1 00:00:00,000 --> 00:00:00,247 2 00:00:00,247 --> 00:00:02,330 The following content is provided under a Creative 3 00:00:02,330 --> 00:00:03,610 Commons license. 4 00:00:03,610 --> 00:00:05,770 Your support will help MIT OpenCourseWare 5 00:00:05,770 --> 00:00:09,930 continue to offer high quality educational resources for free. 6 00:00:09,930 --> 00:00:12,530 To make a donation or to view additional materials 7 00:00:12,530 --> 00:00:16,150 from hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16,150 --> 00:00:19,390 at ocw.mit.edu. 9 00:00:19,390 --> 00:00:26,840 PROFESSOR STRANG: Shall we just start on this review session? 10 00:00:26,840 --> 00:00:31,200 So, any questions on anything from Chapter one, 11 00:00:31,200 --> 00:00:36,050 anything from those first seven lectures is very, very welcome. 12 00:00:36,050 --> 00:00:41,410 So this morning finished the serious part 13 00:00:41,410 --> 00:00:44,880 of what we'll do in the chapter with positive definite 14 00:00:44,880 --> 00:00:46,040 matrices. 15 00:00:46,040 --> 00:00:49,460 And we'll see a lot of those fortunately. 16 00:00:49,460 --> 00:00:52,050 They're the best. 17 00:00:52,050 --> 00:00:57,110 So questions about, I hope you look in the book, 18 00:00:57,110 --> 00:01:00,770 at other problems in the problem sets 19 00:01:00,770 --> 00:01:04,880 as well as the ones I suggest. 20 00:01:04,880 --> 00:01:09,090 And then I can, anyway. 21 00:01:09,090 --> 00:01:13,600 Ready for any questions. 22 00:01:13,600 --> 00:01:14,930 Okay. 23 00:01:14,930 --> 00:01:19,140 Which problem is it? 24 00:01:19,140 --> 00:01:21,740 In section? 25 00:01:21,740 --> 00:01:25,920 Section 1.6, problem 27, what have I done there? 26 00:01:25,920 --> 00:01:28,310 Oh, okay, that's good. 27 00:01:28,310 --> 00:01:30,790 So it's about positive definite matrices. 28 00:01:30,790 --> 00:01:36,890 May I just put on the board what the central question is? 29 00:01:36,890 --> 00:01:40,270 Just put these matrices up. 30 00:01:40,270 --> 00:01:43,910 We're given that H and K are positive definite. 31 00:01:43,910 --> 00:01:47,010 And then the question is, what about these block matrices. 32 00:01:47,010 --> 00:01:50,290 Do I call them M and N? 33 00:01:50,290 --> 00:01:54,630 One is the block matrix that looks like that. 34 00:01:54,630 --> 00:01:59,600 And another one is the block matrix that looks like this. 35 00:01:59,600 --> 00:02:04,330 So those are both symmetric. 36 00:02:04,330 --> 00:02:06,739 We're allowed to ask, are they positive 37 00:02:06,739 --> 00:02:08,530 definite or negative definite, because they 38 00:02:08,530 --> 00:02:10,060 passed the first requirement. 39 00:02:10,060 --> 00:02:10,810 They're symmetric. 40 00:02:10,810 --> 00:02:12,410 We can discuss them. 41 00:02:12,410 --> 00:02:15,620 Because of course H and K each were symmetric. 42 00:02:15,620 --> 00:02:19,530 The transpose of this would bring K transpose down here, 43 00:02:19,530 --> 00:02:22,220 but that's K, so all good. 44 00:02:22,220 --> 00:02:32,520 So the question now. 45 00:02:32,520 --> 00:02:43,030 Of these guys to those guys I guess, yes. 46 00:02:43,030 --> 00:02:44,210 Good question. 47 00:02:44,210 --> 00:02:47,260 So this guy has, let's take eigenvalues first. 48 00:02:47,260 --> 00:02:49,700 So this guy has some eigenvalues, 49 00:02:49,700 --> 00:02:53,000 say lambda_1 to lambda_n. 50 00:02:53,000 --> 00:02:54,650 And this guy, we'll suppose they're 51 00:02:54,650 --> 00:02:57,230 the same size, so they don't have to be. 52 00:02:57,230 --> 00:02:59,490 Maybe I shouldn't, but I will. 53 00:02:59,490 --> 00:03:05,410 This has some other eigenvalues, maybe 54 00:03:05,410 --> 00:03:09,080 e_1 to e_n for eigenvalue. 55 00:03:09,080 --> 00:03:12,780 And then the question is, okay, what about the eigenvalues 56 00:03:12,780 --> 00:03:15,930 of that combination? 57 00:03:15,930 --> 00:03:16,810 And what about this? 58 00:03:16,810 --> 00:03:20,040 So it's a good question, I think for all of us 59 00:03:20,040 --> 00:03:23,840 to practice what just came up in the lecture. 60 00:03:23,840 --> 00:03:29,150 The idea of block matrices. 61 00:03:29,150 --> 00:03:36,970 So looking here at eigenvalues I could also look at pivots. 62 00:03:36,970 --> 00:03:39,530 Pivots would be interesting to look at, too. 63 00:03:39,530 --> 00:03:40,910 Maybe I'll start with pivots. 64 00:03:40,910 --> 00:03:42,700 Can I? 65 00:03:42,700 --> 00:03:43,510 Did you think? 66 00:03:43,510 --> 00:03:44,970 What would be the pivots of M? 67 00:03:44,970 --> 00:03:51,950 If I start elimination on M what will I see for pivots? 68 00:03:51,950 --> 00:03:57,670 Well, I start up in the usual left-hand corner and work down. 69 00:03:57,670 --> 00:04:00,190 So what am I going to see first? 70 00:04:00,190 --> 00:04:03,130 I'm going to see the pivots of H. It won't even know, 71 00:04:03,130 --> 00:04:07,860 by the time I had halfway there, it won't even have seen K. 72 00:04:07,860 --> 00:04:10,680 And then, that'll be fine. 73 00:04:10,680 --> 00:04:16,100 And then this will be, what's going to happen? 74 00:04:16,100 --> 00:04:18,160 This is all zeroes. 75 00:04:18,160 --> 00:04:20,360 So never get touched, right? 76 00:04:20,360 --> 00:04:26,080 So when I get down to the second half I see all zeroes here. 77 00:04:26,080 --> 00:04:28,250 K is still going to be sitting right there. 78 00:04:28,250 --> 00:04:29,320 Nothing happened. 79 00:04:29,320 --> 00:04:33,390 Because when I did these eliminations nothing changed 80 00:04:33,390 --> 00:04:35,970 with K. So the rest of the pivots 81 00:04:35,970 --> 00:04:40,210 will be the pivots of K. Good. 82 00:04:40,210 --> 00:04:42,900 Now, we might hope for the same thing with eigenvalues 83 00:04:42,900 --> 00:04:46,720 and probably that's going to happen. 84 00:04:46,720 --> 00:04:49,430 This is like a diagonal matrix. 85 00:04:49,430 --> 00:04:51,610 And actually, what words would I use? 86 00:04:51,610 --> 00:04:53,000 Block diagonal. 87 00:04:53,000 --> 00:04:55,560 I'd call that matrix block diagonal. 88 00:04:55,560 --> 00:04:58,040 And those are very nice matrices. 89 00:04:58,040 --> 00:05:01,030 That tells us that the big matrix, 90 00:05:01,030 --> 00:05:03,690 for all practical purposes, is breaking up 91 00:05:03,690 --> 00:05:06,390 into these smaller blocks. 92 00:05:06,390 --> 00:05:10,270 Actually MATLAB will search for a way 93 00:05:10,270 --> 00:05:14,930 to reorder the rows and columns to get that in case it's 94 00:05:14,930 --> 00:05:15,870 possible. 95 00:05:15,870 --> 00:05:21,730 So here it's in front of us. 96 00:05:21,730 --> 00:05:24,100 Let's see if we can figure out. 97 00:05:24,100 --> 00:05:31,330 That lambda_1, I believe, is also an eigenvalue of M. 98 00:05:31,330 --> 00:05:34,530 So it was an eigenvalue of H. So that this, 99 00:05:34,530 --> 00:05:38,560 the fact that it has that eigenvalue lambda_1 means what? 100 00:05:38,560 --> 00:05:48,160 That H times this times some vector y is lambda_1*y, right? 101 00:05:48,160 --> 00:05:51,210 If that's an eigenvalue it's got an eigenvector 102 00:05:51,210 --> 00:05:53,270 and let's call it y. 103 00:05:53,270 --> 00:05:56,720 Now this is a good question. 104 00:05:56,720 --> 00:06:01,630 I believe this block matrix also has eigenvalue lambda_1, 105 00:06:01,630 --> 00:06:03,670 and what's its eigenvector? 106 00:06:03,670 --> 00:06:08,830 What could I multiply M by to get 107 00:06:08,830 --> 00:06:13,040 lambda_1 times the same thing? 108 00:06:13,040 --> 00:06:14,050 Can you see what? 109 00:06:14,050 --> 00:06:17,620 Of course I'm thinking that y is going to help 110 00:06:17,620 --> 00:06:20,460 but it's grown now. 111 00:06:20,460 --> 00:06:23,830 So what would be the eigenvector here? 112 00:06:23,830 --> 00:06:26,950 When I multiply by M it'll just come out right 113 00:06:26,950 --> 00:06:33,840 with the same eigenvalue? y_1, or y rather, and then? 114 00:06:33,840 --> 00:06:36,540 And then zero, good. [y; 0]. 115 00:06:36,540 --> 00:06:41,210 Because if I multiply, can I put in what M really is? 116 00:06:41,210 --> 00:06:45,990 The H and K. H there, K there. 117 00:06:45,990 --> 00:06:49,300 When I do that multiplication I get lambda_1*y. 118 00:06:49,300 --> 00:06:51,850 When I do this multiplication, see I've just, 119 00:06:51,850 --> 00:06:55,290 that's a zero block, zero, so I got a zero. 120 00:06:55,290 --> 00:06:56,620 Perfect. 121 00:06:56,620 --> 00:07:05,510 So the eigenvectors of H just sit with a zero in the K part 122 00:07:05,510 --> 00:07:08,730 and produce an eigenvector of the block 123 00:07:08,730 --> 00:07:11,260 matrix with the same lambda_1. 124 00:07:11,260 --> 00:07:14,060 So you can see then, we get the whole picture. 125 00:07:14,060 --> 00:07:17,870 The eigenvalues are just sitting there 126 00:07:17,870 --> 00:07:20,240 and the eigenvectors are there. 127 00:07:20,240 --> 00:07:24,170 Now maybe you got all that and wanted-- well 128 00:07:24,170 --> 00:07:27,450 I haven't said anything about N, Sorry. 129 00:07:27,450 --> 00:07:31,750 Everybody thinks more about N. So what's the thing with N? 130 00:07:31,750 --> 00:07:34,000 What would you say about N? 131 00:07:34,000 --> 00:07:37,400 If you look at that matrix, suppose I don't even tell you 132 00:07:37,400 --> 00:07:40,270 it's positive definite at first, would you 133 00:07:40,270 --> 00:07:45,080 say that looks like a invertible or singular matrix? 134 00:07:45,080 --> 00:07:48,550 Everybody's going to say singular. 135 00:07:48,550 --> 00:07:55,040 And why would you say that's singular? 136 00:07:55,040 --> 00:08:05,160 Well, the determinant of a block matrix, this morning I said 137 00:08:05,160 --> 00:08:07,420 do whatever you like with block matrices. 138 00:08:07,420 --> 00:08:13,890 But I have to admit that if I had a bunch of general blocks, 139 00:08:13,890 --> 00:08:17,270 if I had to take the determinant of that, and of course 140 00:08:17,270 --> 00:08:19,830 everybody's remembering Professor Strang doesn't like 141 00:08:19,830 --> 00:08:24,250 determinants, if I had to take the determinant, 142 00:08:24,250 --> 00:08:27,940 I'd have to do the whole thing. 143 00:08:27,940 --> 00:08:31,900 The separate determinants would not tell me the story, usually. 144 00:08:31,900 --> 00:08:33,990 So determinants are a bit tricky. 145 00:08:33,990 --> 00:08:37,700 But up here the determinant will come out zero. 146 00:08:37,700 --> 00:08:45,460 I guess what I would hope your internal test for a singular 147 00:08:45,460 --> 00:08:50,050 matrix is, are the columns independent? 148 00:08:50,050 --> 00:08:52,280 And then the matrix is invertible. 149 00:08:52,280 --> 00:08:53,710 Or are they dependent? 150 00:08:53,710 --> 00:08:59,070 Do you have some columns that are 151 00:08:59,070 --> 00:09:01,860 in the same direction as other columns, same direction 152 00:09:01,860 --> 00:09:03,910 as combinations of other columns? 153 00:09:03,910 --> 00:09:08,640 If you look at the columns of that, say column one, 154 00:09:08,640 --> 00:09:14,150 so column one is the first column of K repeated. 155 00:09:14,150 --> 00:09:17,290 What do you think about the columns of that matrix, that 156 00:09:17,290 --> 00:09:18,540 block matrix N? 157 00:09:18,540 --> 00:09:24,200 Do you see that same column showing up again? 158 00:09:24,200 --> 00:09:25,550 Yeah. 159 00:09:25,550 --> 00:09:28,760 That very same column, which is the first column of K, 160 00:09:28,760 --> 00:09:32,970 again twice, is going to show up right there, first column of K 161 00:09:32,970 --> 00:09:33,880 again. 162 00:09:33,880 --> 00:09:39,140 So this matrix has two identical columns. 163 00:09:39,140 --> 00:09:41,370 No way it could be invertible. 164 00:09:41,370 --> 00:09:44,650 And in fact, you can tell me what vector, 165 00:09:44,650 --> 00:09:48,810 I'm always saying are the columns independent? 166 00:09:48,810 --> 00:09:50,870 Here, no, they're dependent. 167 00:09:50,870 --> 00:09:56,600 And then you can tell me an x. 168 00:09:56,600 --> 00:09:59,640 So this is my block matrix N. I want 169 00:09:59,640 --> 00:10:06,830 to know an x so that the result is zero. 170 00:10:06,830 --> 00:10:13,880 That's really my same indication. 171 00:10:13,880 --> 00:10:15,830 We found two identical columns. 172 00:10:15,830 --> 00:10:19,640 What would be the x? 173 00:10:19,640 --> 00:10:22,970 Well, you have to tell me more than one, minus one 174 00:10:22,970 --> 00:10:30,790 because I've got a big x there. 175 00:10:30,790 --> 00:10:32,360 Yeah I've gotta make it big enough, 176 00:10:32,360 --> 00:10:34,620 but essentially it's the one, minus one, thanks. 177 00:10:34,620 --> 00:10:38,930 And enough zeroes in there and enough zeroes in there. 178 00:10:38,930 --> 00:10:45,210 So the fact that that vector gets taken to zero 179 00:10:45,210 --> 00:10:49,700 is the same thing as saying that one of this column minus one 180 00:10:49,700 --> 00:10:51,810 of this column gives zero. 181 00:10:51,810 --> 00:10:53,900 In other words, the columns are the same. 182 00:10:53,900 --> 00:10:55,770 And of course, by doing this we're 183 00:10:55,770 --> 00:11:01,010 seeing the one and minus one could have gone into position 184 00:11:01,010 --> 00:11:02,950 two there, position three. 185 00:11:02,950 --> 00:11:06,140 So we've got a whole bunch of vectors. 186 00:11:06,140 --> 00:11:13,860 This matrix N, this [K, K; K, K] has got a whole lot of vectors 187 00:11:13,860 --> 00:11:15,610 that it takes to zero. 188 00:11:15,610 --> 00:11:18,850 What I would say it has a large null space. 189 00:11:18,850 --> 00:11:22,510 A large space of vectors that it takes to zero. 190 00:11:22,510 --> 00:11:25,290 So that's a really useful exercise. 191 00:11:25,290 --> 00:11:26,440 I'm delighted you asked it. 192 00:11:26,440 --> 00:11:34,050 Now I'm ready for more. 193 00:11:34,050 --> 00:11:34,850 Could do. 194 00:11:34,850 --> 00:11:37,060 Exactly, row reduction. 195 00:11:37,060 --> 00:11:39,740 I should look to see what would happen in elimination. 196 00:11:39,740 --> 00:11:43,470 Well, elimination would go swimmingly along 197 00:11:43,470 --> 00:11:46,200 for the first part because it's only looking here. 198 00:11:46,200 --> 00:11:56,390 But then what would I have after the first half of elimination? 199 00:11:56,390 --> 00:12:04,012 Well I'd have I suppose whatever that K changed to, elimination. 200 00:12:04,012 --> 00:12:04,970 What should we call it? 201 00:12:04,970 --> 00:12:09,360 U or something? 202 00:12:09,360 --> 00:12:12,950 When I did these row steps that matrix 203 00:12:12,950 --> 00:12:15,360 turned into this upper triangular matrix. 204 00:12:15,360 --> 00:12:18,110 And maybe you can tell me what will have happened 205 00:12:18,110 --> 00:12:20,900 at the same time to the rest? 206 00:12:20,900 --> 00:12:25,990 What will I see sitting here if I just do ordinary elimination 207 00:12:25,990 --> 00:12:29,470 and I'm just looking there and using the pivots 208 00:12:29,470 --> 00:12:31,720 and so on, I'll see? 209 00:12:31,720 --> 00:12:35,470 It'll be U because whenever I do on the left side 210 00:12:35,470 --> 00:12:37,410 I'm doing to the whole row. 211 00:12:37,410 --> 00:12:40,100 And now, the main point is, what will I see? 212 00:12:40,100 --> 00:12:42,970 Now elimination, keep going, keep going. 213 00:12:42,970 --> 00:12:46,920 Do elimination to clear out this column, 214 00:12:46,920 --> 00:12:49,020 this whole bunch, right? 215 00:12:49,020 --> 00:12:52,110 Elimination. 216 00:12:52,110 --> 00:12:54,700 And now what am I going to see in that corner? 217 00:12:54,700 --> 00:12:57,980 All zeroes, right. 218 00:12:57,980 --> 00:13:05,910 So that's telling me that the matrix has just 219 00:13:05,910 --> 00:13:11,070 got half of the eigenvalues positive, half of the pivots 220 00:13:11,070 --> 00:13:12,000 are positive. 221 00:13:12,000 --> 00:13:16,700 The second half all zeroes. 222 00:13:16,700 --> 00:13:19,490 So I guess, here I've found an eigenvector 223 00:13:19,490 --> 00:13:22,330 with what eigenvalue? 224 00:13:22,330 --> 00:13:24,720 That's looking like an eigenvector to me 225 00:13:24,720 --> 00:13:26,720 if we're thinking eigenvectors. 226 00:13:26,720 --> 00:13:30,050 And what's the eigenvalue that goes with it? 227 00:13:30,050 --> 00:13:30,730 Zero. 228 00:13:30,730 --> 00:13:34,150 Because Nx is 0x. 229 00:13:34,150 --> 00:13:38,680 You can either think of it as Nx=0 if you're thinking about 230 00:13:38,680 --> 00:13:40,200 systems of equations. 231 00:13:40,200 --> 00:13:46,860 Or Nx=0x if you're thinking that that guy is an eigenvector with 232 00:13:46,860 --> 00:13:49,550 eigenvalue zero. 233 00:13:49,550 --> 00:13:51,230 So I'm pretty happy. 234 00:13:51,230 --> 00:13:55,370 I mean many of you will have spotted this. 235 00:13:55,370 --> 00:13:56,640 Probably perhaps all. 236 00:13:56,640 --> 00:13:59,250 But I'm happy that's an example that 237 00:13:59,250 --> 00:14:03,870 just shows how you have to think big 238 00:14:03,870 --> 00:14:06,690 with block matrices I guess. 239 00:14:06,690 --> 00:14:08,030 Good. 240 00:14:08,030 --> 00:14:11,470 Okay on that? 241 00:14:11,470 --> 00:14:27,231 What else, thanks. 242 00:14:27,231 --> 00:14:27,730 That's true. 243 00:14:27,730 --> 00:14:31,530 And that's really all I've done so far is those four examples. 244 00:14:31,530 --> 00:14:35,910 I think that language of fixed-fixed and fixed-free 245 00:14:35,910 --> 00:14:40,180 really comes, I mean I used it early about those four 246 00:14:40,180 --> 00:14:43,680 matrices, but it's really going to show up 247 00:14:43,680 --> 00:14:45,670 at the next lecture, Friday, when 248 00:14:45,670 --> 00:14:53,170 I have a line of springs and the matrices that come out of that. 249 00:14:53,170 --> 00:15:00,050 So Friday we'll finally be on those first four. 250 00:15:00,050 --> 00:15:05,500 A fifth matrix will appear in this course finally. 251 00:15:05,500 --> 00:15:08,770 Of course, it's going to be related to the first ones, 252 00:15:08,770 --> 00:15:14,630 naturally, but we'll move to, we'll see something new 253 00:15:14,630 --> 00:15:17,860 and then we'll see the fixed-free idea again 254 00:15:17,860 --> 00:15:18,780 for those. 255 00:15:18,780 --> 00:15:20,965 So if that can wait until Friday, 256 00:15:20,965 --> 00:15:24,160 you'll see some different ones. 257 00:15:24,160 --> 00:15:26,320 Good. 258 00:15:26,320 --> 00:15:28,680 Questions, thoughts. 259 00:15:28,680 --> 00:15:31,110 You can ask about anything. 260 00:15:31,110 --> 00:15:35,720 Maybe I can ask. 261 00:15:35,720 --> 00:15:40,660 Any thoughts about the pace of the course? 262 00:15:40,660 --> 00:15:49,640 This is sort of a heavy dose of linear algebra, right? 263 00:15:49,640 --> 00:15:53,320 Of course, the answer maybe depends on how much 264 00:15:53,320 --> 00:15:56,160 you had seen before. 265 00:15:56,160 --> 00:15:59,920 So those who haven't seen very much linear algebra at all 266 00:15:59,920 --> 00:16:04,950 really got quite a bit quickly here. 267 00:16:04,950 --> 00:16:06,940 Because many courses on linear algebra 268 00:16:06,940 --> 00:16:12,620 never reach this key idea of positive definiteness 269 00:16:12,620 --> 00:16:16,980 that ties it all together. 270 00:16:16,980 --> 00:16:19,620 So you've seen quite a bit, really. 271 00:16:19,620 --> 00:16:23,040 Of course, we've concentrated on symmetric matrices 272 00:16:23,040 --> 00:16:27,790 and there's a whole garden or forest or zoo 273 00:16:27,790 --> 00:16:32,870 of matrices of different types. 274 00:16:32,870 --> 00:16:34,720 So what matrices have we seen? 275 00:16:34,720 --> 00:16:39,760 Symmetric matrices and then their eigenvectors 276 00:16:39,760 --> 00:16:44,130 were orthogonal and we could say orthonormal. 277 00:16:44,130 --> 00:16:47,990 So that gave us, I don't know if you 278 00:16:47,990 --> 00:16:53,650 remember this part, which when we wrote it down I said, 279 00:16:53,650 --> 00:16:54,860 big deal. 280 00:16:54,860 --> 00:16:56,370 That's very important. 281 00:16:56,370 --> 00:16:59,750 That's this principal axis theorem. 282 00:16:59,750 --> 00:17:04,810 These Q's, what kind of a matrix is Q? 283 00:17:04,810 --> 00:17:06,730 It's the eigenvector matrix. 284 00:17:06,730 --> 00:17:12,370 And for symmetric matrix, so this is the eigenvector matrix. 285 00:17:12,370 --> 00:17:14,820 And what do we know about it? 286 00:17:14,820 --> 00:17:22,540 In the special case of symmetric K? 287 00:17:22,540 --> 00:17:27,260 What do we know especially about the eigenvectors then? 288 00:17:27,260 --> 00:17:28,440 They're orthogonal. 289 00:17:28,440 --> 00:17:29,890 We can make them orthonormal. 290 00:17:29,890 --> 00:17:34,710 So this will be an orthogonal matrix. 291 00:17:34,710 --> 00:17:39,040 And that was a matrix with Q transpose 292 00:17:39,040 --> 00:17:41,400 was the same as Q inverse. 293 00:17:41,400 --> 00:17:43,410 Normally we would see the inverse there, 294 00:17:43,410 --> 00:17:47,350 but for these we can put the transpose. 295 00:17:47,350 --> 00:17:52,140 Here's one type of matrix, symmetric, very important. 296 00:17:52,140 --> 00:17:56,120 Here's another type of matrix, orthogonal matrices. 297 00:17:56,120 --> 00:17:58,550 And of course, many, many other varieties. 298 00:17:58,550 --> 00:18:00,610 Well here we have a very nice matrix, 299 00:18:00,610 --> 00:18:03,870 so that matrix is diagonal. 300 00:18:03,870 --> 00:18:06,380 Right, that's just the eigenvalues, 301 00:18:06,380 --> 00:18:08,710 so that's a diagonal matrix. 302 00:18:08,710 --> 00:18:12,350 And what do we know, if K is positive definite, let's just, 303 00:18:12,350 --> 00:18:14,690 this was for any symmetric one. 304 00:18:14,690 --> 00:18:19,020 So what's special if K is positive definite? 305 00:18:19,020 --> 00:18:20,860 Somehow the positive definiteness 306 00:18:20,860 --> 00:18:22,820 should show up here. 307 00:18:22,820 --> 00:18:26,520 And where does it show? 308 00:18:26,520 --> 00:18:29,150 Positive eigenvalues, exactly. 309 00:18:29,150 --> 00:18:33,790 The Q could be any, any Q would be fine. 310 00:18:33,790 --> 00:18:37,710 But we would see positive eigenvalues. 311 00:18:37,710 --> 00:18:41,760 Oh, here's a little point about eigenvalues. 312 00:18:41,760 --> 00:18:47,080 Suppose I have my matrix K. And it's got some eigenvalues. 313 00:18:47,080 --> 00:18:57,220 Now let me add four times the identity to it. 314 00:18:57,220 --> 00:18:59,320 What are the eigenvalues now? 315 00:18:59,320 --> 00:19:02,230 What are the eigenvectors now? 316 00:19:02,230 --> 00:19:08,710 What's changed and how and what hasn't changed? 317 00:19:08,710 --> 00:19:11,710 Because that's a pretty easy, the identity matrix 318 00:19:11,710 --> 00:19:15,910 is always the easy one for us to know what's happening. 319 00:19:15,910 --> 00:19:20,270 So what is happening to the eigenvalues now? 320 00:19:20,270 --> 00:19:23,380 If K had these eigenvalues lambda, 321 00:19:23,380 --> 00:19:25,010 what are the eigenvalues of K+4I? 322 00:19:25,010 --> 00:19:31,010 323 00:19:31,010 --> 00:19:31,849 You add? 324 00:19:31,849 --> 00:19:32,640 You add four, yeah. 325 00:19:32,640 --> 00:19:38,620 The eigenvalues of this are the eigenvalues of K plus four. 326 00:19:38,620 --> 00:19:41,830 That is just like shifting the matrix, 327 00:19:41,830 --> 00:19:44,630 you could think of it is adding four 328 00:19:44,630 --> 00:19:48,340 along the diagonal will add four. 329 00:19:48,340 --> 00:19:54,300 And the eigenvectors would be exactly the same ones. 330 00:19:54,300 --> 00:19:57,570 I would have Kx would agree with lambda*x. 331 00:19:57,570 --> 00:20:00,440 And 4Ix would agree with 4x. 332 00:20:00,440 --> 00:20:05,250 So that proves it. 333 00:20:05,250 --> 00:20:11,680 Good to see what you can do, the limited number of things 334 00:20:11,680 --> 00:20:14,640 that you're allowed to do without changing 335 00:20:14,640 --> 00:20:18,300 the eigenvectors, and therefore you can spot the eigenvalues 336 00:20:18,300 --> 00:20:19,280 right away. 337 00:20:19,280 --> 00:20:21,330 The limited things you can invert, 338 00:20:21,330 --> 00:20:25,820 you can shift like this, you could square it, cube it, 339 00:20:25,820 --> 00:20:34,530 take powers, things like that. 340 00:20:34,530 --> 00:20:38,200 I'm going to look to you now for giving me a lead on something 341 00:20:38,200 --> 00:20:42,960 that is interesting or not. 342 00:20:42,960 --> 00:20:48,120 Yes, thanks. 343 00:20:48,120 --> 00:20:52,900 Go ahead. 344 00:20:52,900 --> 00:21:02,700 Oh, I see okay, yes. 345 00:21:02,700 --> 00:21:03,540 I see. 346 00:21:03,540 --> 00:21:05,370 Alright. 347 00:21:05,370 --> 00:21:08,350 So that's page 64 of the book. 348 00:21:08,350 --> 00:21:18,220 Well, so that's a problem that physicists love. 349 00:21:18,220 --> 00:21:21,360 I don't know how much I can say about it here, 350 00:21:21,360 --> 00:21:23,010 to tell the truth. 351 00:21:23,010 --> 00:21:26,300 Just to mention. 352 00:21:26,300 --> 00:21:28,730 Do they use a minus sign? 353 00:21:28,730 --> 00:21:30,230 Probably they do. 354 00:21:30,230 --> 00:21:40,120 So their equation is minus the second derivative of u plus (x 355 00:21:40,120 --> 00:21:46,460 squared)*u, and they are interested in the eigenvalues, 356 00:21:46,460 --> 00:21:54,080 equal lambda*u. 357 00:21:54,080 --> 00:21:58,880 The case that we've done in class was without this (x 358 00:21:58,880 --> 00:22:01,680 squared)*u term, right? 359 00:22:01,680 --> 00:22:07,600 The absolutely most important case is the second derivative 360 00:22:07,600 --> 00:22:09,990 of u equal lambda*u. 361 00:22:09,990 --> 00:22:12,300 The eigenvalues were, or what were 362 00:22:12,300 --> 00:22:15,740 the eigenvectors in that case? 363 00:22:15,740 --> 00:22:21,070 What were the eigenvectors of the second derivative before 364 00:22:21,070 --> 00:22:27,760 there was any (x squared)*u, any potential showing up? 365 00:22:27,760 --> 00:22:30,760 They were just sines and cosines, right? 366 00:22:30,760 --> 00:22:32,690 Sines and cosines have the property 367 00:22:32,690 --> 00:22:35,600 that if you take two derivatives you get them back 368 00:22:35,600 --> 00:22:40,330 with some factor lambda. 369 00:22:40,330 --> 00:22:44,680 Now let me just look at that problem 370 00:22:44,680 --> 00:22:49,550 without saying much about it. 371 00:22:49,550 --> 00:22:52,920 First of all, the first thing I want to know 372 00:22:52,920 --> 00:22:55,250 is have I got a linear problem here? 373 00:22:55,250 --> 00:22:57,740 Have I got a linear equation? 374 00:22:57,740 --> 00:23:00,730 Because that's where I talk about eigenvalues. 375 00:23:00,730 --> 00:23:05,120 So in the matrix case, I'd say I have a matrix. 376 00:23:05,120 --> 00:23:09,550 K times an eigenvector. 377 00:23:09,550 --> 00:23:14,210 That matrix represents something linear. 378 00:23:14,210 --> 00:23:18,280 It's just, all the rules of addition work here. 379 00:23:18,280 --> 00:23:20,310 Here it is linear. 380 00:23:20,310 --> 00:23:27,110 It is linear. 381 00:23:27,110 --> 00:23:33,570 What I'm trying to say is, I just 382 00:23:33,570 --> 00:23:36,330 call that a variable coefficient and that's what we're 383 00:23:36,330 --> 00:23:38,480 going to see in Chapter two. 384 00:23:38,480 --> 00:23:44,090 The material or something could lead to some dependence on x. 385 00:23:44,090 --> 00:23:49,810 But u is still there, just linearly. 386 00:23:49,810 --> 00:23:55,520 In other words, this is a perfectly okay linear operator 387 00:23:55,520 --> 00:24:00,100 and am I imagining that it's positive definite? 388 00:24:00,100 --> 00:24:00,900 Let's see. 389 00:24:00,900 --> 00:24:08,090 This part with the minus sign was positive definite, right? 390 00:24:08,090 --> 00:24:12,100 Well, at least semi-definite. 391 00:24:12,100 --> 00:24:15,660 So let me just remember the most important case. 392 00:24:15,660 --> 00:24:20,720 If I look at this equation, d second u/dx squared equals 393 00:24:20,720 --> 00:24:22,970 lambda*u. 394 00:24:22,970 --> 00:24:28,550 So that's the eigenvalue, eigenfunction problem 395 00:24:28,550 --> 00:24:30,990 for our good friend. 396 00:24:30,990 --> 00:24:35,570 What do I say about the eigenvalues now? 397 00:24:35,570 --> 00:24:40,010 What can you tell me about the eigenvalues of that? 398 00:24:40,010 --> 00:24:41,160 Mostly positive. 399 00:24:41,160 --> 00:24:44,680 Because they were sort of omega squares. 400 00:24:44,680 --> 00:24:47,510 But I mean zero could be an eigenvalue, right? 401 00:24:47,510 --> 00:24:55,620 What would the eigenfunction be for lambda equal zero? 402 00:24:55,620 --> 00:24:58,330 If I wanted to get zero here, if I 403 00:24:58,330 --> 00:25:02,990 wanted a zero on the right side, what functions u 404 00:25:02,990 --> 00:25:06,780 could give me zero? 405 00:25:06,780 --> 00:25:08,840 Constant function. 406 00:25:08,840 --> 00:25:12,250 Yeah, the constant function is certainly there 407 00:25:12,250 --> 00:25:14,030 as a possibility. 408 00:25:14,030 --> 00:25:19,490 But anyway, I would say this is positive semi-definite 409 00:25:19,490 --> 00:25:20,820 at least. 410 00:25:20,820 --> 00:25:23,150 And this part? 411 00:25:23,150 --> 00:25:26,290 How do I think about that as a big matrix? 412 00:25:26,290 --> 00:25:29,670 I think of it sort of like a big matrix with x squared 413 00:25:29,670 --> 00:25:36,060 running down the diagonal. 414 00:25:36,060 --> 00:25:38,800 With a matrix, you could say walking down the diagonal 415 00:25:38,800 --> 00:25:41,670 because it's n steps. 416 00:25:41,670 --> 00:25:45,300 For differential equations, maybe running 417 00:25:45,300 --> 00:25:46,420 is the right word. 418 00:25:46,420 --> 00:25:53,250 Because it doesn't jump, it's just bzzz all the way 419 00:25:53,250 --> 00:25:56,250 from zero squared to whatever. 420 00:25:56,250 --> 00:26:05,430 Anyway, that would correspond to a diagonal matrix, but not 421 00:26:05,430 --> 00:26:07,490 constant diagonal. 422 00:26:07,490 --> 00:26:09,070 Diagonal, but not constant diagonal. 423 00:26:09,070 --> 00:26:13,150 Because this x squared number is changing. 424 00:26:13,150 --> 00:26:16,590 It's like a spring, it's like a bunch 425 00:26:16,590 --> 00:26:22,730 of springs in which the first spring maybe 426 00:26:22,730 --> 00:26:24,940 has a spring constant of one. 427 00:26:24,940 --> 00:26:27,840 And then we have a tighter spring and then a very tight 428 00:26:27,840 --> 00:26:31,400 spring and so on, more and more, higher and higher 429 00:26:31,400 --> 00:26:32,720 constants there. 430 00:26:32,720 --> 00:26:39,720 Well, I'm just speaking very roughly here. 431 00:26:39,720 --> 00:26:44,490 Because variable coefficient, variable material properties, 432 00:26:44,490 --> 00:26:48,800 springs of different elasticities, 433 00:26:48,800 --> 00:26:51,810 we're ready to move to that. 434 00:26:51,810 --> 00:26:57,720 Our problems up to now, the springs were all the same. 435 00:26:57,720 --> 00:27:01,270 The bar, if it was a bar, was uniform. 436 00:27:01,270 --> 00:27:05,380 And now this would be a step forward. 437 00:27:05,380 --> 00:27:09,840 But now, of course, this specific problem 438 00:27:09,840 --> 00:27:15,060 just happens to have a solution that physicists love. 439 00:27:15,060 --> 00:27:19,590 It has a meaning to physicists, not to me. 440 00:27:19,590 --> 00:27:23,080 And the eigenfunctions have a meaning 441 00:27:23,080 --> 00:27:26,450 and they're famous functions. 442 00:27:26,450 --> 00:27:28,100 It's just glorious. 443 00:27:28,100 --> 00:27:31,510 So you could say that's the special problem, the way we 444 00:27:31,510 --> 00:27:35,540 had four special matrices in 18.085, 445 00:27:35,540 --> 00:27:44,310 that would be a similar special problem in quantum mechanics. 446 00:27:44,310 --> 00:27:48,390 Let's turn to something entirely different. 447 00:27:48,390 --> 00:27:51,590 Questions about any topic. 448 00:27:51,590 --> 00:27:54,900 Or I can ask some and you can take this, maybe 449 00:27:54,900 --> 00:27:56,350 that's one way to review. 450 00:27:56,350 --> 00:27:57,020 Go ahead. 451 00:27:57,020 --> 00:27:59,010 Thanks. 452 00:27:59,010 --> 00:28:03,570 Number 20 of 1.6. 453 00:28:03,570 --> 00:28:06,360 1.6 is a section, oh, no. 454 00:28:06,360 --> 00:28:11,930 That's positive definite notes so I'm okay with that. 455 00:28:11,930 --> 00:28:15,590 I see that I did ask you a question on the homework 456 00:28:15,590 --> 00:28:22,320 from 1.7 which I may not get to cover in lecture, 457 00:28:22,320 --> 00:28:24,900 but give it a shot anyway. 458 00:28:24,900 --> 00:28:28,620 So what's 20? 459 00:28:28,620 --> 00:28:32,620 Oh, okay, that's good. 460 00:28:32,620 --> 00:28:36,570 Without multiplying out the matrix. 461 00:28:36,570 --> 00:28:41,120 So it's this Q*lambda*Q transpose. 462 00:28:41,120 --> 00:28:43,680 So I'm telling you in that question what 463 00:28:43,680 --> 00:28:48,040 Q, lambda, and Q transpose are. 464 00:28:48,040 --> 00:28:53,970 The Q is this [cosine, minus sine; sine, cosine]. 465 00:28:53,970 --> 00:28:57,940 The lambda is two and five, I think, in that question. 466 00:28:57,940 --> 00:29:02,240 And the Q transpose of course is [cosine, sine; minus sine, 467 00:29:02,240 --> 00:29:05,310 cosine]. 468 00:29:05,310 --> 00:29:12,100 And if I've told you that those are the numbers then you 469 00:29:12,100 --> 00:29:18,060 could multiply those together to get K. But you can tell me, 470 00:29:18,060 --> 00:29:21,140 this is like K exposed. 471 00:29:21,140 --> 00:29:27,070 The matrix is like, we're told more than we would know. 472 00:29:27,070 --> 00:29:28,990 If I multiply it all together, I wouldn't 473 00:29:28,990 --> 00:29:32,620 see that the eigenvectors are these guys, 474 00:29:32,620 --> 00:29:34,990 that the eigenvalues are these guys. 475 00:29:34,990 --> 00:29:41,960 So what, without looking to see, what 476 00:29:41,960 --> 00:29:44,370 are the eigenvalues of this matrix K if we 477 00:29:44,370 --> 00:29:46,810 multiplied it all together? 478 00:29:46,810 --> 00:29:49,930 What would the eigenvalues actually be? 479 00:29:49,930 --> 00:29:53,640 Two and five, right, because we built it up that way. 480 00:29:53,640 --> 00:29:56,520 What would the determinant be? 481 00:29:56,520 --> 00:29:59,930 Now what do we know about determinants? 482 00:29:59,930 --> 00:30:02,600 It would be ten is the right answer. 483 00:30:02,600 --> 00:30:06,810 What's the right way to see that? 484 00:30:06,810 --> 00:30:10,113 Well, the determinant is always the product of the eigenvalues, 485 00:30:10,113 --> 00:30:11,720 isn't it? 486 00:30:11,720 --> 00:30:15,100 These guys have determinant ten anyway. 487 00:30:15,100 --> 00:30:18,620 And if I hadn't normalized, so this had some bigger 488 00:30:18,620 --> 00:30:23,130 determinant, this would have some smaller determinant. 489 00:30:23,130 --> 00:30:25,080 Their inverses, their determinants 490 00:30:25,080 --> 00:30:28,210 will give me the one and there's the ten. 491 00:30:28,210 --> 00:30:33,360 What else could I ask about or did I ask about for that? 492 00:30:33,360 --> 00:30:38,250 The eigenvectors, okay. 493 00:30:38,250 --> 00:30:41,720 The eigenvectors of the matrix, what are they? 494 00:30:41,720 --> 00:30:44,210 They're these columns that are sitting here for us, 495 00:30:44,210 --> 00:30:46,290 they're those two columns, right. 496 00:30:46,290 --> 00:30:49,470 And would you like to just check that if the-- I believe 497 00:30:49,470 --> 00:30:51,450 that column is an eigenvector. 498 00:30:51,450 --> 00:30:56,260 And which do you think, two or five, is its eigenvalue? 499 00:30:56,260 --> 00:30:59,080 That goes with this first column. 500 00:30:59,080 --> 00:31:02,510 Everybody's going to say two and that's right. 501 00:31:02,510 --> 00:31:08,120 And do you want me to just take that matrix times this proposed 502 00:31:08,120 --> 00:31:12,340 eigenvector and just see if it's going to work? 503 00:31:12,340 --> 00:31:17,390 Suppose I just do all and just see, 504 00:31:17,390 --> 00:31:20,340 sure enough this will be an eigenvector. 505 00:31:20,340 --> 00:31:23,100 So what do I have at this point? 506 00:31:23,100 --> 00:31:25,010 Can you do this times this first? 507 00:31:25,010 --> 00:31:31,840 What do I get? c squared plus s squared is one. 508 00:31:31,840 --> 00:31:36,050 And -cs plus cs is zero. 509 00:31:36,050 --> 00:31:38,720 So at that point I have [1, 0]. 510 00:31:38,720 --> 00:31:40,140 Now comes this matrix. 511 00:31:40,140 --> 00:31:46,630 So what do I have after that matrix speaks up? [2, 0]. 512 00:31:46,630 --> 00:31:51,310 And now I take two times this and what do I get? 513 00:31:51,310 --> 00:31:55,260 Or that matrix times the [2, 0]. 514 00:31:55,260 --> 00:31:59,142 How do you multiply a matrix times that [2, 0] vector. 515 00:31:59,142 --> 00:32:00,600 Here's the good way to think of it. 516 00:32:00,600 --> 00:32:04,240 It's two times the first column. 517 00:32:04,240 --> 00:32:06,430 And zero times the second. 518 00:32:06,430 --> 00:32:09,160 So the net result of the whole deal 519 00:32:09,160 --> 00:32:12,810 was two times that first column. 520 00:32:12,810 --> 00:32:16,640 Which is exactly saying that this is an eigenvector. 521 00:32:16,640 --> 00:32:20,350 When I did all that it came back again. 522 00:32:20,350 --> 00:32:24,770 Scaled by two. 523 00:32:24,770 --> 00:32:27,100 So that's a good example. 524 00:32:27,100 --> 00:32:29,260 And then, is the matrix positive definite? 525 00:32:29,260 --> 00:32:33,860 That connects to today's lecture. 526 00:32:33,860 --> 00:32:36,850 What test would you use to show that the matrix is 527 00:32:36,850 --> 00:32:39,110 positive definite? 528 00:32:39,110 --> 00:32:40,700 The eigenvalues, yeah. 529 00:32:40,700 --> 00:32:42,360 The eigenvalues are sitting there. 530 00:32:42,360 --> 00:32:44,410 Two and five, both positive. 531 00:32:44,410 --> 00:32:47,760 If I changed one of those signs, then it would no longer 532 00:32:47,760 --> 00:32:50,780 be positive definite. 533 00:32:50,780 --> 00:32:54,200 It would still be symmetric, I'd still have the eigenvectors, 534 00:32:54,200 --> 00:33:00,850 but the eigenvalue would have jumped to minus five. 535 00:33:00,850 --> 00:33:02,470 I think this sort of helps out. 536 00:33:02,470 --> 00:33:05,570 I guess I hope that as I'm doing these things, 537 00:33:05,570 --> 00:33:10,230 you're ahead of me or with me in the calculation 538 00:33:10,230 --> 00:33:13,730 and you just have to do a bunch of these 539 00:33:13,730 --> 00:33:17,270 to get confidence that you've got the right thing. 540 00:33:17,270 --> 00:33:22,890 Okay, yes? 541 00:33:22,890 --> 00:33:24,350 1.6, 24. 542 00:33:24,350 --> 00:33:26,680 Is that also a homework problem? 543 00:33:26,680 --> 00:33:29,200 Alright, but you guys are reading the rest of the book, 544 00:33:29,200 --> 00:33:30,720 right? 545 00:33:30,720 --> 00:33:32,380 Not only the homework questions. 546 00:33:32,380 --> 00:33:34,320 Ah. 547 00:33:34,320 --> 00:33:35,050 Oh, dear. 548 00:33:35,050 --> 00:33:41,100 24, that's a very good question. 549 00:33:41,100 --> 00:33:43,370 About this, yeah. 550 00:33:43,370 --> 00:33:54,650 Right. 551 00:33:54,650 --> 00:33:55,850 It's a good question. 552 00:33:55,850 --> 00:33:59,810 And if today's lecture had been, well it ran a little late. 553 00:33:59,810 --> 00:34:05,760 But if we ran another 20 minutes late, I could have done this. 554 00:34:05,760 --> 00:34:09,440 I'll just say what's in that problem. 555 00:34:09,440 --> 00:34:16,080 And then we'll see it again. 556 00:34:16,080 --> 00:34:21,340 So what's in that question? 557 00:34:21,340 --> 00:34:23,540 Let me write down what it is. 558 00:34:23,540 --> 00:34:28,830 So I have a positive definite matrix K, right? 559 00:34:28,830 --> 00:34:33,580 And then I've got its energy. 560 00:34:33,580 --> 00:34:39,980 I'm using u rather than x, so let's use u. 561 00:34:39,980 --> 00:34:46,110 So my u transpose Ku, or like x transpose Kx today. 562 00:34:46,110 --> 00:34:52,580 That is this bowl-shaped figure, right? 563 00:34:52,580 --> 00:35:01,140 If I graph this on the u_1, u_2 maybe up 564 00:35:01,140 --> 00:35:02,780 to u_n, all in the base. 565 00:35:02,780 --> 00:35:04,040 And now I have the picture. 566 00:35:04,040 --> 00:35:07,550 So I'm in n+1 dimensions. 567 00:35:07,550 --> 00:35:09,410 The other dimension is this one. 568 00:35:09,410 --> 00:35:15,950 Then that's the one where I might get this bowl-shaped guy. 569 00:35:15,950 --> 00:35:17,930 And I've called that energy. 570 00:35:17,930 --> 00:35:22,570 In many, many physical problems there is a factor of 1/2. 571 00:35:22,570 --> 00:35:27,810 And it's going to be nice to have that factor of 1/2. 572 00:35:27,810 --> 00:35:35,610 So that won't change anything, just half as big. 573 00:35:35,610 --> 00:35:41,630 So what is the minimum value of that energy? 574 00:35:41,630 --> 00:35:46,400 And what is the minimum value of this, if I said minimize that, 575 00:35:46,400 --> 00:35:48,040 you could do it right away. 576 00:35:48,040 --> 00:35:51,570 It'd be a zero. 577 00:35:51,570 --> 00:35:57,470 Now I'm going to introduce a linear term. 578 00:35:57,470 --> 00:36:01,030 This was a quadratic term and it had u squareds in it. 579 00:36:01,030 --> 00:36:04,980 So the linear term is going to be u transpose f 580 00:36:04,980 --> 00:36:06,652 is the shorthand for it. 581 00:36:06,652 --> 00:36:08,985 And of course, we all know that that stands for u_1*f_1, 582 00:36:08,985 --> 00:36:16,540 u_2 all minus, u_2*f_2 and so on. 583 00:36:16,540 --> 00:36:19,760 However many dimensions I'm in. 584 00:36:19,760 --> 00:36:21,660 You can imagine I'm in two dimensions. 585 00:36:21,660 --> 00:36:23,850 So it's -u_1*f_1 - u_2*f_2. 586 00:36:23,850 --> 00:36:30,850 587 00:36:30,850 --> 00:36:34,300 So what I'm saying is that minimizing just this 588 00:36:34,300 --> 00:36:35,720 was like, too easy, right? 589 00:36:35,720 --> 00:36:36,650 The answer was zero. 590 00:36:36,650 --> 00:36:39,810 Nobody's interested in that for very long. 591 00:36:39,810 --> 00:36:42,730 But now it is much more interesting 592 00:36:42,730 --> 00:36:48,380 when I get a linear term in there. 593 00:36:48,380 --> 00:36:51,680 So what happens now? 594 00:36:51,680 --> 00:36:54,190 Well, the effect of that linear term 595 00:36:54,190 --> 00:37:02,540 is to shift that bowl sorta over and down a little. 596 00:37:02,540 --> 00:37:09,590 So that instead of sitting where I drew it, let me erase it. 597 00:37:09,590 --> 00:37:14,870 If I now graph this function, this is my function of u, 598 00:37:14,870 --> 00:37:19,240 this is still the most important part, 599 00:37:19,240 --> 00:37:24,240 but now I have a first order term. 600 00:37:24,240 --> 00:37:27,130 And the result is, it still goes through here. 601 00:37:27,130 --> 00:37:27,630 Right? 602 00:37:27,630 --> 00:37:31,800 Why does it still go through that same point? 603 00:37:31,800 --> 00:37:37,020 Because if I take u_1 and u_2 to be zero, I get zero. 604 00:37:37,020 --> 00:37:38,260 So I still get zero there. 605 00:37:38,260 --> 00:37:40,500 But the bowl has shifted. 606 00:37:40,500 --> 00:37:44,040 It's more like something here. 607 00:37:44,040 --> 00:37:48,690 And it still has a minimum because this is still 608 00:37:48,690 --> 00:37:50,730 the all-important term. 609 00:37:50,730 --> 00:37:52,900 But it's just moved over and down. 610 00:37:52,900 --> 00:37:55,360 So it has the minimum value. 611 00:37:55,360 --> 00:38:00,550 It actually goes below zero, but if I look at it 612 00:38:00,550 --> 00:38:04,130 if I'm sitting at the minimum and looking 613 00:38:04,130 --> 00:38:06,730 I'm seeing a bowl going up, right. 614 00:38:06,730 --> 00:38:14,160 So I hope that picture shows-- And now, of course, that's 615 00:38:14,160 --> 00:38:16,480 the geometry. 616 00:38:16,480 --> 00:38:18,280 In other words, the same geometry just 617 00:38:18,280 --> 00:38:20,360 moved the thing over and down. 618 00:38:20,360 --> 00:38:24,160 But the algebra is, where is the minimum? 619 00:38:24,160 --> 00:38:26,000 What is the value of that minimum? 620 00:38:26,000 --> 00:38:35,180 And this problem, 24, is one way to do the minimum. 621 00:38:35,180 --> 00:38:37,750 One way to do it. 622 00:38:37,750 --> 00:38:42,060 But actually, if you didn't like linear-- well 623 00:38:42,060 --> 00:38:44,560 I won't say didn't like linear algebra, that's 624 00:38:44,560 --> 00:38:46,560 against my religion. 625 00:38:46,560 --> 00:38:51,910 So if you like calculus and you said, wait a minute, 626 00:38:51,910 --> 00:38:54,170 if you give me something you want me to minimize, 627 00:38:54,170 --> 00:38:55,750 what will I do? 628 00:38:55,750 --> 00:38:59,290 I'll set derivatives to zero. 629 00:38:59,290 --> 00:39:03,920 And can I just jump to the answer? 630 00:39:03,920 --> 00:39:12,370 Oh, what derivatives do I set to zero now, for the minimum here? 631 00:39:12,370 --> 00:39:14,500 It's the first derivatives. 632 00:39:14,500 --> 00:39:21,770 And they're first derivatives with respect to? 633 00:39:21,770 --> 00:39:24,330 I look at df/d what? 634 00:39:24,330 --> 00:39:27,250 You see I've already given it away. 635 00:39:27,250 --> 00:39:29,060 These are going to be partial derivatives. 636 00:39:29,060 --> 00:39:30,950 Why's that? 637 00:39:30,950 --> 00:39:32,390 Because I've got two directions. 638 00:39:32,390 --> 00:39:36,200 So I have a df/du_1=0 and a df/du_2=0. 639 00:39:36,200 --> 00:39:39,330 640 00:39:39,330 --> 00:39:42,240 In other words, when I sit here at the bottom 641 00:39:42,240 --> 00:39:46,420 I'm seeing this whole bowl above me. 642 00:39:46,420 --> 00:39:50,920 If I go along the u_2 direction it should go up 643 00:39:50,920 --> 00:39:54,450 and if I come along the u_1 direction, goes up. 644 00:39:54,450 --> 00:40:00,700 But it's flat at the bottom both ways. 645 00:40:00,700 --> 00:40:04,320 So what's my point here? 646 00:40:04,320 --> 00:40:09,560 If you like calculus, you'll get to two equations. 647 00:40:09,560 --> 00:40:12,540 And I just want to say what those equations are, 648 00:40:12,540 --> 00:40:19,230 because they're all important. 649 00:40:19,230 --> 00:40:22,630 Suppose we only had u_1 and nothing else. 650 00:40:22,630 --> 00:40:26,530 Then this would just be a parabola and the derivative 651 00:40:26,530 --> 00:40:29,160 of this would be at 1/2 K*u squared. 652 00:40:29,160 --> 00:40:31,790 Suppose n is one. 653 00:40:31,790 --> 00:40:34,530 I'm only in one. 654 00:40:34,530 --> 00:40:39,410 So what's the derivative of 1/2 K*u squared? 655 00:40:39,410 --> 00:40:40,200 The derivative. 656 00:40:40,200 --> 00:40:44,250 So I'm looking for, if this was 1/2 K*u squared and I took 657 00:40:44,250 --> 00:40:47,480 the derivative with respect to u, it would be? 658 00:40:47,480 --> 00:40:48,610 It'd be Ku. 659 00:40:48,610 --> 00:40:52,130 And it works here in the matrix case. 660 00:40:52,130 --> 00:40:56,766 And what would be the derivative of u, transpose of u times 661 00:40:56,766 --> 00:41:03,930 f, if u was just a number and if u was just one thing and f 662 00:41:03,930 --> 00:41:08,070 was a single number, the derivative would be? f, yeah. 663 00:41:08,070 --> 00:41:12,140 It'd be f. 664 00:41:12,140 --> 00:41:18,560 That's the system. 665 00:41:18,560 --> 00:41:20,400 I've jumped to the answer. 666 00:41:20,400 --> 00:41:27,130 That this set of two or n equations in matrix language 667 00:41:27,130 --> 00:41:33,100 would just be, and I'll even write it better as Ku=f. 668 00:41:33,100 --> 00:41:35,960 That tells me where the minimum is. 669 00:41:35,960 --> 00:41:40,720 The minimizing guy is, so this is in the base 670 00:41:40,720 --> 00:41:43,680 and then the thing is dropping down. 671 00:41:43,680 --> 00:41:49,350 I still have to figure out what's the bottom value. 672 00:41:49,350 --> 00:41:54,290 But I've now identified where the minimum occurs. 673 00:41:54,290 --> 00:41:57,260 So you get two questions about a minimum. 674 00:41:57,260 --> 00:41:59,040 Where is it? 675 00:41:59,040 --> 00:42:01,300 What value of u gives the minimum? 676 00:42:01,300 --> 00:42:09,240 And at that point, at that lowest point, how low is it? 677 00:42:09,240 --> 00:42:11,820 The one thing you've gotta remember 678 00:42:11,820 --> 00:42:15,910 is that when you minimize that quadratic, 679 00:42:15,910 --> 00:42:19,570 you get that system of equations. 680 00:42:19,570 --> 00:42:22,430 And then, of course, the answer, you have to solve that system. 681 00:42:22,430 --> 00:42:25,450 But this goes back to what I said 682 00:42:25,450 --> 00:42:29,260 at the first minute of today. 683 00:42:29,260 --> 00:42:34,030 That we have two ways of looking at a problem. 684 00:42:34,030 --> 00:42:38,080 Usually we go directly to the equations. 685 00:42:38,080 --> 00:42:44,080 Sometimes the problem comes naturally to us 686 00:42:44,080 --> 00:42:46,520 as a minimum problem. 687 00:42:46,520 --> 00:42:48,360 Like we have to minimize the cost, 688 00:42:48,360 --> 00:42:53,030 we want to build a new school or something. 689 00:42:53,030 --> 00:42:56,300 So we've got some cost function that we 690 00:42:56,300 --> 00:43:00,240 minimize that will lead, through calculus or linear algebra, 691 00:43:00,240 --> 00:43:04,980 to this. 692 00:43:04,980 --> 00:43:14,680 So I've done everything but answer the question 24. 693 00:43:14,680 --> 00:43:16,770 We only checked the one by one case 694 00:43:16,770 --> 00:43:21,780 to see that that's the right equations, 695 00:43:21,780 --> 00:43:23,590 derivative equal zero. 696 00:43:23,590 --> 00:43:27,710 And now you could use calculus as I said. 697 00:43:27,710 --> 00:43:36,220 But if I answer that question, well let me just do a little. 698 00:43:36,220 --> 00:43:41,120 The idea of that question 24, so that was what? 699 00:43:41,120 --> 00:43:45,770 1.6, 24, or something. 700 00:43:45,770 --> 00:43:46,740 Is that right? 701 00:43:46,740 --> 00:43:47,770 Yeah. 702 00:43:47,770 --> 00:43:55,130 Is that I could rewrite this to make it clear. 703 00:43:55,130 --> 00:44:00,280 I think it's u minus K inverse f, 704 00:44:00,280 --> 00:44:07,560 transpose K times u minus K inverse f. 705 00:44:07,560 --> 00:44:17,230 And then a minus 1/2 f transpose K inverse f. 706 00:44:17,230 --> 00:44:21,060 Actually, my best friend in China told me this trick. 707 00:44:21,060 --> 00:44:25,600 And I didn't give him credit for it in the book. 708 00:44:25,600 --> 00:44:27,500 But I should have done. 709 00:44:27,500 --> 00:44:31,560 I just think that if you multiply all this out, 710 00:44:31,560 --> 00:44:33,460 you'll get this. 711 00:44:33,460 --> 00:44:35,980 It's what I would call an identity. 712 00:44:35,980 --> 00:44:40,190 That just simply means that it's just true for every u. 713 00:44:40,190 --> 00:44:42,230 It's true for everything. 714 00:44:42,230 --> 00:44:45,700 Can I try to multiply some of that out? 715 00:44:45,700 --> 00:44:53,930 Just so you kind of see it. 716 00:44:53,930 --> 00:44:57,060 Yeah, that's what I mean, multiply it out. 717 00:44:57,060 --> 00:44:59,330 You've got it. 718 00:44:59,330 --> 00:45:01,870 This thing would give me four terms. 719 00:45:01,870 --> 00:45:05,140 It'd be this transpose times that times that. 720 00:45:05,140 --> 00:45:07,560 Which is my guy here. 721 00:45:07,560 --> 00:45:09,270 And then I'll have something. 722 00:45:09,270 --> 00:45:10,610 It's just like numbers. 723 00:45:10,610 --> 00:45:13,350 Then this thing times that times this. 724 00:45:13,350 --> 00:45:15,700 And this thing times that times that. 725 00:45:15,700 --> 00:45:18,410 And this thing times that times that. 726 00:45:18,410 --> 00:45:19,760 Let me do that last one. 727 00:45:19,760 --> 00:45:26,990 What happens when I do the 1/2 and this transpose 728 00:45:26,990 --> 00:45:28,880 times the K times this. 729 00:45:28,880 --> 00:45:32,910 So I'm using the distributive, whatever, laws. 730 00:45:32,910 --> 00:45:35,960 Let's just do that particular term 731 00:45:35,960 --> 00:45:37,370 and see what we're getting. 732 00:45:37,370 --> 00:45:42,380 So I have 1/2 of the minus K inverse f transpose. 733 00:45:42,380 --> 00:45:45,740 So how do I write that? 734 00:45:45,740 --> 00:45:47,920 Shoot. 735 00:45:47,920 --> 00:45:50,560 Well, it's something times something transpose. 736 00:45:50,560 --> 00:45:53,460 So what do I have to do? 737 00:45:53,460 --> 00:45:54,890 Opposite order. 738 00:45:54,890 --> 00:45:58,151 So I have a minus, an F transpose and the K inverse 739 00:45:58,151 --> 00:45:58,650 transpose. 740 00:45:58,650 --> 00:46:02,730 You're seeing all this stuff. 741 00:46:02,730 --> 00:46:06,440 And then comes the K and then comes the minus, 742 00:46:06,440 --> 00:46:07,470 oh again the minus. 743 00:46:07,470 --> 00:46:08,900 So that'd be a plus, right? 744 00:46:08,900 --> 00:46:13,190 Times K inverse times f. 745 00:46:13,190 --> 00:46:15,470 So that's one of the terms. 746 00:46:15,470 --> 00:46:18,550 That's one of the terms that shows up. 747 00:46:18,550 --> 00:46:22,960 And what good is that one? 748 00:46:22,960 --> 00:46:25,120 So that's one. 749 00:46:25,120 --> 00:46:27,340 You could say that's the longest term. 750 00:46:27,340 --> 00:46:30,470 That's the one with the messiest term. 751 00:46:30,470 --> 00:46:31,900 But you can fix it. 752 00:46:31,900 --> 00:46:34,670 What would you do with that? 753 00:46:34,670 --> 00:46:37,350 K times K inverse is? 754 00:46:37,350 --> 00:46:38,030 Identity. 755 00:46:38,030 --> 00:46:40,390 So we can forget that. 756 00:46:40,390 --> 00:46:42,210 And now we're there. 757 00:46:42,210 --> 00:46:45,520 That's 1/2, f transpose, f on this side. 758 00:46:45,520 --> 00:46:49,200 Oh, what's K inverse transpose? 759 00:46:49,200 --> 00:46:53,420 It's the same as K inverse because K is symmetric, 760 00:46:53,420 --> 00:46:54,850 so its inverse is symmetric. 761 00:46:54,850 --> 00:46:57,880 So that transpose doesn't change the matrix. 762 00:46:57,880 --> 00:47:08,600 In other words, this term will show up and this term is oh! 763 00:47:08,600 --> 00:47:11,110 Nope, sorry. 764 00:47:11,110 --> 00:47:13,030 I was going to goof here. 765 00:47:13,030 --> 00:47:15,050 I was going to say this is the same as this, 766 00:47:15,050 --> 00:47:16,830 but it's not, right? 767 00:47:16,830 --> 00:47:18,880 Why not? 768 00:47:18,880 --> 00:47:20,610 Because it's positive. 769 00:47:20,610 --> 00:47:24,470 And this guy is negative. 770 00:47:24,470 --> 00:47:27,980 Has my good friend Professor Lin messed up? 771 00:47:27,980 --> 00:47:32,680 Nope. 772 00:47:32,680 --> 00:47:35,890 What's going to happen now? 773 00:47:35,890 --> 00:47:39,900 The two that I didn't do, you see, the 1/2 u 774 00:47:39,900 --> 00:47:43,070 transpose K u is here. 775 00:47:43,070 --> 00:47:46,540 Then comes this one, which I didn't 776 00:47:46,540 --> 00:47:49,450 do, and then another one that I didn't do, 777 00:47:49,450 --> 00:47:51,760 and then this one that I did. 778 00:47:51,760 --> 00:47:53,960 They'll all be the same. 779 00:47:53,960 --> 00:47:57,600 So they'll all contribute with their plus sign or minus sign 780 00:47:57,600 --> 00:48:02,170 and the net result will be a perfect match, yeah. 781 00:48:02,170 --> 00:48:08,310 So I won't wear out your patience by doing that. 782 00:48:08,310 --> 00:48:11,590 But I do want to make the point. 783 00:48:11,590 --> 00:48:16,880 What was Professor Lin's point in suggesting to write it 784 00:48:16,880 --> 00:48:21,010 in this more complicated way? 785 00:48:21,010 --> 00:48:27,710 His point was we could see this is just a constant. 786 00:48:27,710 --> 00:48:29,860 Doesn't depend on u. 787 00:48:29,860 --> 00:48:34,630 And now I can see what value of u would make this as small 788 00:48:34,630 --> 00:48:35,410 as possible. 789 00:48:35,410 --> 00:48:38,260 Remember, I'm still trying to minimize. 790 00:48:38,260 --> 00:48:43,190 This part, I can't make it bigger or smaller, it's fixed. 791 00:48:43,190 --> 00:48:44,950 It's u that I can play with. 792 00:48:44,950 --> 00:48:51,700 So what u should I choose to make this part smaller? 793 00:48:51,700 --> 00:48:53,180 Bear with me. 794 00:48:53,180 --> 00:48:58,580 What u will make this big mess as small as I can get it 795 00:48:58,580 --> 00:49:01,850 and how small can I get it? 796 00:49:01,850 --> 00:49:07,350 If I take u to be K inverse f, then this is zero, 797 00:49:07,350 --> 00:49:10,900 this is zero, I get zero. 798 00:49:10,900 --> 00:49:15,060 And that's my claim, that u equal K inverse f is the best 799 00:49:15,060 --> 00:49:17,930 possible, is the minimizer. 800 00:49:17,930 --> 00:49:21,460 And how do I know that I can't make this 801 00:49:21,460 --> 00:49:25,210 more negative than the zero? 802 00:49:25,210 --> 00:49:29,790 I can get it down to zero by making 803 00:49:29,790 --> 00:49:31,900 that to be the zero vector. 804 00:49:31,900 --> 00:49:41,090 But how do I know I can't make it below zero? 805 00:49:41,090 --> 00:49:44,090 The K is positive definite and I'm sitting here 806 00:49:44,090 --> 00:49:48,370 with some x transpose and some x. 807 00:49:48,370 --> 00:49:51,980 The x has this sort of messy form but it's an x 808 00:49:51,980 --> 00:49:53,690 and here's its transpose. 809 00:49:53,690 --> 00:50:00,070 So this is an x transpose Kx and can't be brought below zero 810 00:50:00,070 --> 00:50:02,230 when K is positive definite. 811 00:50:02,230 --> 00:50:05,370 Good. 812 00:50:05,370 --> 00:50:08,890 So we've said a good bit about positive definite here, 813 00:50:08,890 --> 00:50:19,970 but happy to think-- Yeah, thanks. 814 00:50:19,970 --> 00:50:27,020 In fact, finally a fifth. 815 00:50:27,020 --> 00:50:29,060 Exactly. 816 00:50:29,060 --> 00:50:30,470 Thanks, perfect question. 817 00:50:30,470 --> 00:50:33,070 And let me answer it clearly. 818 00:50:33,070 --> 00:50:36,560 Each of those five tests completely 819 00:50:36,560 --> 00:50:38,760 decides positive definite. 820 00:50:38,760 --> 00:50:42,590 So the five tests are all equivalent. 821 00:50:42,590 --> 00:50:46,760 If a matrix passes one test, it passes all five. 822 00:50:46,760 --> 00:50:48,140 So that's great, right? 823 00:50:48,140 --> 00:50:52,700 So we just do whichever test we want. 824 00:50:52,700 --> 00:50:55,820 Or whichever way we want to understand the matrix. 825 00:50:55,820 --> 00:51:02,720 I was going to add, I didn't say a lot about this one. 826 00:51:02,720 --> 00:51:09,790 Can I just add a note about a MATLAB command? 827 00:51:09,790 --> 00:51:18,410 The command chol(K). 828 00:51:18,410 --> 00:51:23,150 That's the first letters in the name Cholesky. 829 00:51:23,150 --> 00:51:28,450 So chol is the first four letters of this name. 830 00:51:28,450 --> 00:51:34,650 And that's a MATLAB command. 831 00:51:34,650 --> 00:51:37,670 If I've defined a matrix that's positive definite 832 00:51:37,670 --> 00:51:42,470 and I use that command, out will pop an A, 833 00:51:42,470 --> 00:51:45,540 one particular A that works. 834 00:51:45,540 --> 00:51:50,580 Out will pop an A that makes this work. 835 00:51:50,580 --> 00:51:55,060 It'll be a square A and it'll be upper triangular. 836 00:51:55,060 --> 00:52:03,150 So out will pop, so this command is very, very close to the LU 837 00:52:03,150 --> 00:52:06,230 but it's just sort of the appropriate version, 838 00:52:06,230 --> 00:52:12,510 symmetrized version of elimination 839 00:52:12,510 --> 00:52:16,560 when you have a positive definite symmetric matrix. 840 00:52:16,560 --> 00:52:19,690 If your matrix is not positive definite, 841 00:52:19,690 --> 00:52:22,240 MATLAB will tell you so. 842 00:52:22,240 --> 00:52:24,700 So it produces one particular A. There 843 00:52:24,700 --> 00:52:27,970 are many A's that would work, but there's 844 00:52:27,970 --> 00:52:30,310 one particular upper triangular one. 845 00:52:30,310 --> 00:52:40,010 It's just related to the usual U, but yes, thanks. 846 00:52:40,010 --> 00:52:43,070 No, I only even get into that ballpark 847 00:52:43,070 --> 00:52:45,110 if the matrix is symmetric. 848 00:52:45,110 --> 00:52:47,510 I don't touch it otherwise. 849 00:52:47,510 --> 00:52:51,510 So my matrix is symmetric before I begin. 850 00:52:51,510 --> 00:52:54,670 So I know good things about it. 851 00:52:54,670 --> 00:52:57,440 And here I'm asking for more. 852 00:52:57,440 --> 00:53:01,460 Here I'm asking are the pivots all positive? 853 00:53:01,460 --> 00:53:03,810 Are the eigenvalues all positive? 854 00:53:03,810 --> 00:53:06,060 So that's more. 855 00:53:06,060 --> 00:53:10,460 But I could think of some interpretation that 856 00:53:10,460 --> 00:53:12,890 would, for non-symmetric matrices, 857 00:53:12,890 --> 00:53:17,590 but it has problems, so I'd rather just leave it. 858 00:53:17,590 --> 00:53:22,470 Stay with symmetric. 859 00:53:22,470 --> 00:53:27,210 Well that's two hours of lots of linear algebra. 860 00:53:27,210 --> 00:53:31,220 I'm hoping you're going to like the MATLAB problem. 861 00:53:31,220 --> 00:53:41,640 Would you like to see what it'll be? 862 00:53:41,640 --> 00:53:44,850 I'll just tell you what the equation will be. 863 00:53:44,850 --> 00:53:49,920 So it'll be a differential equation. 864 00:53:49,920 --> 00:53:53,830 Oh, dear, what is it? 865 00:53:53,830 --> 00:53:57,030 So it's a differential equation with a -u'' 866 00:53:57,030 --> 00:54:00,260 that we know and love. 867 00:54:00,260 --> 00:54:03,110 And what else has it got? 868 00:54:03,110 --> 00:54:05,600 Oh yes, right. 869 00:54:05,600 --> 00:54:09,300 So here's the problem. 870 00:54:09,300 --> 00:54:13,890 Here's the equation. 871 00:54:13,890 --> 00:54:21,050 So it has the -u'', the second derivative and it has a first 872 00:54:21,050 --> 00:54:24,600 derivative equal whatever. 873 00:54:24,600 --> 00:54:28,570 In fact, the example will choose a delta function there. 874 00:54:28,570 --> 00:54:32,390 So what am I talking about here? 875 00:54:32,390 --> 00:54:37,200 This would be a diffusion and this would be, 876 00:54:37,200 --> 00:54:39,910 anybody met these things before? 877 00:54:39,910 --> 00:54:41,380 That would be a convection. 878 00:54:41,380 --> 00:54:46,890 So that's a first derivative, that's an anti-symmetric. 879 00:54:46,890 --> 00:54:50,800 The MATLAB problem is now going to create the difference matrix 880 00:54:50,800 --> 00:54:51,370 for that. 881 00:54:51,370 --> 00:54:55,070 So the symmetric part will be our old friend K. 882 00:54:55,070 --> 00:55:02,340 But now we've got the convection term is appearing. 883 00:55:02,340 --> 00:55:04,900 And it's going to be anti-symmetric. 884 00:55:04,900 --> 00:55:09,450 And if v is big, it gets more and more important. 885 00:55:09,450 --> 00:55:10,520 So what happens? 886 00:55:10,520 --> 00:55:13,360 What happens with equations like this? 887 00:55:13,360 --> 00:55:16,980 Really this is like the first time in the course 888 00:55:16,980 --> 00:55:22,390 that we've allowed this first derivative term to pop up. 889 00:55:22,390 --> 00:55:29,970 But nevertheless we can see a lot of what's happening. 890 00:55:29,970 --> 00:55:32,570 And how to deal with those equations? 891 00:55:32,570 --> 00:55:37,130 I mean, if you ask a chemical engineer or anybody, they're 892 00:55:37,130 --> 00:55:41,920 always dealing with a flow, like the Charles River is flowing 893 00:55:41,920 --> 00:55:44,820 along, that's coming from the velocity there, 894 00:55:44,820 --> 00:55:48,670 but at the same time stuff is diffusing in it. 895 00:55:48,670 --> 00:55:53,070 It's just a constant problem in true, true applications. 896 00:55:53,070 --> 00:55:56,450 And this is the best model, I think. 897 00:55:56,450 --> 00:56:03,780 So you'll see that and I'm pleased about that. 898 00:56:03,780 --> 00:56:06,120 As you'd see. 899 00:56:06,120 --> 00:56:07,460 Any last question? 900 00:56:07,460 --> 00:56:09,580 I'm always happy. 901 00:56:09,580 --> 00:56:10,830 Well I'll see you Friday then. 902 00:56:10,830 --> 00:56:12,610 Thanks for coming.