1 00:00:00 --> 00:00:01 2 00:00:01 --> 00:00:02 The following content is provided under a Creative 3 00:00:02 --> 00:00:03 Commons license. 4 00:00:03 --> 00:00:06 Your support will help MIT OpenCourseWare continue to 5 00:00:06 --> 00:00:09 offer high-quality educational resources for free. 6 00:00:09 --> 00:00:13 To make a donation or to view additional materials from 7 00:00:13 --> 00:00:16 hundreds of MIT courses, visit MIt OpenCourseWare 8 00:00:16 --> 00:00:19 at ocw.mit.edu. 9 00:00:19 --> 00:00:26 PROFESSOR STRANG: Shall we just start on this review session? 10 00:00:26 --> 00:00:31 So, any questions on anything from Chapter one, anything from 11 00:00:31 --> 00:00:36 those first seven lectures is very, very welcome. 12 00:00:36 --> 00:00:41 So this morning finished the serious part of what we'll 13 00:00:41 --> 00:00:46 do in the chapter with positive definite matrices. 14 00:00:46 --> 00:00:49 And we'll see a lot of those fortunately. 15 00:00:49 --> 00:00:52 They're the best. 16 00:00:52 --> 00:00:58 So questions about, I hope you look in the book at other 17 00:00:58 --> 00:01:04 problems in the problem sets as well as the ones I suggest. 18 00:01:04 --> 00:01:09 And then I can, anyway. 19 00:01:09 --> 00:01:13 Ready for any questions. 20 00:01:13 --> 00:01:14 Ok. 21 00:01:14 --> 00:01:19 Which problem is it? 22 00:01:19 --> 00:01:21 In section? 23 00:01:21 --> 00:01:25 Section 1.6, problem 27, what have I done there? 24 00:01:25 --> 00:01:28 Oh, ok, that's good. 25 00:01:28 --> 00:01:30 So it's about positive definite matrices. 26 00:01:30 --> 00:01:36 May I just put on the board what the central question is? 27 00:01:36 --> 00:01:40 Just put these matrices up. 28 00:01:40 --> 00:01:43 We're given that H and K are positive definite. 29 00:01:43 --> 00:01:47 And then the question is, what about these block matrices. 30 00:01:47 --> 00:01:50 Do I call them M and N? 31 00:01:50 --> 00:01:54 One is the block matrix that looks like that. 32 00:01:54 --> 00:01:59 And another one is the block matrix that looks like this. 33 00:01:59 --> 00:02:04 So those are both symmetric. 34 00:02:04 --> 00:02:07 We're allowed to ask, are they positive definite or negative 35 00:02:07 --> 00:02:10 definite because they passed the first requirement. 36 00:02:10 --> 00:02:10 They're symmetric. 37 00:02:10 --> 00:02:12 We can discuss them. 38 00:02:12 --> 00:02:15 Because of course H and K each were symmetric. 39 00:02:15 --> 00:02:19 The transpose of this would bring K transpose down here, 40 00:02:19 --> 00:02:22 but that's K, so all good. 41 00:02:22 --> 00:02:32 So the question now. 42 00:02:32 --> 00:02:43 Of these guys to those guys I guess, yes. 43 00:02:43 --> 00:02:44 Good question. 44 00:02:44 --> 00:02:47 So this guy has, let's take eigenvalues first. 45 00:02:47 --> 00:02:50 So this guy has some eigenvalues, say 46 00:02:50 --> 00:02:51 lambda_1 to lambda_n. 47 00:02:53 --> 00:02:55 And this guy, we'll suppose they're the same size, so 48 00:02:55 --> 00:02:57 they don't have to be. 49 00:02:57 --> 00:02:59 Maybe I shouldn't, but I will. 50 00:02:59 --> 00:03:06 This has some other eigenvalues, maybe e_1 51 00:03:06 --> 00:03:09 to e_n for eigenvalue. 52 00:03:09 --> 00:03:12 And then the question is, okay, what about the eigenvalues 53 00:03:12 --> 00:03:15 of that combination? 54 00:03:15 --> 00:03:16 And what about this? 55 00:03:16 --> 00:03:20 So it's a good question, I think for all of us to 56 00:03:20 --> 00:03:23 practice what just came up in the lecture. 57 00:03:23 --> 00:03:29 The idea of block matrices. 58 00:03:29 --> 00:03:36 So looking here at eigenvalues I could also look at pivots. 59 00:03:36 --> 00:03:39 Pivots would be interesting to look at, too. 60 00:03:39 --> 00:03:40 Maybe I'll start with pivots. 61 00:03:40 --> 00:03:42 Can I? 62 00:03:42 --> 00:03:43 Did you think? 63 00:03:43 --> 00:03:44 What would be the pivots of M? 64 00:03:44 --> 00:03:51 If I start elimination on M what will I see for pivots? 65 00:03:51 --> 00:03:57 Well, I start up in the usual left-hand corner and work down. 66 00:03:57 --> 00:04:00 So what am I going to see first? 67 00:04:00 --> 00:04:02 I'm going to see the pivots of H. 68 00:04:02 --> 00:04:05 It won't even know, by the time I had halfway there, 69 00:04:05 --> 00:04:07 it won't even have seen K. 70 00:04:07 --> 00:04:10 And then, that'll be fine. 71 00:04:10 --> 00:04:16 And then this will be, what's going to happen? 72 00:04:16 --> 00:04:18 This is all zeroes. 73 00:04:18 --> 00:04:20 So never get touched, right? 74 00:04:20 --> 00:04:24 So when I get down to the second half I 75 00:04:24 --> 00:04:26 see all zeroes here. 76 00:04:26 --> 00:04:28 K is still going to be sitting right there. 77 00:04:28 --> 00:04:29 Nothing happened. 78 00:04:29 --> 00:04:32 Because when I did these eliminations nothing 79 00:04:32 --> 00:04:34 changed with K. 80 00:04:34 --> 00:04:38 So the rest of the pivots will be the pivots of K. 81 00:04:38 --> 00:04:40 Good. 82 00:04:40 --> 00:04:42 Now, we might hope for the same thing with eigenvalues 83 00:04:42 --> 00:04:46 and probably that's going to happen. 84 00:04:46 --> 00:04:49 This is like a diagonal matrix. 85 00:04:49 --> 00:04:51 And actually, what words would I use? 86 00:04:51 --> 00:04:53 Block diagonal. 87 00:04:53 --> 00:04:55 I'd call that matrix block diagonal. 88 00:04:55 --> 00:04:58 And those are very nice matrices. 89 00:04:58 --> 00:05:02 That tells us that the big matrix, for all practical 90 00:05:02 --> 00:05:06 purposes, is breaking up into these smaller blocks. 91 00:05:06 --> 00:05:12 Actually MATLAB will search for a way to reorder the rows and 92 00:05:12 --> 00:05:15 columns to get that in case it's possible. 93 00:05:15 --> 00:05:21 So here it's in front of us. 94 00:05:21 --> 00:05:24 Let's see if we can figure out. 95 00:05:24 --> 00:05:31 That lambda_1 I believe is also an eigenvalue of M. 96 00:05:31 --> 00:05:33 So it was an eigenvalue of H. 97 00:05:33 --> 00:05:36 So that this, the fact that it has that eigenvalue 98 00:05:36 --> 00:05:38 lambda_1 means what? 99 00:05:38 --> 00:05:48 That H times this times some vector y is lambda_1*y, right? 100 00:05:48 --> 00:05:51 If that's an eigenvalue it's got an eigenvector 101 00:05:51 --> 00:05:53 and let's call it y. 102 00:05:53 --> 00:05:56 Now this is a good question. 103 00:05:56 --> 00:06:01 I believe this block matrix also has eigenvalue lambda_1 104 00:06:01 --> 00:06:03 and what's its eigenvector? 105 00:06:03 --> 00:06:09 What could I multiply M by to get lambda_1 106 00:06:09 --> 00:06:13 times the same thing? 107 00:06:13 --> 00:06:14 Can you see what? 108 00:06:14 --> 00:06:17 Of course I'm thinking that y is going to help 109 00:06:17 --> 00:06:20 but it's grown now. 110 00:06:20 --> 00:06:23 So what would be the eigenvector here? 111 00:06:23 --> 00:06:27 When I multiply by M it'll just come out right with the same 112 00:06:27 --> 00:06:33 eigenvalue? y_1, or y rather, and then? 113 00:06:33 --> 00:06:36 And then zero, good. y_0. 114 00:06:36 --> 00:06:41 Because if I multiply, can I put in what M really is? 115 00:06:41 --> 00:06:43 The H and K. 116 00:06:43 --> 00:06:45 H there, K there. 117 00:06:45 --> 00:06:48 When I do that multiplication I get lambda_1*y. 118 00:06:49 --> 00:06:52 When I do this multiplication, see I've just, that's a zero 119 00:06:52 --> 00:06:55 block, zero, so I got a zero. 120 00:06:55 --> 00:06:56 Perfect. 121 00:06:56 --> 00:07:05 So the eigenvectors of H just sit with a zero in the K part 122 00:07:05 --> 00:07:09 and produce an eigenvector of the block matrix with 123 00:07:09 --> 00:07:10 the same lambda_1. 124 00:07:11 --> 00:07:14 So you can see then, we get the whole picture. 125 00:07:14 --> 00:07:18 The eigenvalues are just sitting there and the 126 00:07:18 --> 00:07:20 eigenvectors are there. 127 00:07:20 --> 00:07:24 Now maybe you got all that and wanted-- well I haven't said 128 00:07:24 --> 00:07:27 anything about N, Sorry. 129 00:07:27 --> 00:07:29 Everybody thinks more about N. 130 00:07:29 --> 00:07:31 So what's the thing with N? 131 00:07:31 --> 00:07:34 What would you say about N? 132 00:07:34 --> 00:07:37 If you look at that matrix, suppose I don't even tell you 133 00:07:37 --> 00:07:40 it's positive definite at first, would you say that 134 00:07:40 --> 00:07:45 looks like a invertible or singular matrix? 135 00:07:45 --> 00:07:48 Everybody's going to say singular. 136 00:07:48 --> 00:07:55 And why would you say that's singular? 137 00:07:55 --> 00:08:04 Well, the determinant of a block matrix, this morning I 138 00:08:04 --> 00:08:07 said do whatever you like with block matrices. 139 00:08:07 --> 00:08:13 But I have to admit that if I had a bunch of general blocks, 140 00:08:13 --> 00:08:16 if I had to take the determinant of that, and of 141 00:08:16 --> 00:08:19 course everybody's remembering Professor Strang doesn't like 142 00:08:19 --> 00:08:24 determinants, if I had to take the determinant, I'd have 143 00:08:24 --> 00:08:27 to do the whole thing. 144 00:08:27 --> 00:08:31 The separate determinants would not tell me the story, usually. 145 00:08:31 --> 00:08:33 So determinants are a bit tricky. 146 00:08:33 --> 00:08:37 But up here the determinant will come out zero. 147 00:08:37 --> 00:08:44 I guess what I would hope your internal test for a 148 00:08:44 --> 00:08:50 singular matrix is, are the columns independent? 149 00:08:50 --> 00:08:52 And then the matrix is invertible. 150 00:08:52 --> 00:08:53 Or are they dependent? 151 00:08:53 --> 00:09:00 Do you have some columns that are in the same direction as 152 00:09:00 --> 00:09:02 other columns, same direction as combinations of 153 00:09:02 --> 00:09:03 other columns? 154 00:09:03 --> 00:09:08 If you look at the columns of that, say column one, 155 00:09:08 --> 00:09:14 so column one is the first column of K repeated. 156 00:09:14 --> 00:09:17 What do you think about the columns of that matrix, 157 00:09:17 --> 00:09:18 that block matrix N? 158 00:09:18 --> 00:09:24 Do you see that same column showing up again? 159 00:09:24 --> 00:09:25 Yeah. 160 00:09:25 --> 00:09:29 That very same column, which is the first column of K, again 161 00:09:29 --> 00:09:32 twice, is going to show up right there, first 162 00:09:32 --> 00:09:33 column of K again. 163 00:09:33 --> 00:09:39 So this matrix has two identical columns. 164 00:09:39 --> 00:09:41 No way it could be invertible. 165 00:09:41 --> 00:09:46 And in fact, you can tell me what vector, I'm always saying 166 00:09:46 --> 00:09:48 are the columns independent? 167 00:09:48 --> 00:09:50 Here, no, they're dependent. 168 00:09:50 --> 00:09:56 And then you can tell me an x. 169 00:09:56 --> 00:09:59 So this is my block matrix N. 170 00:09:59 --> 00:10:06 I want to know an x so that the result is zero. 171 00:10:06 --> 00:10:13 That's really my same indication. 172 00:10:13 --> 00:10:15 We found two identical columns. 173 00:10:15 --> 00:10:19 What would be the x? 174 00:10:19 --> 00:10:23 Well, you have to tell me more than one, minus one because 175 00:10:23 --> 00:10:30 I've got a big x there. 176 00:10:30 --> 00:10:33 Yeah I've gotta make it big enough, but essentially it's 177 00:10:33 --> 00:10:34 the one, minus one, thanks. 178 00:10:34 --> 00:10:38 And enough zeroes in there and enough zeroes in there. 179 00:10:38 --> 00:10:46 So the fact that that vector gets taken to zero is the same 180 00:10:46 --> 00:10:49 thing as saying that one of this column minus one of 181 00:10:49 --> 00:10:51 this column gives zero. 182 00:10:51 --> 00:10:53 In other words, the columns are the same. 183 00:10:53 --> 00:10:58 And of course, by doing this we're seeing the one and minus 184 00:10:58 --> 00:11:01 one could have gone into position two there, 185 00:11:01 --> 00:11:02 position three. 186 00:11:02 --> 00:11:06 So we've got a whole bunch of vectors. 187 00:11:06 --> 00:11:13 This matrix N, this [K, K; K, K] has got a whole lot of 188 00:11:13 --> 00:11:15 vectors that it takes to zero. 189 00:11:15 --> 00:11:18 What I would say it has a large null space. 190 00:11:18 --> 00:11:22 A large space of vectors that it takes to zero. 191 00:11:22 --> 00:11:25 So that's a really useful exercise. 192 00:11:25 --> 00:11:26 I'm delighted you asked it. 193 00:11:26 --> 00:11:34 Now I'm ready for more. 194 00:11:34 --> 00:11:34 Could do. 195 00:11:34 --> 00:11:37 Exactly, row reduction. 196 00:11:37 --> 00:11:39 I should look to see what would happen in elimination. 197 00:11:39 --> 00:11:43 Well, elimination would go swimmingly along for the 198 00:11:43 --> 00:11:46 first part because it's only looking here. 199 00:11:46 --> 00:11:52 But then what would I have after the first half 200 00:11:52 --> 00:11:56 of elimination? 201 00:11:56 --> 00:12:02 Well I'd have I suppose whatever that K changed 202 00:12:02 --> 00:12:04 to, elimination. 203 00:12:04 --> 00:12:04 What should we call it? 204 00:12:04 --> 00:12:09 U or something? 205 00:12:09 --> 00:12:13 When I did these row steps that matrix turned into this 206 00:12:13 --> 00:12:15 upper triangular matrix. 207 00:12:15 --> 00:12:18 And maybe you can tell me what will have happened at the 208 00:12:18 --> 00:12:20 same time to the rest? 209 00:12:20 --> 00:12:25 What will I see sitting here if I just do ordinary elimination 210 00:12:25 --> 00:12:29 and I'm just looking there and using the pivots and 211 00:12:29 --> 00:12:31 so on, I'll see? 212 00:12:31 --> 00:12:35 It'll be U because whenever I do on the left side I'm 213 00:12:35 --> 00:12:37 doing to the whole row. 214 00:12:37 --> 00:12:40 And now, the main point is, what will I see? 215 00:12:40 --> 00:12:42 Now elimination, keep going, keep going. 216 00:12:42 --> 00:12:47 Do elimination to clear out this column, this 217 00:12:47 --> 00:12:49 whole bunch, right? 218 00:12:49 --> 00:12:52 Elimination. 219 00:12:52 --> 00:12:54 And now what am I going to see in that corner? 220 00:12:54 --> 00:12:57 All zeroes, right. 221 00:12:57 --> 00:13:09 So that's telling me that the matrix has just got half of the 222 00:13:09 --> 00:13:12 eigenvalues positive, half of the pivots are positive. 223 00:13:12 --> 00:13:16 The second half all zeroes. 224 00:13:16 --> 00:13:19 So I guess, here I've found an eigenvector 225 00:13:19 --> 00:13:22 with what eigenvalue? 226 00:13:22 --> 00:13:24 That's looking like an eigenvector to me if we're 227 00:13:24 --> 00:13:26 thinking eigenvectors. 228 00:13:26 --> 00:13:30 And what's the eigenvalue that goes with it? 229 00:13:30 --> 00:13:30 Zero. 230 00:13:30 --> 00:13:32 Because Nx is 0x. 231 00:13:34 --> 00:13:37 You can either think of it as Nx=0 if you're thinking 232 00:13:37 --> 00:13:40 about systems of equations. 233 00:13:40 --> 00:13:46 Or Nx=0x if you're thinking that that guy is an eigenvector 234 00:13:46 --> 00:13:49 with eigenvalues here. 235 00:13:49 --> 00:13:51 So I'm pretty happy. 236 00:13:51 --> 00:13:55 I mean many of you will have spotted this. 237 00:13:55 --> 00:13:56 Probably perhaps all. 238 00:13:56 --> 00:14:03 But I'm happy that's an example that just shows how you have to 239 00:14:03 --> 00:14:06 think big with block matrices I guess. 240 00:14:06 --> 00:14:08 Good. 241 00:14:08 --> 00:14:11 Ok on that? 242 00:14:11 --> 00:14:27 What else, thanks. 243 00:14:27 --> 00:14:27 That's true. 244 00:14:27 --> 00:14:31 And that's really all I've done so far is those four examples. 245 00:14:31 --> 00:14:35 I think that language of fixed-fixed and fixed-free 246 00:14:35 --> 00:14:40 really comes, I mean I used it early about those four 247 00:14:40 --> 00:14:44 matrices, but it's really going to show up at the next lecture, 248 00:14:44 --> 00:14:52 Friday, when I have a line of springs and the matrices 249 00:14:52 --> 00:14:53 that come out of that. 250 00:14:53 --> 00:15:00 So Friday we'll finally be on those first four. 251 00:15:00 --> 00:15:05 A fifth matrix will appear in this course finally. 252 00:15:05 --> 00:15:08 Of course, it's going to be related to the first ones, 253 00:15:08 --> 00:15:14 naturally but we'll move to, we'll see something new and 254 00:15:14 --> 00:15:18 then we'll see the fixed-free idea again for those. 255 00:15:18 --> 00:15:21 So if that can wait until Friday, you'll see 256 00:15:21 --> 00:15:24 some different ones. 257 00:15:24 --> 00:15:26 Good. 258 00:15:26 --> 00:15:28 Questions, thoughts. 259 00:15:28 --> 00:15:31 You can ask about anything. 260 00:15:31 --> 00:15:35 Maybe I can ask. 261 00:15:35 --> 00:15:40 Any thoughts about the pace of the course? 262 00:15:40 --> 00:15:49 This is sort of a heavy dose of linear algebra, right? 263 00:15:49 --> 00:15:53 Of course, the answer maybe depends on how much 264 00:15:53 --> 00:15:56 you had seen before. 265 00:15:56 --> 00:15:59 So those who haven't seen very much linear algebra at all 266 00:15:59 --> 00:16:04 really got quite a bit quickly here. 267 00:16:04 --> 00:16:10 Because many courses on linear algebra never reach this key 268 00:16:10 --> 00:16:16 idea of positive definiteness that ties it all together. 269 00:16:16 --> 00:16:19 So you've seen quite a bit, really. 270 00:16:19 --> 00:16:23 Of course, we've concentrated on symmetric matrices and 271 00:16:23 --> 00:16:29 there's a whole garden or forest or zoo of matrices 272 00:16:29 --> 00:16:32 of different types. 273 00:16:32 --> 00:16:34 So what matrices have we seen? 274 00:16:34 --> 00:16:40 Symmetric matrices and then their eigenvectors were 275 00:16:40 --> 00:16:44 orthogonal and we could say orthonormal. 276 00:16:44 --> 00:16:51 So that gave us, I don't know if you remember this part, 277 00:16:51 --> 00:16:54 which when we wrote it down I said, big deal. 278 00:16:54 --> 00:16:56 That's very important. 279 00:16:56 --> 00:16:59 That's this principal axis theorem. 280 00:16:59 --> 00:17:04 These Q's, what kind of a matrix is Q? 281 00:17:04 --> 00:17:06 It's the eigenvector matrix. 282 00:17:06 --> 00:17:12 And for symmetric matrix, so this is the eigenvector matrix. 283 00:17:12 --> 00:17:14 And what do we know about it? 284 00:17:14 --> 00:17:22 In the special case of symmetric K? 285 00:17:22 --> 00:17:27 What do we know especially about the eigenvectors then? 286 00:17:27 --> 00:17:28 They're orthogonal. 287 00:17:28 --> 00:17:29 We can make them orthonormal. 288 00:17:29 --> 00:17:34 So this will be an orthogonal matrix. 289 00:17:34 --> 00:17:39 And that was a matrix with Q transpose was 290 00:17:39 --> 00:17:41 the same as Q inverse. 291 00:17:41 --> 00:17:44 Normally we would see the inverse there, but for these 292 00:17:44 --> 00:17:47 we can put the transpose. 293 00:17:47 --> 00:17:52 Here's one type of matrix, symmetric, very important. 294 00:17:52 --> 00:17:56 Here's another type of matrix, orthogonal matrices. 295 00:17:56 --> 00:17:58 And of course, many, many other varieties. 296 00:17:58 --> 00:18:01 Well here we have a very nice matrix, so that 297 00:18:01 --> 00:18:03 matrix is diagonal. 298 00:18:03 --> 00:18:06 Right, that's just the eigenvalues, so that's 299 00:18:06 --> 00:18:08 a diagonal matrix. 300 00:18:08 --> 00:18:12 And what do we know, if K is positive definite, let's just, 301 00:18:12 --> 00:18:14 this was for any symmetric one. 302 00:18:14 --> 00:18:19 So what's special if K is positive definite? 303 00:18:19 --> 00:18:21 Somehow the positive definiteness should 304 00:18:21 --> 00:18:22 show up here. 305 00:18:22 --> 00:18:26 And where does it show? 306 00:18:26 --> 00:18:29 Positive eigenvalues, exactly. 307 00:18:29 --> 00:18:33 The Q could be any, any Q would be fine. 308 00:18:33 --> 00:18:37 But we would see positive eigenvalues. 309 00:18:37 --> 00:18:41 Oh, here's a little point about eigenvalues. 310 00:18:41 --> 00:18:44 Suppose I have my matrix K. 311 00:18:44 --> 00:18:47 And it's got some eigenvalues. 312 00:18:47 --> 00:18:57 Now let me add four times the identity to it. 313 00:18:57 --> 00:18:59 What are the eigenvalues now? 314 00:18:59 --> 00:19:02 What are the eigenvectors now? 315 00:19:02 --> 00:19:08 What's changed and how and what hasn't changed? 316 00:19:08 --> 00:19:12 Because that's a pretty easy, the identity matrix is always 317 00:19:12 --> 00:19:15 the easy one for us to know what's happening. 318 00:19:15 --> 00:19:20 So what is happening to the eigenvalues now? 319 00:19:20 --> 00:19:23 If K had these eigenvalues lambda, what are the 320 00:19:23 --> 00:19:25 eigenvalues of K+4I? 321 00:19:25 --> 00:19:31 322 00:19:31 --> 00:19:32 You add? 323 00:19:32 --> 00:19:32 You add four, yeah. 324 00:19:32 --> 00:19:36 The eigenvalues of this are the eigenvalues of K+4. 325 00:19:38 --> 00:19:42 That is just like shifting the matrix, you could think of it 326 00:19:42 --> 00:19:48 is adding four along the diagonal will add four. 327 00:19:48 --> 00:19:54 And the eigenvectors would be exactly the same ones. 328 00:19:54 --> 00:19:56 I would have Kx would agree with lambda*S. 329 00:19:57 --> 00:19:59 And 4Ix would agree with 4x. 330 00:20:00 --> 00:20:05 So that proves it. 331 00:20:05 --> 00:20:11 Good to see what you can do, the limited number of things 332 00:20:11 --> 00:20:15 that you're allowed to do without changing the 333 00:20:15 --> 00:20:17 eigenvectors, and therefore you can spot the 334 00:20:17 --> 00:20:19 eigenvalues right away. 335 00:20:19 --> 00:20:22 The limited things you can invert, you can shift like 336 00:20:22 --> 00:20:27 this, you could square it, cube it, take powers, 337 00:20:27 --> 00:20:34 things like that. 338 00:20:34 --> 00:20:37 I'm going to look to you now for giving me a lead 339 00:20:37 --> 00:20:42 on something that is interesting or not. 340 00:20:42 --> 00:20:48 Yes, thanks. 341 00:20:48 --> 00:20:52 Go ahead. 342 00:20:52 --> 00:21:02 Oh, I see okay, yes. 343 00:21:02 --> 00:21:03 I see. 344 00:21:03 --> 00:21:05 Alright. 345 00:21:05 --> 00:21:08 So that's page 64 of the book. 346 00:21:08 --> 00:21:18 Well, so that's a problem that physicists love. 347 00:21:18 --> 00:21:21 I don't know how much I can say about it here, 348 00:21:21 --> 00:21:23 to tell the truth. 349 00:21:23 --> 00:21:26 Just to mention. 350 00:21:26 --> 00:21:28 Do they use a minus sign? 351 00:21:28 --> 00:21:30 Probably they do. 352 00:21:30 --> 00:21:39 So their equation is minus the second derivative of u plus 353 00:21:39 --> 00:21:45 (x squared)*u and they are interested in the 354 00:21:45 --> 00:21:54 eigenvalues equal lambda*u. 355 00:21:54 --> 00:21:58 The case that we've done in class was without this (x 356 00:21:58 --> 00:22:01 squared)*u term, right? 357 00:22:01 --> 00:22:07 The absolutely most important case is the second derivative 358 00:22:07 --> 00:22:08 of u equal lambda*u. 359 00:22:09 --> 00:22:13 The eigenvalues were, or what were the eigenvectors 360 00:22:13 --> 00:22:15 in that case? 361 00:22:15 --> 00:22:21 What were the eigenvectors of the second derivative before 362 00:22:21 --> 00:22:27 there was any (x squared)*u and E potential showing up? 363 00:22:27 --> 00:22:30 They were just sines and cosines, right? 364 00:22:30 --> 00:22:33 Sines and cosines have the property that if you take two 365 00:22:33 --> 00:22:40 derivatives you get them back with some factor lambda. 366 00:22:40 --> 00:22:45 Now let me just look at that problem without 367 00:22:45 --> 00:22:49 saying much about it. 368 00:22:49 --> 00:22:53 First of all, the first thing I want to know is have I 369 00:22:53 --> 00:22:55 got a linear problem here? 370 00:22:55 --> 00:22:57 Have I got a linear equation? 371 00:22:57 --> 00:23:00 Because that's where I talk about eigenvalues. 372 00:23:00 --> 00:23:05 So in the matrix case, I'd say I have a matrix. 373 00:23:05 --> 00:23:09 K times an eigenvector. 374 00:23:09 --> 00:23:14 That matrix represents something linear. 375 00:23:14 --> 00:23:18 It's just, all the rules of addition work here. 376 00:23:18 --> 00:23:20 Here it is linear. 377 00:23:20 --> 00:23:27 It is linear. 378 00:23:27 --> 00:23:34 What I'm trying to say is, I just call that a variable 379 00:23:34 --> 00:23:36 coefficient and that's what we're going to see 380 00:23:36 --> 00:23:38 in Chapter two. 381 00:23:38 --> 00:23:44 The material or something could lead to some dependence on x. 382 00:23:44 --> 00:23:49 But u is still there, just linearly. 383 00:23:49 --> 00:23:55 In other words, this is a perfectly ok linear operator 384 00:23:55 --> 00:24:00 and am I imagining that it's positive definite? 385 00:24:00 --> 00:24:00 Let's see. 386 00:24:00 --> 00:24:08 This part with the minus sign was positive definite, right? 387 00:24:08 --> 00:24:12 Well, at least semi-definite. 388 00:24:12 --> 00:24:15 So let me just remember the most important case. 389 00:24:15 --> 00:24:20 If I look at this equation, d second u/dx squared 390 00:24:20 --> 00:24:21 equals lambda*u. 391 00:24:22 --> 00:24:28 So that's the eigenvalue, eigenfunction problem 392 00:24:28 --> 00:24:30 for our good friend. 393 00:24:30 --> 00:24:35 What do I say about the eigenvalues now? 394 00:24:35 --> 00:24:40 What can you tell me about the eigenvalues of that? 395 00:24:40 --> 00:24:41 Mostly positive. 396 00:24:41 --> 00:24:44 Because they were sort of omega squares. 397 00:24:44 --> 00:24:47 But I mean zero could be an eigenvalue, right? 398 00:24:47 --> 00:24:55 What would the eigenfunction be for lambda equal zero? 399 00:24:55 --> 00:24:59 If I wanted to get zero here, if I wanted a zero on the 400 00:24:59 --> 00:25:06 right side, what functions u could give me zero? 401 00:25:06 --> 00:25:08 Constant function. 402 00:25:08 --> 00:25:12 Yeah, the constant function is certainly there 403 00:25:12 --> 00:25:14 as a possibility. 404 00:25:14 --> 00:25:18 But anyway, I would say this is positive 405 00:25:18 --> 00:25:20 semi-definite at least. 406 00:25:20 --> 00:25:23 And this part? 407 00:25:23 --> 00:25:26 How do I think about that as a big matrix? 408 00:25:26 --> 00:25:29 I think of it sort of like a big matrix with x squared 409 00:25:29 --> 00:25:36 running down the diagonal. 410 00:25:36 --> 00:25:38 With a matrix, you could say walking down the diagonal 411 00:25:38 --> 00:25:41 because it's n steps. 412 00:25:41 --> 00:25:45 For differential equations, maybe running is 413 00:25:45 --> 00:25:46 the right word. 414 00:25:46 --> 00:25:54 Because it doesn't jump, it's just bzzz all the way from 415 00:25:54 --> 00:25:56 zero squared to whatever. 416 00:25:56 --> 00:26:04 Anyway, that would correspond to a diagonal matrix, but 417 00:26:04 --> 00:26:07 not constant diagonal. 418 00:26:07 --> 00:26:09 Diagonal, but not constant diagonal. 419 00:26:09 --> 00:26:13 Because this x squared number is changing. 420 00:26:13 --> 00:26:20 It's like a spring, it's like a bunch of springs in which the 421 00:26:20 --> 00:26:24 first spring maybe has a spring constant of one. 422 00:26:24 --> 00:26:27 And then we have a tighter spring and then a very tight 423 00:26:27 --> 00:26:31 spring and so on, more and more, higher and higher 424 00:26:31 --> 00:26:32 constants there. 425 00:26:32 --> 00:26:39 Well, I'm just speaking very roughly here. 426 00:26:39 --> 00:26:44 Because variable coefficient, variable material properties, 427 00:26:44 --> 00:26:48 springs of different elasticities, we're 428 00:26:48 --> 00:26:51 ready to move to that. 429 00:26:51 --> 00:26:57 Our problems up to now, the springs were all the same. 430 00:26:57 --> 00:27:01 The bar, if it was a bar, was uniform. 431 00:27:01 --> 00:27:05 And now this would be a step forward. 432 00:27:05 --> 00:27:10 But now, of course, this specific problem just 433 00:27:10 --> 00:27:15 happens to have a solution that physicists love. 434 00:27:15 --> 00:27:19 It has a meaning to physicists, not to me. 435 00:27:19 --> 00:27:23 And the eigenfunctions have a meaning and they're 436 00:27:23 --> 00:27:26 famous functions. 437 00:27:26 --> 00:27:28 It's just glorious. 438 00:27:28 --> 00:27:31 So you could say that's the special problem, the way we had 439 00:27:31 --> 00:27:36 four special matrices in 18.085, that would be a similar 440 00:27:36 --> 00:27:44 special problem in quantum mechanics. 441 00:27:44 --> 00:27:48 Let's turn to something entirely different. 442 00:27:48 --> 00:27:51 Questions about any topic. 443 00:27:51 --> 00:27:54 Or I can ask some and you can take this, maybe 444 00:27:54 --> 00:27:56 that's one way to review. 445 00:27:56 --> 00:27:57 Go ahead. 446 00:27:57 --> 00:27:59 Thanks. 447 00:27:59 --> 00:28:03 Number 20 of 1.6. 448 00:28:03 --> 00:28:06 1.6 is a section, oh, no. 449 00:28:06 --> 00:28:11 That's positive definite notes so I'm okay with that. 450 00:28:11 --> 00:28:15 I see that I did ask you a question on the homework from 451 00:28:15 --> 00:28:23 1.7 which I may not get to cover in lecture, but 452 00:28:23 --> 00:28:24 give it a shot anyway. 453 00:28:24 --> 00:28:28 So what's 20? 454 00:28:28 --> 00:28:32 Oh, ok, that's good. 455 00:28:32 --> 00:28:36 Without multiplying out the matrix. 456 00:28:36 --> 00:28:39 So it's this Q*lambda*Q transpose. 457 00:28:41 --> 00:28:44 So I'm telling you in that question what Q, lambda, 458 00:28:44 --> 00:28:48 and Q transpose are. 459 00:28:48 --> 00:28:53 The Q is this [cosine, minus sine; sine, cosine]. 460 00:28:53 --> 00:28:57 The lambda is two and five, I think, in that question. 461 00:28:57 --> 00:29:01 And the Q transpose of course is [cosine, sine; 462 00:29:01 --> 00:29:05 minus sine, cosine]. 463 00:29:05 --> 00:29:12 And if I've told you that those are the numbers then you 464 00:29:12 --> 00:29:15 could multiply those together to get K. 465 00:29:15 --> 00:29:21 But you can tell me, this is like K exposed. 466 00:29:21 --> 00:29:27 The matrix is like, we're told more than we would know. 467 00:29:27 --> 00:29:30 If I multiply it all together, I wouldn't see that the 468 00:29:30 --> 00:29:33 eigenvectors are these guys, that the eigenvalues 469 00:29:33 --> 00:29:34 are these guys. 470 00:29:34 --> 00:29:42 So what, without looking to see, what are the eigenvalues 471 00:29:42 --> 00:29:46 of this matrix K if we multiplied it all together? 472 00:29:46 --> 00:29:49 What would the eigenvalues actually be? 473 00:29:49 --> 00:29:53 Two and five, right, because we built it up that way. 474 00:29:53 --> 00:29:56 What would the determinant be? 475 00:29:56 --> 00:29:59 Now what do we know about determinants? 476 00:29:59 --> 00:30:02 It would be ten is the right answer. 477 00:30:02 --> 00:30:06 What's the right way to see that? 478 00:30:06 --> 00:30:09 Well, the determinant is always the product of the 479 00:30:09 --> 00:30:11 eigenvalues, isn't it? 480 00:30:11 --> 00:30:15 These guys have determinant ten anyway. 481 00:30:15 --> 00:30:18 And if I hadn't normalized, so this had some bigger 482 00:30:18 --> 00:30:23 determinant, this would have some smaller determinant. 483 00:30:23 --> 00:30:25 Their inverses, their determinants will give me 484 00:30:25 --> 00:30:28 the 1 and there's the ten. 485 00:30:28 --> 00:30:33 What else could I ask about or did I ask about for that? 486 00:30:33 --> 00:30:38 The eigenvectors, ok. 487 00:30:38 --> 00:30:41 The eigenvectors of the matrix, what are they? 488 00:30:41 --> 00:30:44 They're these columns that are sitting here for us, they're 489 00:30:44 --> 00:30:46 those two columns, right. 490 00:30:46 --> 00:30:49 And would you like to just check that if the, I believe 491 00:30:49 --> 00:30:51 that column is an eigenvector. 492 00:30:51 --> 00:30:56 And which, do you think two or five is it's eigenvalue? 493 00:30:56 --> 00:30:59 That goes with this first column. 494 00:30:59 --> 00:31:02 Everybody's going to say two and that's right. 495 00:31:02 --> 00:31:08 And do you want me to just take that matrix times this proposed 496 00:31:08 --> 00:31:12 eigenvector and just see if it's going to work? 497 00:31:12 --> 00:31:19 Suppose I just do all and just see, sure enough this 498 00:31:19 --> 00:31:20 will be an eigenvector. 499 00:31:20 --> 00:31:23 So what do I have at this point? 500 00:31:23 --> 00:31:25 Can you do this times this first? 501 00:31:25 --> 00:31:31 What do I get? c squared plus s squared is one. 502 00:31:31 --> 00:31:36 And -cs plus cs is zero. 503 00:31:36 --> 00:31:38 So at that point I have . 504 00:31:38 --> 00:31:40 Now comes this matrix. 505 00:31:40 --> 00:31:46 So what do I have after that matrix speaks up? . 506 00:31:46 --> 00:31:51 And now I take two times this and what do I get? 507 00:31:51 --> 00:31:55 Or that matrix times the . 508 00:31:55 --> 00:31:59 How do you multiply a matrix times that vector. 509 00:31:59 --> 00:32:00 Here's the good way to think of it. 510 00:32:00 --> 00:32:04 It's two times the first column. 511 00:32:04 --> 00:32:06 And zero times the second. 512 00:32:06 --> 00:32:10 So the net result of the whole deal was two times 513 00:32:10 --> 00:32:12 that first column. 514 00:32:12 --> 00:32:16 Which is exactly saying that this is an eigenvector. 515 00:32:16 --> 00:32:20 When I did all that it came back again. 516 00:32:20 --> 00:32:24 Scaled by two. 517 00:32:24 --> 00:32:27 So that's a good example. 518 00:32:27 --> 00:32:29 And then, is the matrix positive definite? 519 00:32:29 --> 00:32:33 That connects to today's lecture. 520 00:32:33 --> 00:32:36 What test would you use to show that the matrix 521 00:32:36 --> 00:32:39 is positive definite? 522 00:32:39 --> 00:32:40 The eigenvalues, yeah. 523 00:32:40 --> 00:32:42 The eigenvalues are sitting there. 524 00:32:42 --> 00:32:44 Two and five, both positive. 525 00:32:44 --> 00:32:47 If I changed one of those signs, then it would no 526 00:32:47 --> 00:32:50 longer be positive definite. 527 00:32:50 --> 00:32:53 It would still be symmetric, I'd still have the 528 00:32:53 --> 00:32:56 eigenvectors, but then eigenvalue would have 529 00:32:56 --> 00:33:00 jumped to minus five. 530 00:33:00 --> 00:33:02 I think this sort of helps out. 531 00:33:02 --> 00:33:06 I guess I hope that as I'm doing these things, you're 532 00:33:06 --> 00:33:12 ahead of me or with me in the calculation and you just have 533 00:33:12 --> 00:33:16 to do a bunch of these to get confidence that you've 534 00:33:16 --> 00:33:17 got the right thing. 535 00:33:17 --> 00:33:22 Ok, yes? 536 00:33:22 --> 00:33:24 1.6, 24. 537 00:33:24 --> 00:33:26 Is that also a homework problem? 538 00:33:26 --> 00:33:28 Alright, but you guys are reading the rest 539 00:33:28 --> 00:33:30 of the book, right? 540 00:33:30 --> 00:33:32 Not only the homework questions. 541 00:33:32 --> 00:33:34 Ah. 542 00:33:34 --> 00:33:35 Oh, dear. 543 00:33:35 --> 00:33:41 24, that's a very good question. 544 00:33:41 --> 00:33:43 About this, yeah. 545 00:33:43 --> 00:33:54 Right. 546 00:33:54 --> 00:33:55 It's a good question. 547 00:33:55 --> 00:33:58 And if today's lecture had been, well it 548 00:33:58 --> 00:33:59 ran a little late. 549 00:33:59 --> 00:34:03 But if we ran another 20 minutes late, I could 550 00:34:03 --> 00:34:05 have done this. 551 00:34:05 --> 00:34:09 I'll just say what's in that problem. 552 00:34:09 --> 00:34:16 And then we'll see it again. 553 00:34:16 --> 00:34:21 So what's in that question? 554 00:34:21 --> 00:34:23 Let me write down what it is. 555 00:34:23 --> 00:34:28 So I have a positive definite matrix K, right? 556 00:34:28 --> 00:34:33 And then I've got its energy. 557 00:34:33 --> 00:34:39 I'm using u rather than x, so let's use u. 558 00:34:39 --> 00:34:46 So my u transpose Ku, or like x transpose Kx today. 559 00:34:46 --> 00:34:52 That is this bowl-shaped figure, right? 560 00:34:52 --> 00:35:01 If I graph this on the u_1, u_2 maybe up to 561 00:35:01 --> 00:35:02 u_n, all in the base. 562 00:35:02 --> 00:35:04 And now I have the picture. 563 00:35:04 --> 00:35:07 So I'm in n+1 dimensions. 564 00:35:07 --> 00:35:09 The other dimension is this one. 565 00:35:09 --> 00:35:15 Then that's the one where I might get this bowl-shaped guy. 566 00:35:15 --> 00:35:17 And I've called that energy. 567 00:35:17 --> 00:35:22 In many, many physical problems there is a factor of 1/2. 568 00:35:22 --> 00:35:27 And it's going to be nice to have that factor of 1/2. 569 00:35:27 --> 00:35:35 So that won't change anything, just half as big. 570 00:35:35 --> 00:35:41 So what is the minimum value of that energy? 571 00:35:41 --> 00:35:46 And what is the minimum value of this, if I said minimize 572 00:35:46 --> 00:35:48 that, you could do it right away. 573 00:35:48 --> 00:35:51 It'd be a zero. 574 00:35:51 --> 00:35:57 Now I'm going to introduce a linear term. 575 00:35:57 --> 00:36:01 This was a quadratic term and it had u squareds in it. 576 00:36:01 --> 00:36:05 So the linear term is going to be u transpose f is 577 00:36:05 --> 00:36:06 the shorthand for it. 578 00:36:06 --> 00:36:12 And of course, we all know that that stands for u_1*f_1, u_2 579 00:36:12 --> 00:36:16 all minus, u_2*f_2 and so on. 580 00:36:16 --> 00:36:19 However many dimensions I'm in. 581 00:36:19 --> 00:36:21 You can imagine I'm in two dimensions. 582 00:36:21 --> 00:36:23 So it's -u_1*f_1 - u_2*f_2. 583 00:36:23 --> 00:36:30 584 00:36:30 --> 00:36:34 So what I'm saying is that minimizing just this was 585 00:36:34 --> 00:36:35 like, too easy, right? 586 00:36:35 --> 00:36:36 The answer was zero. 587 00:36:36 --> 00:36:39 Nobody's interested in that for very long. 588 00:36:39 --> 00:36:43 But now it is much more interesting when I get a 589 00:36:43 --> 00:36:48 linear term in there. 590 00:36:48 --> 00:36:51 So what happens now? 591 00:36:51 --> 00:36:59 Well, the effect of that linear term is to shift that bowl 592 00:36:59 --> 00:37:02 sorta over and down a little. 593 00:37:02 --> 00:37:04 So that instead of sitting where I drew 594 00:37:04 --> 00:37:09 it, let me erase it. 595 00:37:09 --> 00:37:15 If I know graph this function, this is my function of u, this 596 00:37:15 --> 00:37:22 is still the most important part, but now I have 597 00:37:22 --> 00:37:24 a first order turn. 598 00:37:24 --> 00:37:27 And the result is, it still goes through here. 599 00:37:27 --> 00:37:27 Right? 600 00:37:27 --> 00:37:31 Why does it still go through that same point? 601 00:37:31 --> 00:37:37 Because if I take u_1 and u_2 to be zero, I get zero. 602 00:37:37 --> 00:37:38 So I still get zero there. 603 00:37:38 --> 00:37:39 But the bowl 604 00:37:39 --> 00:37:40 has shifted. 605 00:37:40 --> 00:37:44 It's more like something here. 606 00:37:44 --> 00:37:48 And it still has a minimum because this is still 607 00:37:48 --> 00:37:50 the all-important term. 608 00:37:50 --> 00:37:52 But it's just moved over and down. 609 00:37:52 --> 00:37:55 So it has the minimum value. 610 00:37:55 --> 00:38:01 It actually goes below zero, but if I look at it if I'm 611 00:38:01 --> 00:38:04 sitting at the minimum and looking I'm seeing a 612 00:38:04 --> 00:38:06 bowl going up, right. 613 00:38:06 --> 00:38:11 So I hope that picture shows. 614 00:38:11 --> 00:38:16 And now, of course, that's the geometry. 615 00:38:16 --> 00:38:18 In other words, the same geometry just moved the 616 00:38:18 --> 00:38:20 thing over and down. 617 00:38:20 --> 00:38:24 But the algebra is, where is the minimum? 618 00:38:24 --> 00:38:26 What is the value of that minimum? 619 00:38:26 --> 00:38:35 And this problem, 24, is one way to do the minimum. 620 00:38:35 --> 00:38:37 One way to do it. 621 00:38:37 --> 00:38:42 But actually, if you doesn't like linear, well I won't say 622 00:38:42 --> 00:38:46 didn't like linear algebra, that's against my religion. 623 00:38:46 --> 00:38:52 So if you like calculus and you said, wait a minute, if you 624 00:38:52 --> 00:38:55 give me something you want me to minimize, what will I do? 625 00:38:55 --> 00:38:59 I'll set derivatives to zero. 626 00:38:59 --> 00:39:03 And can I just jump to the answer? 627 00:39:03 --> 00:39:07 Oh, what derivatives do I set to zero now, 628 00:39:07 --> 00:39:12 for the minimum here? 629 00:39:12 --> 00:39:14 It's the first derivatives. 630 00:39:14 --> 00:39:21 And they're first derivatives with respect to? 631 00:39:21 --> 00:39:24 I look at df/d what? 632 00:39:24 --> 00:39:27 You see I've already given it away. 633 00:39:27 --> 00:39:29 These are going to be partial derivatives. 634 00:39:29 --> 00:39:30 Why's that? 635 00:39:30 --> 00:39:32 Because I've got two directions. 636 00:39:32 --> 00:39:36 So I have a df/du_1=0 and a df/du_2=0. 637 00:39:36 --> 00:39:39 638 00:39:39 --> 00:39:42 In other words, when I sit here at the bottom I'm seeing 639 00:39:42 --> 00:39:46 this whole bowl above me. 640 00:39:46 --> 00:39:51 If I go along the u_2 direction it should go up and if 641 00:39:51 --> 00:39:54 I come along the u_1 direction, goes up. 642 00:39:54 --> 00:40:00 But it's flat at the bottom both ways. 643 00:40:00 --> 00:40:04 So what's my point here? 644 00:40:04 --> 00:40:09 If you like calculus, you'll get to two equations. 645 00:40:09 --> 00:40:12 And I just want to say what those equations are, because 646 00:40:12 --> 00:40:19 they're all important. 647 00:40:19 --> 00:40:22 Suppose we only had u_1 and nothing else. 648 00:40:22 --> 00:40:26 Then this would just be a parabola and the derivative 649 00:40:26 --> 00:40:29 of this would be at 1/2 K*u squared. 650 00:40:29 --> 00:40:31 Suppose n is one. 651 00:40:31 --> 00:40:34 I'm only in one. 652 00:40:34 --> 00:40:39 So what's the derivative of 1/2 K*u squared? 653 00:40:39 --> 00:40:40 The derivative. 654 00:40:40 --> 00:40:44 So I'm looking for if this was 1/2 K*u squared and I took the 655 00:40:44 --> 00:40:47 derivative with respect to u, it would be? 656 00:40:47 --> 00:40:47 It'd be Ku. 657 00:40:48 --> 00:40:52 And it works here in the matrix case. 658 00:40:52 --> 00:40:56 And what would be the derivative of u, transpose of u 659 00:40:56 --> 00:41:03 times f, if u was just a number and if u was just one thing and 660 00:41:03 --> 00:41:08 f was a single number, the derivative would be? f, yeah. 661 00:41:08 --> 00:41:12 It'd be f. 662 00:41:12 --> 00:41:18 That's the system. 663 00:41:18 --> 00:41:20 I've jumped to the answer. 664 00:41:20 --> 00:41:27 That this set of two or n equations in matrix language 665 00:41:27 --> 00:41:30 would just be, and I'll even write it better as Ku=f. 666 00:41:33 --> 00:41:35 That tells me where the minimum is. 667 00:41:35 --> 00:41:42 The minimizing guy is, so this is in the base and then the 668 00:41:42 --> 00:41:43 thing is dropping down. 669 00:41:43 --> 00:41:49 I still have to figure out what's the bottom value. 670 00:41:49 --> 00:41:54 But I've now identified where the minimum occurs. 671 00:41:54 --> 00:41:57 So you get two questions about a minimum. 672 00:41:57 --> 00:41:59 Where is it? 673 00:41:59 --> 00:42:01 What value of u gives the minimum? 674 00:42:01 --> 00:42:09 And at that point, at that lowest point, how low is it? 675 00:42:09 --> 00:42:13 The one thing you've gotta remember is that when you 676 00:42:13 --> 00:42:19 minimize that quadratic, you get that system of equations. 677 00:42:19 --> 00:42:21 And then, of course, the answer, you have to 678 00:42:21 --> 00:42:22 solve that system. 679 00:42:22 --> 00:42:25 But this goes back to what I said at the 680 00:42:25 --> 00:42:29 first minute of today. 681 00:42:29 --> 00:42:34 That we have two ways of looking at a problem. 682 00:42:34 --> 00:42:38 Usually we go directly to the equations. 683 00:42:38 --> 00:42:44 Sometimes the problem comes naturally to us 684 00:42:44 --> 00:42:46 as a minimum problem. 685 00:42:46 --> 00:42:49 Like we have to minimize the cost, we want to build a 686 00:42:49 --> 00:42:53 new school or something. 687 00:42:53 --> 00:42:57 So we've got some cost function that we minimize that will 688 00:42:57 --> 00:43:04 lead, through calculus or linear algebra, to this. 689 00:43:04 --> 00:43:14 So I've done everything but answer the question 24. 690 00:43:14 --> 00:43:19 We only checked the one by one case to see that that's 691 00:43:19 --> 00:43:23 the right equations, derivative equal zero. 692 00:43:23 --> 00:43:27 And now you could use calculus as I said. 693 00:43:27 --> 00:43:36 But if I answer that question, well let me just do a little. 694 00:43:36 --> 00:43:41 The idea of that question 24, so that was what? 695 00:43:41 --> 00:43:45 1.6, 24, or something. 696 00:43:45 --> 00:43:46 Is that right? 697 00:43:46 --> 00:43:47 Yeah. 698 00:43:47 --> 00:43:55 Is that I could rewrite this to make it clear. 699 00:43:55 --> 00:43:56 I think it's u minus K inverse f. 700 00:43:56 --> 00:44:00 701 00:44:00 --> 00:44:07 Transpose K times u minus K inverse f. 702 00:44:07 --> 00:44:17 And then a minus f 1/2, f transpose K inverse 1/2. 703 00:44:17 --> 00:44:21 Actually, my best friend in China told me this trick. 704 00:44:21 --> 00:44:25 And I didn't give him credit for it in the book. 705 00:44:25 --> 00:44:27 But I should have done. 706 00:44:27 --> 00:44:30 I just think that if you multiply all this 707 00:44:30 --> 00:44:33 out, you'll get this. 708 00:44:33 --> 00:44:35 It's what I would call an identity. 709 00:44:35 --> 00:44:40 That just simply means that it's just true for every u. 710 00:44:40 --> 00:44:42 It's true for everything. 711 00:44:42 --> 00:44:45 Can I try to multiply some of that out? 712 00:44:45 --> 00:44:53 Just so you kind of see it. 713 00:44:53 --> 00:44:57 Yeah, that's what I mean, multiply it out. 714 00:44:57 --> 00:44:59 You've got it. 715 00:44:59 --> 00:45:01 This thing would give me four terms. 716 00:45:01 --> 00:45:05 It'd be this transpose times that times that. 717 00:45:05 --> 00:45:07 Which is my guy here. 718 00:45:07 --> 00:45:09 And then I'll have something. 719 00:45:09 --> 00:45:10 It's just like numbers. 720 00:45:10 --> 00:45:13 Then this thing times that times this. 721 00:45:13 --> 00:45:15 And this thing times that times that. 722 00:45:15 --> 00:45:18 And this thing times that times that. 723 00:45:18 --> 00:45:19 Let me do that last one. 724 00:45:19 --> 00:45:26 What happens when I do the 1/2 and this transpose 725 00:45:26 --> 00:45:28 times the K times this. 726 00:45:28 --> 00:45:32 So I'm using the distributive, whatever, laws. 727 00:45:32 --> 00:45:36 Let's just do that particular term and see what 728 00:45:36 --> 00:45:37 we're getting. 729 00:45:37 --> 00:45:42 So I have 1/2 of the minus K inverse f transpose. 730 00:45:42 --> 00:45:45 So how do I write that? 731 00:45:45 --> 00:45:47 Shoot. 732 00:45:47 --> 00:45:50 Well, it's something times something transpose. 733 00:45:50 --> 00:45:53 So what do I have to do? 734 00:45:53 --> 00:45:54 Opposite order. 735 00:45:54 --> 00:45:57 So I have a minus, an F transpose and the 736 00:45:57 --> 00:45:58 K inverse transpose. 737 00:45:58 --> 00:46:02 You're seeing all this stuff. 738 00:46:02 --> 00:46:06 And then comes the K and then comes the minus, 739 00:46:06 --> 00:46:07 oh again the minus. 740 00:46:07 --> 00:46:08 So that'd be a plus, right? 741 00:46:08 --> 00:46:13 Times K inverse times f. 742 00:46:13 --> 00:46:15 So that's one of the terms. 743 00:46:15 --> 00:46:18 That's one of the terms that shows up. 744 00:46:18 --> 00:46:22 And what good is that one? 745 00:46:22 --> 00:46:25 So that's one. 746 00:46:25 --> 00:46:27 You could say that's the longest term. 747 00:46:27 --> 00:46:30 That's the one with the messiest term. 748 00:46:30 --> 00:46:31 But you can fix it. 749 00:46:31 --> 00:46:34 What would you do with that? 750 00:46:34 --> 00:46:37 K times K inverse is? 751 00:46:37 --> 00:46:38 Identity. 752 00:46:38 --> 00:46:40 So we can forget that. 753 00:46:40 --> 00:46:42 And now we're there. 754 00:46:42 --> 00:46:45 That's 1/2, f transpose, f on this side. 755 00:46:45 --> 00:46:49 Oh, what's K inverse transpose? 756 00:46:49 --> 00:46:53 It's the same as K inverse because K is symmetric, so 757 00:46:53 --> 00:46:54 its inverse is symmetric. 758 00:46:54 --> 00:46:57 So that transpose doesn't change the matrix. 759 00:46:57 --> 00:47:08 In other words, this term will show up and this term is oh! 760 00:47:08 --> 00:47:11 Nope, sorry. 761 00:47:11 --> 00:47:13 I was going to goof here. 762 00:47:13 --> 00:47:15 I was going to say this is the same as this, 763 00:47:15 --> 00:47:16 but it's not, right? 764 00:47:16 --> 00:47:18 Why not? 765 00:47:18 --> 00:47:20 Because it's positive. 766 00:47:20 --> 00:47:24 And this guy is negative. 767 00:47:24 --> 00:47:27 Has my good friend Professor Lin messed up? 768 00:47:27 --> 00:47:32 Nope. 769 00:47:32 --> 00:47:35 What's going to happen now? 770 00:47:35 --> 00:47:39 The two that I didn't do, you see, the 1/2 u 771 00:47:39 --> 00:47:43 transpose K u is here. 772 00:47:43 --> 00:47:48 Then comes this one, which I didn't do, and then another 773 00:47:48 --> 00:47:51 one that I didn't do, and then this one that I did. 774 00:47:51 --> 00:47:53 They'll all be the same. 775 00:47:53 --> 00:47:57 So they'll all contribute with their plus sign or minus sign 776 00:47:57 --> 00:48:02 and the net result will be a perfect match, yeah. 777 00:48:02 --> 00:48:08 So I won't wear out your patience by doing that. 778 00:48:08 --> 00:48:11 But I do want to make the point. 779 00:48:11 --> 00:48:17 What was Professor Lin's point in suggesting to write it in 780 00:48:17 --> 00:48:21 this more complicated way? 781 00:48:21 --> 00:48:27 His point was we could see this is just a constant. 782 00:48:27 --> 00:48:29 Doesn't depend on u. 783 00:48:29 --> 00:48:34 And now I can see what value of u would make this 784 00:48:34 --> 00:48:35 as small as possible. 785 00:48:35 --> 00:48:38 Remember, I'm still trying to minimize. 786 00:48:38 --> 00:48:43 This part, I can't make it bigger or smaller, it's fixed. 787 00:48:43 --> 00:48:44 It's u that I can play with. 788 00:48:44 --> 00:48:51 So what u should I choose to make this part smaller? 789 00:48:51 --> 00:48:53 Bear with me. 790 00:48:53 --> 00:48:58 What u will make this big mess as small as I can get it 791 00:48:58 --> 00:49:01 and how small can I get it? 792 00:49:01 --> 00:49:07 If I take u to B K inverse f, then this is zero, 793 00:49:07 --> 00:49:10 this is zero, I get zero. 794 00:49:10 --> 00:49:15 And that's my claim, that u equal K inverse f is the best 795 00:49:15 --> 00:49:17 possible, is the minimizer. 796 00:49:17 --> 00:49:24 And how do I know that I can't make this more negative 797 00:49:24 --> 00:49:25 than the zero? 798 00:49:25 --> 00:49:30 I can get it down to zero by making that to 799 00:49:30 --> 00:49:31 be the zero vector. 800 00:49:31 --> 00:49:41 But how do I know I can't make it below zero? 801 00:49:41 --> 00:49:44 The K is positive definite and I'm sitting here with some 802 00:49:44 --> 00:49:48 x transpose and some x. 803 00:49:48 --> 00:49:52 The X hax this sort of messy form but it's an x and 804 00:49:52 --> 00:49:53 here's its transpose. 805 00:49:53 --> 00:50:00 So this is an x transpose, Kx and can't be brought below zero 806 00:50:00 --> 00:50:02 when K is positive definite. 807 00:50:02 --> 00:50:05 Good. 808 00:50:05 --> 00:50:09 So we've said a good bit about positive definite here, but 809 00:50:09 --> 00:50:19 happy to think-- Yeah, thanks. 810 00:50:19 --> 00:50:27 In fact, finally a fifth. 811 00:50:27 --> 00:50:29 Exactly. 812 00:50:29 --> 00:50:30 Thanks, perfect question. 813 00:50:30 --> 00:50:33 And let me answer it clearly. 814 00:50:33 --> 00:50:37 Each of those five tests completely decides 815 00:50:37 --> 00:50:38 positive definite. 816 00:50:38 --> 00:50:42 So the five tests are all equivalent. 817 00:50:42 --> 00:50:46 If a matrix passes one test, it passes all five. 818 00:50:46 --> 00:50:48 So that's great, right? 819 00:50:48 --> 00:50:52 So we just do whichever test we want. 820 00:50:52 --> 00:50:55 Or whichever way we want to understand the matrix. 821 00:50:55 --> 00:51:02 I was going to add, I didn't say a lot about this one. 822 00:51:02 --> 00:51:09 Can I just add a note about a MATLAB command? 823 00:51:09 --> 00:51:18 The command chol(K). 824 00:51:18 --> 00:51:23 That's the first letters in the name Cholesky. 825 00:51:23 --> 00:51:28 So chol is the first four letters of this name. 826 00:51:28 --> 00:51:34 And that's a MATLAB command. 827 00:51:34 --> 00:51:38 If I've defined a matrix that's positive definite and I use 828 00:51:38 --> 00:51:45 that command, out will pop an A, one particular A that works. 829 00:51:45 --> 00:51:50 Out will pop on A that makes this work. 830 00:51:50 --> 00:51:55 It'll be a square A and it'll be upper triangular. 831 00:51:55 --> 00:52:00 So out will pop, so this command is very, very close to 832 00:52:00 --> 00:52:06 the LU but it's just sort of the appropriate version, 833 00:52:06 --> 00:52:13 symmetrized version of elimination when you have a 834 00:52:13 --> 00:52:16 positive definite symmetric matrix. 835 00:52:16 --> 00:52:20 If your matrix is not positive definite, the MATLAB 836 00:52:20 --> 00:52:22 will tell you so. 837 00:52:22 --> 00:52:24 So it produces one particular A. 838 00:52:24 --> 00:52:28 There are many A's that would work, but there's one 839 00:52:28 --> 00:52:30 particular upper triangular one. 840 00:52:30 --> 00:52:40 It's just related to the usual u, but yes, thanks. 841 00:52:40 --> 00:52:43 No, I only even get into that ballpark if the 842 00:52:43 --> 00:52:45 matrix is symmetric. 843 00:52:45 --> 00:52:47 I don't touch it otherwise. 844 00:52:47 --> 00:52:51 So my matrix is symmetric before I begin. 845 00:52:51 --> 00:52:54 So I know good things about it. 846 00:52:54 --> 00:52:57 And here I'm asking for more. 847 00:52:57 --> 00:53:01 Here I'm asking are the pivots all positive? 848 00:53:01 --> 00:53:03 Are the eigenvalues all positive? 849 00:53:03 --> 00:53:06 So that's more. 850 00:53:06 --> 00:53:11 But I could think of some interpretation that would, for 851 00:53:11 --> 00:53:15 non-symmetric matrices, but it has problems, so I'd 852 00:53:15 --> 00:53:17 rather just leave it. 853 00:53:17 --> 00:53:22 Stay with symmetric. 854 00:53:22 --> 00:53:27 Well that's two hours of lots of linear algebra. 855 00:53:27 --> 00:53:31 I'm hoping you're going to like the MATLAB problem. 856 00:53:31 --> 00:53:41 Would you like to see what it'll be? 857 00:53:41 --> 00:53:44 I'll just tell you what the equation will be. 858 00:53:44 --> 00:53:49 So it'll be a differential equation. 859 00:53:49 --> 00:53:53 Oh, dear, what is it? 860 00:53:53 --> 00:53:57 So it's a differential equation with a -u'' 861 00:53:57 --> 00:54:00 that we know and love. 862 00:54:00 --> 00:54:03 And what else has it got? 863 00:54:03 --> 00:54:05 Oh yes, right. 864 00:54:05 --> 00:54:09 So here's the problem. 865 00:54:09 --> 00:54:13 Here's the equation. 866 00:54:13 --> 00:54:21 So it has the -u'', the second derivative and it has a first 867 00:54:21 --> 00:54:24 derivative equal whatever. 868 00:54:24 --> 00:54:28 In fact, the example will choose a delta function there. 869 00:54:28 --> 00:54:32 So what am I talking about here? 870 00:54:32 --> 00:54:37 This would be a diffusion and this would be, anybody 871 00:54:37 --> 00:54:39 met these things before? 872 00:54:39 --> 00:54:41 That would be a convection. 873 00:54:41 --> 00:54:46 So that's a first derivative, that's an anti-symmetric. 874 00:54:46 --> 00:54:50 The MATLAB problem is now going to create the difference 875 00:54:50 --> 00:54:51 matrix for that. 876 00:54:51 --> 00:54:55 So the symmetric part will be our old friend K. 877 00:54:55 --> 00:55:02 But now we've got the convection term is appearing. 878 00:55:02 --> 00:55:04 And it's going to be anti-symmetric. 879 00:55:04 --> 00:55:09 And if v is big, it gets more and more important. 880 00:55:09 --> 00:55:10 So what happens? 881 00:55:10 --> 00:55:13 What happens with equations like this? 882 00:55:13 --> 00:55:17 Really this is like the first time in the course that 883 00:55:17 --> 00:55:22 we've allowed this first derivative term to pop up. 884 00:55:22 --> 00:55:29 But nevertheless we can see a lot of what's happening. 885 00:55:29 --> 00:55:32 And how to deal with those equations? 886 00:55:32 --> 00:55:37 I mean, if you ask a chemical engineer or anybody, they're 887 00:55:37 --> 00:55:41 always dealing with a flow, like the Charles River is 888 00:55:41 --> 00:55:45 flowing along, that's coming from the velocity there, but 889 00:55:45 --> 00:55:48 at the same time stuff is diffusing in it. 890 00:55:48 --> 00:55:53 It's just a constant problem in true, true applications. 891 00:55:53 --> 00:55:56 And this is the best model, I think. 892 00:55:56 --> 00:56:03 So you'll see that and I'm pleased about that. 893 00:56:03 --> 00:56:06 As you'd see. 894 00:56:06 --> 00:56:07 Any last question? 895 00:56:07 --> 00:56:09 I'm always happy. 896 00:56:09 --> 00:56:10 Well I'll see you Friday then. 897 00:56:10 --> 00:56:12 Thanks for coming.