in the -- in the lecture description that I 566 00:35:06,410 --> 00:35:13,680 would take the last minutes to start on positive definite 567 00:35:13,680 --> 00:35:15,970 matrixes, because we're right there, 568 00:35:15,970 --> 00:35:22,506 we're ready to say what's a positive definite matrix? 569 00:35:31,740 --> 00:35:34,470 It's symmetric, first of all. 570 00:35:34,470 --> 00:35:37,090 On -- always I will mean symmetric. 571 00:35:40,090 --> 00:35:43,690 So this is the -- this is the next section of the book. 572 00:35:43,690 --> 00:35:45,420 It's about this -- 573 00:35:45,420 --> 00:35:50,030 if symmetric matrixes are good, which was, like, 574 00:35:50,030 --> 00:35:54,020 the point of my lecture so far, then positive, 575 00:35:54,020 --> 00:35:57,650 definite matrixes are -- 576 00:35:57,650 --> 00:36:03,180 a subclass that are excellent, okay. 577 00:36:03,180 --> 00:36:05,430 Just the greatest. 578 00:36:05,430 --> 00:36:07,380 so what are they? 579 00:36:07,380 --> 00:36:10,930 They're matrixes -- they're symmetric matrixes, 580 00:36:10,930 --> 00:36:13,240 so all their eigenvalues are real. 581 00:36:13,240 --> 00:36:15,420 You can guess what they are. 582 00:36:15,420 --> 00:36:20,070 These are symmetric matrixes with all -- 583 00:36:20,070 --> 00:36:21,190 the eigenvalues are -- 584 00:36:25,790 --> 00:36:27,270 okay, tell me what to write. 585 00:36:31,240 --> 00:36:34,040 What -- well, it -- it's hinted, of course, 586 00:36:34,040 --> 00:36:36,200 by the name for these things. 587 00:36:36,200 --> 00:36:39,750 All the eigenvalues are positive. 588 00:36:39,750 --> 00:36:40,250 Okay. 589 00:36:45,080 --> 00:36:46,900 Tell me about the pivots. 590 00:36:46,900 --> 00:36:50,200 We can check the eigenvalues or we can check the pivots. 591 00:36:50,200 --> 00:36:53,270 All the pivots are what? 592 00:36:58,430 --> 00:36:59,730 And then I'll -- 593 00:36:59,730 --> 00:37:01,230 then I'll finally give an example. 594 00:37:01,230 --> 00:37:04,460 I feel awful that I have got to this point in the lecture 595 00:37:04,460 --> 00:37:05,920 and I haven't given you a single 596 00:37:05,920 --> 00:37:06,790 example. 597 00:37:06,790 --> 00:37:08,690 So let me give you one. 598 00:37:08,690 --> 00:37:13,880 Five three two two. 599 00:37:13,880 --> 00:37:17,460 That's symmetric, fine. 600 00:37:17,460 --> 00:37:21,620 It's eigenvalues are real, for sure. 601 00:37:21,620 --> 00:37:27,850 But more than that, I know the sines of those eigenvalues. 602 00:37:27,850 --> 00:37:32,440 And also I know the sines of those pivots, 603 00:37:32,440 --> 00:37:34,430 so what's the deal with the pivots? 604 00:37:34,430 --> 00:37:39,800 The Ei- if the eigenvalues are all positive and if this little 605 00:37:39,800 --> 00:37:44,240 fact is true that the pivots and eigenvalues have the same 606 00:37:44,240 --> 00:37:48,680 sines, then this must be true -- all the pivots are positive. 607 00:37:51,450 --> 00:37:54,330 And that's the good way to test. 608 00:37:54,330 --> 00:37:56,090 This is the good test, because I can -- 609 00:37:56,090 --> 00:37:59,090 what are the pivots for that matrix? 610 00:37:59,090 --> 00:38:02,610 The pivots for that matrix are five. 611 00:38:02,610 --> 00:38:08,840 So pivots are five and what's the second pivot? 612 00:38:08,840 --> 00:38:13,570 Have we, like, noticed the formula for the second pivot 613 00:38:13,570 --> 00:38:14,560 in a matrix? 614 00:38:18,332 --> 00:38:19,790 It doesn't necessarily -- you know, 615 00:38:19,790 --> 00:38:22,200 it may come out a fraction for sure, 616 00:38:22,200 --> 00:38:24,200 but what is that fraction? 617 00:38:24,200 --> 00:38:25,070 Can you tell me? 618 00:38:25,070 --> 00:38:30,070 Well, here, the product of the pivots is the determinant. 619 00:38:30,070 --> 00:38:31,790 What's the determinant of this matrix? 620 00:38:34,840 --> 00:38:36,460 Eleven? 621 00:38:36,460 --> 00:38:41,180 So the second pivot must be eleven over five, 622 00:38:41,180 --> 00:38:44,600 so that the product is eleven. 623 00:38:44,600 --> 00:38:47,430 They're both positive. 624 00:38:47,430 --> 00:38:50,150 Then I know that the eigenvalues of that matrix 625 00:38:50,150 --> 00:38:51,820 are both positive. 626 00:38:51,820 --> 00:38:53,140 What are the eigenvalues? 627 00:38:53,140 --> 00:38:55,550 Well, I've got to take the roots of -- you know, 628 00:38:55,550 --> 00:38:57,770 do I put in a minus lambda? 629 00:38:57,770 --> 00:39:03,800 You mentally do this -- lambda squared minus how many lambdas? 630 00:39:03,800 --> 00:39:04,310 Eight? 631 00:39:04,310 --> 00:39:04,810 Right. 632 00:39:04,810 --> 00:39:07,510 Five and three, the trace comes in there, 633 00:39:07,510 --> 00:39:11,410 plus what number comes here? 634 00:39:11,410 --> 00:39:14,185 The determinant, the eleven, so I set that to 635 00:39:14,185 --> 00:39:14,685 zero. 636 00:39:17,190 --> 00:39:20,190 So the eigenvalues are -- 637 00:39:20,190 --> 00:39:24,040 let's see, half of that is four, look at that positive number, 638 00:39:24,040 --> 00:39:28,600 plus or minus the square root of sixteen minus eleven, 639 00:39:28,600 --> 00:39:29,360 I think five. 640 00:39:32,480 --> 00:39:35,450 The eigenvalues -- well, two by two they're not so terrible, 641 00:39:35,450 --> 00:39:37,750 but they're not so perfect. 642 00:39:37,750 --> 00:39:40,235 Pivots are really simple. 643 00:39:44,580 --> 00:39:48,370 And this is a -- this is the family of matrixes that you 644 00:39:48,370 --> 00:39:51,180 really want in differential equations, 645 00:39:51,180 --> 00:39:55,720 because you know the sines of the eigenvalues, 646 00:39:55,720 --> 00:39:58,520 so you know the stability or not. 647 00:39:58,520 --> 00:39:59,340 Okay. 648 00:39:59,340 --> 00:40:03,700 There's one other related fact I can pop in here in -- 649 00:40:03,700 --> 00:40:07,745 in the time available for positive definite matrixes. 650 00:40:10,380 --> 00:40:14,737 The related fact is to ask you about determinants. 651 00:40:14,737 --> 00:40:15,820 So what's the determinant? 652 00:40:24,990 --> 00:40:27,470 What can you tell me if I -- remember, 653 00:40:27,470 --> 00:40:32,090 positive definite means all eigenvalues are positive, 654 00:40:32,090 --> 00:40:34,710 all pivots are positive, so what can you tell me about 655 00:40:34,710 --> 00:40:36,890 the determinant? 656 00:40:36,890 --> 00:40:40,240 It's positive, too. 657 00:40:40,240 --> 00:40:45,070 But somehow that -- that's not quite enough. 658 00:40:45,070 --> 00:40:50,730 Here -- here's a matrix minus one minus three, 659 00:40:50,730 --> 00:40:54,330 what's the determinant of that guy? 660 00:40:54,330 --> 00:40:55,880 It's positive, right? 661 00:40:55,880 --> 00:40:58,010 Is this a positive, definite matrix? 662 00:40:58,010 --> 00:41:00,000 Are the pivots -- what are the pivots? 663 00:41:00,000 --> 00:41:02,230 Well, negative. 664 00:41:02,230 --> 00:41:03,400 What are the eigenvalues? 665 00:41:03,400 --> 00:41:05,470 Well, they're also the same. 666 00:41:05,470 --> 00:41:12,240 So somehow I don't just want the determinant of the whole 667 00:41:12,240 --> 00:41:12,850 matrix. 668 00:41:12,850 --> 00:41:14,770 Here is eleven, that's great. 669 00:41:14,770 --> 00:41:16,540 Here the determinant of the whole matrix 670 00:41:16,540 --> 00:41:20,220 is three, that's positive. 671 00:41:20,220 --> 00:41:26,450 I also -- I've got to check, like, little sub-determinants, 672 00:41:26,450 --> 00:41:29,330 say maybe coming down from the left. 673 00:41:29,330 --> 00:41:32,950 So the one by one and the two by two have to be positive. 674 00:41:32,950 --> 00:41:36,840 So there -- that's where I get the all. 675 00:41:36,840 --> 00:41:41,140 All -- can I call them sub-determinants -- 676 00:41:41,140 --> 00:41:43,400 are -- see, I have to -- 677 00:41:43,400 --> 00:41:45,670 I need to make the thing plural. 678 00:41:45,670 --> 00:41:51,230 I need to test n things, not just the big determinant. 679 00:41:51,230 --> 00:41:55,130 All sub-determinants are positive. 680 00:41:55,130 --> 00:41:58,110 Then I'm okay. 681 00:41:58,110 --> 00:42:00,610 Then I'm okay. 682 00:42:00,610 --> 00:42:03,310 This passes the test. 683 00:42:03,310 --> 00:42:06,830 Five is positive and eleven is positive. 684 00:42:06,830 --> 00:42:12,220 This fails the test because that minus one there is negative. 685 00:42:12,220 --> 00:42:16,050 And then the big determinant is positive three. 686 00:42:16,050 --> 00:42:18,800 So t- this -- 687 00:42:18,800 --> 00:42:23,320 these -- this fact -- you see that actually the course, like, 688 00:42:23,320 --> 00:42:24,110 coming together. 689 00:42:27,210 --> 00:42:29,000 And that's really my point now. 690 00:42:29,000 --> 00:42:33,850 In the next -- in this lecture and particularly next Wednesday 691 00:42:33,850 --> 00:42:38,150 and Friday, the course comes together. 692 00:42:38,150 --> 00:42:41,810 These pivots that we met in the first week, 693 00:42:41,810 --> 00:42:46,180 these determinants that we met in the middle of the course, 694 00:42:46,180 --> 00:42:50,810 these eigenvalues that we met most recently -- 695 00:42:50,810 --> 00:42:56,010 all matrixes are square here, so coming together for square 696 00:42:56,010 --> 00:43:00,150 matrixes means these three pieces come together and they 697 00:43:00,150 --> 00:43:05,210 come together in that beautiful fact, that if -- 698 00:43:05,210 --> 00:43:07,430 that all the -- that if I have one of these, 699 00:43:07,430 --> 00:43:09,150 I have the others. 700 00:43:09,150 --> 00:43:10,140 That if I -- 701 00:43:10,140 --> 00:43:12,180 but for symmetric matrixes. 702 00:43:12,180 --> 00:43:17,570 So that -- this will be the positive definite section 703 00:43:17,570 --> 00:43:22,540 and then the real climax of the course is to make everything 704 00:43:22,540 --> 00:43:27,150 come together for n by n matrixes, 705 00:43:27,150 --> 00:43:30,330 not necessarily symmetric -- 706 00:43:30,330 --> 00:43:33,070 bring everything together there and that 707 00:43:33,070 --> 00:43:34,850 will be the final thing. 708 00:43:34,850 --> 00:43:35,490 Okay. 709 00:43:35,490 --> 00:43:38,970 So have a great weekend and don't 710 00:43:38,970 --> 00:43:40,710 forget symmetric matrixes. 711 00:43:40,710 --> 00:43:42,260 Thanks.