1 00:00:00,000 --> 00:00:00,047 2 00:00:00,047 --> 00:00:02,130 The following content is provided under a Creative 3 00:00:02,130 --> 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 --> 00:00:10,050 continue to offer high quality educational resources for free. 6 00:00:10,050 --> 00:00:12,590 To make a donation or to view additional materials 7 00:00:12,590 --> 00:00:16,180 from hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16,180 --> 00:00:22,230 at ocw.mit.edu. 9 00:00:22,230 --> 00:00:27,580 PROFESSOR STRANG: Just to give an overview in three lines: 10 00:00:27,580 --> 00:00:32,780 the text is the book of that name, Computational Science 11 00:00:32,780 --> 00:00:33,570 and Engineering. 12 00:00:33,570 --> 00:00:36,390 That was completed just last year, 13 00:00:36,390 --> 00:00:41,040 so it really ties pretty well with the course. 14 00:00:41,040 --> 00:00:43,890 I don't cover everything in the book, by all means. 15 00:00:43,890 --> 00:00:47,630 And I don't, certainly, don't stand here and read the book. 16 00:00:47,630 --> 00:00:50,290 That would be no good. 17 00:00:50,290 --> 00:00:55,430 But you'll be able, if you miss a class -- well, 18 00:00:55,430 --> 00:00:56,530 don't miss a class. 19 00:00:56,530 --> 00:01:01,420 But if you miss a class, you'll be able, probably, 20 00:01:01,420 --> 00:01:05,190 to see roughly what we did. 21 00:01:05,190 --> 00:01:08,100 OK, so the first part of the semester 22 00:01:08,100 --> 00:01:10,660 is applied linear algebra. 23 00:01:10,660 --> 00:01:13,990 And I don't know how many of you have had a linear algebra 24 00:01:13,990 --> 00:01:16,470 course, and that's why I thought I would 25 00:01:16,470 --> 00:01:19,490 start with a quick review. 26 00:01:19,490 --> 00:01:23,500 And you'll catch on. 27 00:01:23,500 --> 00:01:26,670 I want matrices to come to life, actually. 28 00:01:26,670 --> 00:01:30,754 You know, instead of just being a four 29 00:01:30,754 --> 00:01:33,610 by four array of numbers, there are four by four, 30 00:01:33,610 --> 00:01:36,970 or n by n or m by n array of special numbers. 31 00:01:36,970 --> 00:01:38,400 They have a meaning. 32 00:01:38,400 --> 00:01:41,780 When they multiply a vector, they do something. 33 00:01:41,780 --> 00:01:45,700 And so it's just, part of this first step 34 00:01:45,700 --> 00:01:49,190 is just, like, getting to recognize, 35 00:01:49,190 --> 00:01:50,760 what's that matrix doing? 36 00:01:50,760 --> 00:01:52,180 Where does it come from? 37 00:01:52,180 --> 00:01:53,910 What are its properties? 38 00:01:53,910 --> 00:01:57,210 So that's a theme at the start. 39 00:01:57,210 --> 00:02:04,050 Then differential equations, like Laplace's equation, 40 00:02:04,050 --> 00:02:06,530 are beautiful examples. 41 00:02:06,530 --> 00:02:11,300 So here we get, especially, to numerical methods, 42 00:02:11,300 --> 00:02:14,770 finite differences, finite elements, above all. 43 00:02:14,770 --> 00:02:16,510 So I think in this class you'll really 44 00:02:16,510 --> 00:02:20,390 see how finite elements work, and other ideas. 45 00:02:20,390 --> 00:02:21,890 All sorts of ideas. 46 00:02:21,890 --> 00:02:25,990 And then the last part of the course is about Fourier. 47 00:02:25,990 --> 00:02:29,400 That's Fourier series, that you may have seen, 48 00:02:29,400 --> 00:02:30,540 and Fourier integrals. 49 00:02:30,540 --> 00:02:35,000 But also, highly important, Discrete Fourier Transform, 50 00:02:35,000 --> 00:02:36,190 DFT. 51 00:02:36,190 --> 00:02:39,410 That's a fundamental step for understanding 52 00:02:39,410 --> 00:02:42,770 what a signal contains. 53 00:02:42,770 --> 00:02:46,060 Yeah, so that's great stuff, Fourier. 54 00:02:46,060 --> 00:02:52,620 OK, what else should I say before I start? 55 00:02:52,620 --> 00:02:55,010 I said this was my favorite course, 56 00:02:55,010 --> 00:03:01,730 and maybe I'll elaborate a little. 57 00:03:01,730 --> 00:03:06,190 Well, I think what I want to say is that I really 58 00:03:06,190 --> 00:03:12,740 feel my life is here to teach you and not to grade you. 59 00:03:12,740 --> 00:03:16,040 I'm not going to spend this semester worrying about grades, 60 00:03:16,040 --> 00:03:18,040 and please don't. 61 00:03:18,040 --> 00:03:19,610 They come out fine. 62 00:03:19,610 --> 00:03:22,200 We've got lots to learn. 63 00:03:22,200 --> 00:03:26,630 And I'll do my very best to explain it clearly. 64 00:03:26,630 --> 00:03:30,090 And I know you'll do your best. 65 00:03:30,090 --> 00:03:31,380 I know from experience. 66 00:03:31,380 --> 00:03:36,400 This class goes for it and does it right. 67 00:03:36,400 --> 00:03:40,510 So that's what makes it so good. 68 00:03:40,510 --> 00:03:41,410 OK. 69 00:03:41,410 --> 00:03:45,840 Homeworks, by the way, well, the first homework 70 00:03:45,840 --> 00:03:50,870 will simply be a way to get a grade list, a list of everybody 71 00:03:50,870 --> 00:03:52,180 taking the course. 72 00:03:52,180 --> 00:03:55,820 They won't be graded in great detail. 73 00:03:55,820 --> 00:03:59,610 Too large a class. 74 00:03:59,610 --> 00:04:03,520 And you're allowed to talk to each other about homework. 75 00:04:03,520 --> 00:04:05,970 So homework is not an exam at all. 76 00:04:05,970 --> 00:04:10,450 So let me just leave any discussion of exams and grades 77 00:04:10,450 --> 00:04:12,180 for the future. 78 00:04:12,180 --> 00:04:16,010 I'll tell you, you'll see how informally the first homework 79 00:04:16,010 --> 00:04:18,370 will be. 80 00:04:18,370 --> 00:04:21,120 And I hope it'll go up on the website. 81 00:04:21,120 --> 00:04:23,700 The first homework will be for Monday. 82 00:04:23,700 --> 00:04:29,340 So it's a bit early, but it's pretty open-ended. 83 00:04:29,340 --> 00:04:33,070 If you could take three problems from 1.1, 84 00:04:33,070 --> 00:04:36,920 the first section of the book, any three, 85 00:04:36,920 --> 00:04:43,960 and any three problems from 1.2, and print your name 86 00:04:43,960 --> 00:04:46,920 on the homework -- because we're going to use that to create 87 00:04:46,920 --> 00:04:50,030 the grade list -- I'll be completely happy. 88 00:04:50,030 --> 00:04:51,810 Well, especially if you get them right 89 00:04:51,810 --> 00:04:53,820 and do them neatly and so on. 90 00:04:53,820 --> 00:04:59,110 But actually we won't know. 91 00:04:59,110 --> 00:05:02,760 So that's for Monday. 92 00:05:02,760 --> 00:05:03,320 OK. 93 00:05:03,320 --> 00:05:05,430 And we'll talk more about it. 94 00:05:05,430 --> 00:05:10,750 I'll announce the TA on the website and the TA hours, 95 00:05:10,750 --> 00:05:12,950 the office hours, and everything. 96 00:05:12,950 --> 00:05:17,210 There'll be a Friday afternoon office hour, 97 00:05:17,210 --> 00:05:20,241 because homeworks will typically come Monday. 98 00:05:20,241 --> 00:05:20,740 OK. 99 00:05:20,740 --> 00:05:27,050 Questions about the course before I just start? 100 00:05:27,050 --> 00:05:30,140 OK. 101 00:05:30,140 --> 00:05:31,660 Another time for questions, too. 102 00:05:31,660 --> 00:05:41,450 OK, so can we just start with that matrix? 103 00:05:41,450 --> 00:05:45,130 So I said about matrices, I'm interested in their properties. 104 00:05:45,130 --> 00:05:47,990 Like, I'm going to ask you about that. 105 00:05:47,990 --> 00:05:51,050 And then, I'm interested in their meaning. 106 00:05:51,050 --> 00:05:53,200 Where do they come from? 107 00:05:53,200 --> 00:05:56,530 You know, why that matrix instead of some other? 108 00:05:56,530 --> 00:06:01,090 And then, the numerical part is how do we deal with them? 109 00:06:01,090 --> 00:06:05,400 How do we solve a linear system with that coefficient matrix? 110 00:06:05,400 --> 00:06:07,400 What can we say about the solution? 111 00:06:07,400 --> 00:06:09,830 So the purpose. 112 00:06:09,830 --> 00:06:10,330 Right. 113 00:06:10,330 --> 00:06:15,040 OK, now help me out. 114 00:06:15,040 --> 00:06:17,520 So I guess my plan with the video taping 115 00:06:17,520 --> 00:06:20,410 is, whatever you say, I'll repeat. 116 00:06:20,410 --> 00:06:26,140 So say it as clearly as possible, and it's 117 00:06:26,140 --> 00:06:30,810 fantastic to have discussion, conversation here. 118 00:06:30,810 --> 00:06:34,130 So I'll just repeat it so that it safely gets on the tape. 119 00:06:34,130 --> 00:06:35,880 So tell me its properties. 120 00:06:35,880 --> 00:06:41,310 Tell me the first property that you notice about that matrix. 121 00:06:41,310 --> 00:06:41,810 Symmetric. 122 00:06:41,810 --> 00:06:42,910 Symmetric. 123 00:06:42,910 --> 00:06:44,150 Right. 124 00:06:44,150 --> 00:06:45,730 I could have slowed down a little 125 00:06:45,730 --> 00:06:48,310 and everybody probably would have said that at once. 126 00:06:48,310 --> 00:06:51,790 So that's a symmetric matrix. 127 00:06:51,790 --> 00:06:54,770 Now we might as well pick up some matrix notation. 128 00:06:54,770 --> 00:06:58,970 How do I express the fact that this a symmetric matrix? 129 00:06:58,970 --> 00:07:01,890 In simple matrix notation, I would 130 00:07:01,890 --> 00:07:07,260 say that K is the same as K transpose. 131 00:07:07,260 --> 00:07:11,710 The transpose, everybody knows, it comes from -- oh, 132 00:07:11,710 --> 00:07:14,870 I shouldn't say this -- flipping it across the diagonal. 133 00:07:14,870 --> 00:07:17,830 That's not a very "math" thing to do. 134 00:07:17,830 --> 00:07:21,960 But that's the way to visualize it. 135 00:07:21,960 --> 00:07:27,560 And let me use a capital T for transpose. 136 00:07:27,560 --> 00:07:30,460 So it's symmetric. 137 00:07:30,460 --> 00:07:31,340 Very important. 138 00:07:31,340 --> 00:07:32,660 Very, very important. 139 00:07:32,660 --> 00:07:34,850 That's the most important class of matrices, 140 00:07:34,850 --> 00:07:35,840 symmetric matrices. 141 00:07:35,840 --> 00:07:37,890 We'll see them all the time, because they 142 00:07:37,890 --> 00:07:41,310 come from equilibrium problems. 143 00:07:41,310 --> 00:07:44,870 They come from all sorts of -- they come everywhere 144 00:07:44,870 --> 00:07:47,060 in applications. 145 00:07:47,060 --> 00:07:49,800 And we will be doing applications. 146 00:07:49,800 --> 00:07:53,180 The first week or week and a half, 147 00:07:53,180 --> 00:07:56,540 you'll see pretty much discussion 148 00:07:56,540 --> 00:08:00,710 of matrices and the reasons, what their meaning is. 149 00:08:00,710 --> 00:08:03,440 And then we'll get to physical applications: 150 00:08:03,440 --> 00:08:05,750 mechanics and more. 151 00:08:05,750 --> 00:08:06,680 OK. 152 00:08:06,680 --> 00:08:08,720 All right. 153 00:08:08,720 --> 00:08:12,230 Now I'm looking for properties, other properties, 154 00:08:12,230 --> 00:08:13,430 of that matrix. 155 00:08:13,430 --> 00:08:18,340 Let me write "2" here so that you got a spot to put it. 156 00:08:18,340 --> 00:08:21,420 What are you going to tell me next about that matrix? 157 00:08:21,420 --> 00:08:22,040 Periodic. 158 00:08:22,040 --> 00:08:23,270 Well, okay. 159 00:08:23,270 --> 00:08:25,200 Actually, that's a good question. 160 00:08:25,200 --> 00:08:31,280 Let me write periodic down here. 161 00:08:31,280 --> 00:08:36,110 You're using that word, because somehow that pattern 162 00:08:36,110 --> 00:08:37,480 is suggesting something. 163 00:08:37,480 --> 00:08:40,150 But you'll see I have a little more 164 00:08:40,150 --> 00:08:44,420 to add before I would use the word periodic. 165 00:08:44,420 --> 00:08:47,060 So that's great to see that here. 166 00:08:47,060 --> 00:08:47,560 What else? 167 00:08:47,560 --> 00:08:50,260 Somebody else was going to say something. 168 00:08:50,260 --> 00:08:51,370 Please. 169 00:08:51,370 --> 00:08:52,040 Sparse! 170 00:08:52,040 --> 00:08:53,240 Oh, very good. 171 00:08:53,240 --> 00:08:54,500 Sparse. 172 00:08:54,500 --> 00:08:59,350 That's also an obvious property that you 173 00:08:59,350 --> 00:09:01,220 see from looking at the matrix. 174 00:09:01,220 --> 00:09:03,550 What does sparse mean? 175 00:09:03,550 --> 00:09:05,200 Mostly zeros. 176 00:09:05,200 --> 00:09:07,870 Well that isn't mostly zeros, I guess. 177 00:09:07,870 --> 00:09:11,840 I mean, that's got what, out of sixteen entries, 178 00:09:11,840 --> 00:09:13,560 it's got six zeros. 179 00:09:13,560 --> 00:09:14,930 That doesn't sound like sparse. 180 00:09:14,930 --> 00:09:20,466 But when I grow the matrix -- because this is just a four 181 00:09:20,466 --> 00:09:21,040 by four. 182 00:09:21,040 --> 00:09:24,550 I would even call this one K_4. 183 00:09:24,550 --> 00:09:29,580 When the matrix grows to 100 by 100, 184 00:09:29,580 --> 00:09:31,910 then you really see it as sparse. 185 00:09:31,910 --> 00:09:36,490 So if that matrix was 100 by 100, how many non-zeros 186 00:09:36,490 --> 00:09:37,430 would it have? 187 00:09:37,430 --> 00:09:44,010 So if n is 100, then the number of non-zeros -- wow, 188 00:09:44,010 --> 00:09:46,530 that's the first MATLAB command I've written. 189 00:09:46,530 --> 00:09:50,620 A number of non-zeros of K would be -- 190 00:09:50,620 --> 00:09:53,160 anybody know what it would be? 191 00:09:53,160 --> 00:10:00,750 I'm just asking to go up to five by five. 192 00:10:00,750 --> 00:10:03,860 I'm asking you to keep that pattern alive. 193 00:10:03,860 --> 00:10:08,330 Twos on the diagonal, minus ones above and below. 194 00:10:08,330 --> 00:10:13,640 So yeah, so 298, would it be? 195 00:10:13,640 --> 00:10:20,020 A hundred diagonal entries, 99 and 99, maybe 298? 196 00:10:20,020 --> 00:10:27,590 298 out of 100 by 100 would be what? 197 00:10:27,590 --> 00:10:29,140 It's been a long summer. 198 00:10:29,140 --> 00:10:32,090 Yeah, a lot of zeros. 199 00:10:32,090 --> 00:10:32,710 A lot. 200 00:10:32,710 --> 00:10:33,210 Right. 201 00:10:33,210 --> 00:10:37,760 Because the matrix has got what 100 by 100, 10,000 entries. 202 00:10:37,760 --> 00:10:39,620 Out of 10,000. 203 00:10:39,620 --> 00:10:42,300 So that's sparse. 204 00:10:42,300 --> 00:10:46,090 But we see those all the time, and fortunately we do. 205 00:10:46,090 --> 00:10:49,050 Because, of course, this matrix, or even 100 by 100, 206 00:10:49,050 --> 00:10:53,000 we could deal with if it was dense. 207 00:10:53,000 --> 00:10:57,640 But 10,000, 100,000, or 1 million, 208 00:10:57,640 --> 00:11:02,390 which happens all the time now in scientific computation. 209 00:11:02,390 --> 00:11:04,510 A million by million dense matrix 210 00:11:04,510 --> 00:11:07,170 is not a nice thing to think about. 211 00:11:07,170 --> 00:11:13,780 A million by million matrix like this is a cinch. 212 00:11:13,780 --> 00:11:14,280 OK. 213 00:11:14,280 --> 00:11:15,940 So sparse. 214 00:11:15,940 --> 00:11:18,550 What else do you want to say? 215 00:11:18,550 --> 00:11:19,490 Toeplitz. 216 00:11:19,490 --> 00:11:22,600 Holy Moses. 217 00:11:22,600 --> 00:11:23,920 Exactly right. 218 00:11:23,920 --> 00:11:28,550 But I want to say, before I use that word, 219 00:11:28,550 --> 00:11:30,540 so that'll be my second MATLAB command. 220 00:11:30,540 --> 00:11:31,060 Thanks. 221 00:11:31,060 --> 00:11:33,780 Toeplitz. 222 00:11:33,780 --> 00:11:35,850 What's that mean? 223 00:11:35,850 --> 00:11:39,680 So this matrix has a property that we 224 00:11:39,680 --> 00:11:46,940 see right away, which is? 225 00:11:46,940 --> 00:11:51,450 I want to stay with Toeplitz but everybody tell me 226 00:11:51,450 --> 00:11:54,860 something more about properties of that matrix. 227 00:11:54,860 --> 00:11:56,100 Tridiagonal. 228 00:11:56,100 --> 00:12:02,880 Tridiagonal, so that's almost a special subcase of sparse. 229 00:12:02,880 --> 00:12:05,530 It has just three diagonals. 230 00:12:05,530 --> 00:12:08,440 Tridiagonal matrices are truly important. 231 00:12:08,440 --> 00:12:10,070 They come in all the time, we'll see 232 00:12:10,070 --> 00:12:14,080 that they come from second order differential equations, which 233 00:12:14,080 --> 00:12:17,360 are, thanks to Newton, the big ones. 234 00:12:17,360 --> 00:12:23,090 Ok, now it's more than tridiagonal and what more? 235 00:12:23,090 --> 00:12:26,850 So what further, we're getting deeper now. 236 00:12:26,850 --> 00:12:31,970 What patterns do you see beyond just tridiagonal, 237 00:12:31,970 --> 00:12:34,950 because tridiagonal would allow any numbers there 238 00:12:34,950 --> 00:12:39,150 but those are not, there's more of a pattern than just three 239 00:12:39,150 --> 00:12:42,490 diagonals, what is it? 240 00:12:42,490 --> 00:12:45,770 Those diagonals are constant. 241 00:12:45,770 --> 00:12:48,780 If I run down each of those three diagonals, 242 00:12:48,780 --> 00:12:50,350 I see the same number. 243 00:12:50,350 --> 00:12:53,370 Twos, minus ones, minus ones, and that's 244 00:12:53,370 --> 00:12:55,670 what the word Toeplitz means. 245 00:12:55,670 --> 00:13:05,280 Toeplitz is constant diagonal. 246 00:13:05,280 --> 00:13:05,850 Ok. 247 00:13:05,850 --> 00:13:09,840 And that kind of matrix is so important. 248 00:13:09,840 --> 00:13:17,730 It corresponds, yeah, if we were in EE, 249 00:13:17,730 --> 00:13:23,240 I would use the words time-invariant filter, linear, 250 00:13:23,240 --> 00:13:23,920 time-invariant. 251 00:13:23,920 --> 00:13:27,840 So it's linear because we're dealing with a matrix. 252 00:13:27,840 --> 00:13:31,750 And it's time-invariant, shift-invariant. 253 00:13:31,750 --> 00:13:34,900 I just use all these equivalent words 254 00:13:34,900 --> 00:13:38,300 to mean that we're seeing the same thing row 255 00:13:38,300 --> 00:13:44,000 by row, except of course, at, shall I call that the boundary? 256 00:13:44,000 --> 00:13:46,370 That's like, the end of the system and this 257 00:13:46,370 --> 00:13:51,740 is like the other end and there it's chopped off. 258 00:13:51,740 --> 00:13:56,070 But if it was ten by ten I would see that row eight times. 259 00:13:56,070 --> 00:13:58,980 100 by 100 I'd see it 98 times. 260 00:13:58,980 --> 00:14:05,030 So it's constant diagonals and the guy who first 261 00:14:05,030 --> 00:14:08,350 studied that was Toeplitz. 262 00:14:08,350 --> 00:14:13,470 And we wouldn't need that great historical information 263 00:14:13,470 --> 00:14:18,700 except that MATLAB created a command to create that matrix. 264 00:14:18,700 --> 00:14:25,750 K, MATLAB is all set to create Toeplitz matrices. 265 00:14:25,750 --> 00:14:30,110 Yeah, so I'll have to put what MATLAB would put. 266 00:14:30,110 --> 00:14:37,460 I realize I'm already using the word MATLAB. 267 00:14:37,460 --> 00:14:42,030 I think that MATLAB language is really convenient 268 00:14:42,030 --> 00:14:44,050 to talk about linear algebra. 269 00:14:44,050 --> 00:14:46,550 And how many know MATLAB or have used it? 270 00:14:46,550 --> 00:14:49,120 Yeah. 271 00:14:49,120 --> 00:14:53,650 You know it better than I. I talk a good line with MATLAB 272 00:14:53,650 --> 00:14:56,760 but I -- the code never runs. 273 00:14:56,760 --> 00:14:58,300 Never! 274 00:14:58,300 --> 00:15:02,560 I always forget some stupid semicolon. 275 00:15:02,560 --> 00:15:04,970 You may have had that experience. 276 00:15:04,970 --> 00:15:11,519 And I just want to say it now that there are other languages, 277 00:15:11,519 --> 00:15:13,185 and if you want to do homeworks and want 278 00:15:13,185 --> 00:15:18,340 to do your own work in other languages, that makes sense. 279 00:15:18,340 --> 00:15:21,290 So the older established alternatives 280 00:15:21,290 --> 00:15:25,800 were Mathematica and Maple and those two 281 00:15:25,800 --> 00:15:30,230 have symbolic-- they can deal with algebra as well 282 00:15:30,230 --> 00:15:33,320 as numbers. 283 00:15:33,320 --> 00:15:34,990 But there are newer languages. 284 00:15:34,990 --> 00:15:37,040 I don't know if you know them. 285 00:15:37,040 --> 00:15:41,270 I just know my friends say, yes they're terrific. 286 00:15:41,270 --> 00:15:46,050 Python is one. 287 00:15:46,050 --> 00:15:49,150 And R. I've just had a email saying, 288 00:15:49,150 --> 00:15:53,290 tell your class about R. And others. 289 00:15:53,290 --> 00:15:59,670 Ok, so but we'll use MATLAB language because that's really 290 00:15:59,670 --> 00:16:01,060 a good common language. 291 00:16:01,060 --> 00:16:03,370 Ok, so what is a Toeplitz matrix? 292 00:16:03,370 --> 00:16:06,050 A Toeplitz matrix is one with constant diagonals. 293 00:16:06,050 --> 00:16:09,220 You could use the word time-invariant, 294 00:16:09,220 --> 00:16:11,260 linear time-invariant filter. 295 00:16:11,260 --> 00:16:16,860 And to create K, this is an 18.085 command. 296 00:16:16,860 --> 00:16:19,400 It's just set up for us. 297 00:16:19,400 --> 00:16:26,350 I can create K by telling the system the first row. 298 00:16:26,350 --> 00:16:32,360 Two, minus one, zero, zero. 299 00:16:32,360 --> 00:16:37,340 That would, then if it wasn't symmetric 300 00:16:37,340 --> 00:16:40,370 I would have to give the first column also. 301 00:16:40,370 --> 00:16:41,990 Toeplitz would be constant diagonal, 302 00:16:41,990 --> 00:16:44,130 it doesn't have to be symmetric. 303 00:16:44,130 --> 00:16:47,050 But if it's symmetric, then the first row and first column 304 00:16:47,050 --> 00:16:50,470 are the same vector, so I just have to give that vector. 305 00:16:50,470 --> 00:16:55,570 Okay, so that's the quickest way to create K. 306 00:16:55,570 --> 00:17:00,100 And of course, if it was bigger then I would, 307 00:17:00,100 --> 00:17:08,120 rather than writing 100 zeros, I could put zeros of 98 and one. 308 00:17:08,120 --> 00:17:09,930 Wouldn't I have to say that? 309 00:17:09,930 --> 00:17:11,410 Or is it one and 98? 310 00:17:11,410 --> 00:17:15,910 You see why it doesn't run. 311 00:17:15,910 --> 00:17:18,740 Well I guess I'm thinking of that as a row. 312 00:17:18,740 --> 00:17:19,610 I don't know. 313 00:17:19,610 --> 00:17:24,560 Anyway. 314 00:17:24,560 --> 00:17:26,830 I realize getting this videotaped means I'm 315 00:17:26,830 --> 00:17:28,090 supposed to get things right! 316 00:17:28,090 --> 00:17:31,410 Usually it's like, we'll get it right later. 317 00:17:31,410 --> 00:17:36,170 But anyway, that might work. 318 00:17:36,170 --> 00:17:37,340 Okay. 319 00:17:37,340 --> 00:17:39,870 So there's a command that you know. 320 00:17:39,870 --> 00:17:44,250 "zeros", that creates a matrix of this size with all zeros. 321 00:17:44,250 --> 00:17:45,560 Okay. 322 00:17:45,560 --> 00:17:48,380 That would create the 100 by 100. 323 00:17:48,380 --> 00:17:48,910 Good. 324 00:17:48,910 --> 00:17:50,110 Ok. 325 00:17:50,110 --> 00:17:53,350 Oh, by the way, as long as we're speaking about computation 326 00:17:53,350 --> 00:17:55,710 I've gotta say something more. 327 00:17:55,710 --> 00:17:59,290 We said that the matrix is sparse. 328 00:17:59,290 --> 00:18:02,360 And this 100 by 100 matrix is certainly sparse. 329 00:18:02,360 --> 00:18:07,880 But if I create it this way, I've created all those zeros 330 00:18:07,880 --> 00:18:13,550 and if I ask MATLAB to work with that matrix, to square it 331 00:18:13,550 --> 00:18:18,810 or whatever, it would carry all those zeros 332 00:18:18,810 --> 00:18:21,330 and do all those zero computations. 333 00:18:21,330 --> 00:18:24,837 In other words, it would treat K like a dense matrix 334 00:18:24,837 --> 00:18:27,170 and it would just, it wouldn't know the zeros were there 335 00:18:27,170 --> 00:18:29,420 until it looked. 336 00:18:29,420 --> 00:18:33,840 So I just want to say that if you have really big systems 337 00:18:33,840 --> 00:18:38,740 sparse MATLAB is the way to go. 338 00:18:38,740 --> 00:18:42,930 Because sparse MATLAB keeps track only of the non-zeros. 339 00:18:42,930 --> 00:18:45,420 So it knows-- and their locations, of course. 340 00:18:45,420 --> 00:18:47,970 What the numbers are and their location. 341 00:18:47,970 --> 00:18:50,790 So I could create a sparse matrix out of that, 342 00:18:50,790 --> 00:18:53,700 like KS for K sparse. 343 00:18:53,700 --> 00:18:59,620 I think if I just did sparse(K) that 344 00:18:59,620 --> 00:19:01,400 would create a sparse matrix. 345 00:19:01,400 --> 00:19:04,740 And then if I do stuff to it, MATLAB 346 00:19:04,740 --> 00:19:08,280 would automatically know those zeros were there 347 00:19:08,280 --> 00:19:12,620 and not spend it's time multiplying by zero. 348 00:19:12,620 --> 00:19:14,220 But of course, this isn't perfect 349 00:19:14,220 --> 00:19:17,960 because I've created the big matrix before sparsifying it. 350 00:19:17,960 --> 00:19:20,700 And better to have created it in the first place 351 00:19:20,700 --> 00:19:22,090 as a sparse matrix. 352 00:19:22,090 --> 00:19:27,510 Ok. 353 00:19:27,510 --> 00:19:32,010 So those were properties that you could see. 354 00:19:32,010 --> 00:19:36,280 Now I'm looking for little deeper. 355 00:19:36,280 --> 00:19:39,790 What's the first question I would ask about a matrix if I 356 00:19:39,790 --> 00:19:41,830 have to solve a system of equations, 357 00:19:41,830 --> 00:19:46,100 say KU=F or something. 358 00:19:46,100 --> 00:19:53,770 I got a 4 by 4 matrix, four equations, four unknowns. 359 00:19:53,770 --> 00:19:57,070 What would I want to know next? 360 00:19:57,070 --> 00:19:59,780 Is it invertible? 361 00:19:59,780 --> 00:20:04,140 Is the matrix invertible? 362 00:20:04,140 --> 00:20:07,030 And that's an important question and how 363 00:20:07,030 --> 00:20:10,640 do you recognize an invertible matrix? 364 00:20:10,640 --> 00:20:12,100 This one is invertible. 365 00:20:12,100 --> 00:20:15,360 So let me say K is invertible. 366 00:20:15,360 --> 00:20:17,110 And what does that mean? 367 00:20:17,110 --> 00:20:19,410 That means that there's another matrix, 368 00:20:19,410 --> 00:20:27,630 K inverse such that K times K inverse is the identity matrix. 369 00:20:27,630 --> 00:20:32,190 The identity matrix in MATLAB would be eye(n) 370 00:20:32,190 --> 00:20:35,720 and it's the diagonal matrix of ones. 371 00:20:35,720 --> 00:20:40,480 It's the unit matrix; it's the matrix that doesn't do anything 372 00:20:40,480 --> 00:20:43,360 to a vector. 373 00:20:43,360 --> 00:20:48,510 So this K has an inverse. 374 00:20:48,510 --> 00:20:49,910 But how do you know? 375 00:20:49,910 --> 00:20:53,440 How can you recognize that a matrix is invertible? 376 00:20:53,440 --> 00:20:56,530 Because obviously that's a critical question and many, 377 00:20:56,530 --> 00:20:59,070 many-- since our matrices are not-- 378 00:20:59,070 --> 00:21:03,030 a random matrix would be invertible, for sure, 379 00:21:03,030 --> 00:21:06,670 but our matrices have patterns, they're 380 00:21:06,670 --> 00:21:10,150 created out of a problem and the question 381 00:21:10,150 --> 00:21:13,850 of whether that matrix is invertible is fundamental. 382 00:21:13,850 --> 00:21:17,260 I mean finite elements has these, 383 00:21:17,260 --> 00:21:20,900 zero-energy modes that you have to watch out for because, what 384 00:21:20,900 --> 00:21:24,580 are they? 385 00:21:24,580 --> 00:21:28,130 They produce non-invertible stiffness matrix. 386 00:21:28,130 --> 00:21:28,670 Ok. 387 00:21:28,670 --> 00:21:30,940 So how did we know, or how could we 388 00:21:30,940 --> 00:21:34,480 know that this K is invertible? 389 00:21:34,480 --> 00:21:37,640 Somebody said invertible and I wrote it down. 390 00:21:37,640 --> 00:21:39,600 Yeah? 391 00:21:39,600 --> 00:21:41,880 Well ok. 392 00:21:41,880 --> 00:21:45,080 Now I get to make a speech about determinants. 393 00:21:45,080 --> 00:21:46,890 Don't deal with them! 394 00:21:46,890 --> 00:21:49,010 Don't touch determinants. 395 00:21:49,010 --> 00:21:53,180 I mean this particular four by four 396 00:21:53,180 --> 00:21:56,230 happens to have a nice determinant. 397 00:21:56,230 --> 00:21:58,100 I think it's five. 398 00:21:58,100 --> 00:22:02,480 But if it was a 100 by 100 how would 399 00:22:02,480 --> 00:22:06,900 we show that the matrix was invertible? 400 00:22:06,900 --> 00:22:10,980 And what I mean by this is the whole family is invertible. 401 00:22:10,980 --> 00:22:13,650 All sizes are invertible. 402 00:22:13,650 --> 00:22:16,640 K_n is invertible for every n, not just 403 00:22:16,640 --> 00:22:20,010 this particular guy, whose determinant we could take. 404 00:22:20,010 --> 00:22:22,370 But as five by five, six by six, we 405 00:22:22,370 --> 00:22:28,100 would be up in the-- but you're completely right. 406 00:22:28,100 --> 00:22:33,930 The determinant is a test. 407 00:22:33,930 --> 00:22:35,990 Alright. 408 00:22:35,990 --> 00:22:45,530 But I guess I'm saying that it's not the test that I would use. 409 00:22:45,530 --> 00:22:49,180 So what I do? 410 00:22:49,180 --> 00:22:52,540 I would row reduce. 411 00:22:52,540 --> 00:22:58,330 That's the default option in linear algebra. 412 00:22:58,330 --> 00:23:00,660 If you don't know what to do with a matrix, 413 00:23:00,660 --> 00:23:03,780 if you want to see what's going on, row reduce. 414 00:23:03,780 --> 00:23:04,910 What does that mean? 415 00:23:04,910 --> 00:23:09,050 That means-- shall I try it? 416 00:23:09,050 --> 00:23:15,090 So let me just start it, just so I'm not using 417 00:23:15,090 --> 00:23:24,590 a word that we don't need. 418 00:23:24,590 --> 00:23:25,390 Ok. 419 00:23:25,390 --> 00:23:29,120 And actually, maybe the third lecture, maybe next Monday 420 00:23:29,120 --> 00:23:33,120 we'll come back to row reduce. 421 00:23:33,120 --> 00:23:38,180 So I won't make heavy weather of that, certainly not now. 422 00:23:38,180 --> 00:23:39,710 So what is row reduce? 423 00:23:39,710 --> 00:23:43,510 Just so you know. 424 00:23:43,510 --> 00:23:46,880 I want to get that minus one to be a zero. 425 00:23:46,880 --> 00:23:50,210 I'm aiming for a triangular matrix. 426 00:23:50,210 --> 00:23:53,960 I want to clean out below the diagonal 427 00:23:53,960 --> 00:23:56,210 because if my matrix is triangular then 428 00:23:56,210 --> 00:23:59,560 I can see immediately everything. 429 00:23:59,560 --> 00:24:01,330 Right? 430 00:24:01,330 --> 00:24:06,890 Ultimately I'll reach a matrix U that'll be upper triangular 431 00:24:06,890 --> 00:24:11,790 and that first row won't change but the second row will change. 432 00:24:11,790 --> 00:24:13,340 And what does it change to? 433 00:24:13,340 --> 00:24:17,460 How do I clean out, get a zero in that, where 434 00:24:17,460 --> 00:24:21,930 the minus one is right now? 435 00:24:21,930 --> 00:24:29,890 Well I want to use the first row, the first equation. 436 00:24:29,890 --> 00:24:33,330 I want to add some multiple of the first row 437 00:24:33,330 --> 00:24:36,130 to the second row. 438 00:24:36,130 --> 00:24:38,970 And what should that multiple be? 439 00:24:38,970 --> 00:24:41,330 I want to multiply that row by something. 440 00:24:41,330 --> 00:24:43,590 And I'll say "add" today. 441 00:24:43,590 --> 00:24:46,280 Later I'll say "subtract." 442 00:24:46,280 --> 00:24:47,360 But what shall I do? 443 00:24:47,360 --> 00:24:50,130 Just tell me what the heck to do. 444 00:24:50,130 --> 00:24:52,470 I've got that row and I want to use it, 445 00:24:52,470 --> 00:24:55,380 I want to take a combination of these two rows. 446 00:24:55,380 --> 00:24:59,010 This row and some multiple of this one that'll 447 00:24:59,010 --> 00:25:00,930 produce a zero. 448 00:25:00,930 --> 00:25:02,670 This is called the pivot. 449 00:25:02,670 --> 00:25:07,330 That's the first pivot P-I-V-O-T. Pivot. 450 00:25:07,330 --> 00:25:11,290 And then that's the pivot row. 451 00:25:11,290 --> 00:25:14,170 And what do I do? 452 00:25:14,170 --> 00:25:15,930 Tell me what to do. 453 00:25:15,930 --> 00:25:18,180 Add half this row to this one. 454 00:25:18,180 --> 00:25:21,630 When I add half of that row to that one, what do I get? 455 00:25:21,630 --> 00:25:22,610 I get that zero. 456 00:25:22,610 --> 00:25:26,440 What do I get here for the second pivot? 457 00:25:26,440 --> 00:25:27,820 What is it? 458 00:25:27,820 --> 00:25:30,690 1.5, 3/2. 459 00:25:30,690 --> 00:25:32,670 Because half of that is, so 3/2. 460 00:25:32,670 --> 00:25:39,920 And the rest won't change. 461 00:25:39,920 --> 00:25:43,040 So I'm happy with that zero. 462 00:25:43,040 --> 00:25:48,140 Now I've got a couple more entries below that first pivot, 463 00:25:48,140 --> 00:25:49,340 but they're already zero. 464 00:25:49,340 --> 00:25:52,070 That's where the sparseness pays off. 465 00:25:52,070 --> 00:25:54,890 The tridiagonal really pays off. 466 00:25:54,890 --> 00:25:59,410 So those zeros say the first column is finished. 467 00:25:59,410 --> 00:26:02,670 So I'm ready to go on to the second column. 468 00:26:02,670 --> 00:26:08,560 It's like I got to this smaller problem with the 3/2 here. 469 00:26:08,560 --> 00:26:12,170 And a zero there. 470 00:26:12,170 --> 00:26:13,870 What do I do now? 471 00:26:13,870 --> 00:26:16,200 There is the second pivot, 3/2. 472 00:26:16,200 --> 00:26:17,720 Below it is a non-zero. 473 00:26:17,720 --> 00:26:20,030 I gotta get rid of it. 474 00:26:20,030 --> 00:26:23,400 What do I multiply by now? 475 00:26:23,400 --> 00:26:24,620 2/3. 476 00:26:24,620 --> 00:26:28,460 2/3 of that new second row added to the third row 477 00:26:28,460 --> 00:26:30,820 will clean out the third row. 478 00:26:30,820 --> 00:26:32,850 This was already cleaned out. 479 00:26:32,850 --> 00:26:34,590 This is already a zero. 480 00:26:34,590 --> 00:26:38,830 But I want to have 2/3 of this row added to this one so 481 00:26:38,830 --> 00:26:41,290 what's my new third row? 482 00:26:41,290 --> 00:26:43,990 Starts with zero and what's the third pivot now? 483 00:26:43,990 --> 00:26:46,870 You see the pivots appearing? 484 00:26:46,870 --> 00:26:52,470 The third pivot will be 4/3 because I've got 2/3 this -1 485 00:26:52,470 --> 00:26:59,370 and 2 is 6/3 so I have 6/3, I'm taking 2/3 away, I get 4/3 486 00:26:59,370 --> 00:27:01,770 and that -1 is still there. 487 00:27:01,770 --> 00:27:07,060 So you see that I'm-- this is fast. 488 00:27:07,060 --> 00:27:08,900 This is really fast. 489 00:27:08,900 --> 00:27:12,510 And the next step, maybe you can see the beautiful patterns 490 00:27:12,510 --> 00:27:13,300 that are coming. 491 00:27:13,300 --> 00:27:16,990 Do you want to just guess the fourth pivot? 492 00:27:16,990 --> 00:27:21,070 5/4, good guess, right. 493 00:27:21,070 --> 00:27:24,950 5/4. 494 00:27:24,950 --> 00:27:29,880 Now this is actually how MATLAB would find the determinant. 495 00:27:29,880 --> 00:27:32,660 It would do elimination. 496 00:27:32,660 --> 00:27:34,390 I call that elimination because it 497 00:27:34,390 --> 00:27:37,850 eliminated all those numbers below the diagonal 498 00:27:37,850 --> 00:27:39,870 and got zeros. 499 00:27:39,870 --> 00:27:42,510 Now what's the determinant? 500 00:27:42,510 --> 00:27:45,000 If I asked you for the determinant, 501 00:27:45,000 --> 00:27:50,820 and I will very rarely use the word determinant, 502 00:27:50,820 --> 00:27:55,620 but I guess I'm into it now, so tell me the determinant. 503 00:27:55,620 --> 00:27:58,020 Five. 504 00:27:58,020 --> 00:27:59,510 Why's that? 505 00:27:59,510 --> 00:28:01,650 I guess I did say five earlier. 506 00:28:01,650 --> 00:28:06,420 But how do you know it's five? 507 00:28:06,420 --> 00:28:10,360 Whatever the determinant of that matrix is, why is it five? 508 00:28:10,360 --> 00:28:12,530 Because it's a triangular matrix. 509 00:28:12,530 --> 00:28:16,660 Triangular matrices, you've got all these zeros. 510 00:28:16,660 --> 00:28:18,340 You can see what's happening. 511 00:28:18,340 --> 00:28:21,290 And the determinant of a triangular matrix 512 00:28:21,290 --> 00:28:24,310 is just the product down the diagonal. 513 00:28:24,310 --> 00:28:25,920 The product of these pivots. 514 00:28:25,920 --> 00:28:29,270 The determinant is the product of the pivots. 515 00:28:29,270 --> 00:28:32,350 And that's how MATLAB would compute a determinant. 516 00:28:32,350 --> 00:28:35,890 And it would take 2 times 3/2 times 4/3 times 5/4 517 00:28:35,890 --> 00:28:40,120 and it would give the answer five. 518 00:28:40,120 --> 00:28:45,540 My friend Alan Edelman told me something yesterday. 519 00:28:45,540 --> 00:28:54,610 MATLAB computes in floating point. 520 00:28:54,610 --> 00:29:02,960 So 4/3, that's 1.3333, etc. 521 00:29:02,960 --> 00:29:06,660 So MATLAB would not, when it does that multiplication, 522 00:29:06,660 --> 00:29:08,570 get a whole number. 523 00:29:08,570 --> 00:29:09,900 Right? 524 00:29:09,900 --> 00:29:14,210 Because in MATLAB that would be 1.333 and probably 525 00:29:14,210 --> 00:29:18,290 it would make that last pivot a decimal, a long decimal. 526 00:29:18,290 --> 00:29:22,560 And then when it multiplies that it gets whatever it gets. 527 00:29:22,560 --> 00:29:25,400 But it's not exactly five I think. 528 00:29:25,400 --> 00:29:30,110 Nevertheless MATLAB will print the answer five. 529 00:29:30,110 --> 00:29:31,910 It's cheated actually. 530 00:29:31,910 --> 00:29:35,510 It's done that calculation and I don't 531 00:29:35,510 --> 00:29:39,520 know if it takes the nearest integer when 532 00:29:39,520 --> 00:29:43,080 it knows that the-- I shouldn't tell you this, 533 00:29:43,080 --> 00:29:46,320 this isn't even interesting. 534 00:29:46,320 --> 00:29:49,950 If the determinant of an integer matrix, whole number 535 00:29:49,950 --> 00:29:52,230 is a whole number, so MATLAB says, 536 00:29:52,230 --> 00:29:54,680 better get a whole number. 537 00:29:54,680 --> 00:29:58,060 And somehow it gets one. 538 00:29:58,060 --> 00:30:01,240 Actually, it doesn't always get the right one. 539 00:30:01,240 --> 00:30:09,670 So maybe later I'll know the matrix whose determinant 540 00:30:09,670 --> 00:30:11,530 might not come out right. 541 00:30:11,530 --> 00:30:15,170 But ours is right, five. 542 00:30:15,170 --> 00:30:19,700 Now where was this going? 543 00:30:19,700 --> 00:30:23,420 It got thrown off track by the determinant. 544 00:30:23,420 --> 00:30:25,370 What's the real test? 545 00:30:25,370 --> 00:30:27,830 Well so I said there are two ways to see 546 00:30:27,830 --> 00:30:30,690 that a matrix is invertible. 547 00:30:30,690 --> 00:30:32,160 Or not invertible. 548 00:30:32,160 --> 00:30:34,990 Here we're talking about the first way. 549 00:30:34,990 --> 00:30:39,130 How do I know that this matrix-- I've got an upper triangular 550 00:30:39,130 --> 00:30:39,730 matrix. 551 00:30:39,730 --> 00:30:41,650 When is it invertible? 552 00:30:41,650 --> 00:30:47,310 When is an upper triangular matrix invertible? 553 00:30:47,310 --> 00:30:48,502 Upper triangular is great. 554 00:30:48,502 --> 00:30:49,960 When you've got it in that form you 555 00:30:49,960 --> 00:30:51,890 should be able to see stuff. 556 00:30:51,890 --> 00:30:55,070 So this key question of invertible, 557 00:30:55,070 --> 00:31:03,470 which is not obvious for a typical matrix 558 00:31:03,470 --> 00:31:06,000 is obvious for a triangular matrix. 559 00:31:06,000 --> 00:31:06,780 And why? 560 00:31:06,780 --> 00:31:10,180 What's the test? 561 00:31:10,180 --> 00:31:11,630 Well, we could do the determinant 562 00:31:11,630 --> 00:31:15,330 but we can say it without using that long word. 563 00:31:15,330 --> 00:31:18,730 The diagonal is non-zero. 564 00:31:18,730 --> 00:31:22,340 K as invertible because the diagonal-- no, 565 00:31:22,340 --> 00:31:24,550 it's got a full set of pivots. 566 00:31:24,550 --> 00:31:26,810 It's got four non-zero pivots. 567 00:31:26,810 --> 00:31:28,110 That's what it takes. 568 00:31:28,110 --> 00:31:31,080 That's what it's going to take to solve systems. 569 00:31:31,080 --> 00:31:33,880 So this is the first step in solving this system. 570 00:31:33,880 --> 00:31:38,290 In other words, to decide if a matrix is invertible, 571 00:31:38,290 --> 00:31:41,150 you just go ahead and use it. 572 00:31:41,150 --> 00:31:46,080 You don't stop first necessarily to check invertibility. 573 00:31:46,080 --> 00:31:48,320 You go forward, you get to this point 574 00:31:48,320 --> 00:31:51,070 and you see non-zeros there and then 575 00:31:51,070 --> 00:31:55,800 you're practically got to the answer here. 576 00:31:55,800 --> 00:32:01,530 I'll leave for another day the final back-- going back upwards 577 00:32:01,530 --> 00:32:03,760 that gives you the answer. 578 00:32:03,760 --> 00:32:05,270 So K is invertible. 579 00:32:05,270 --> 00:32:15,160 That means full set of pivots. n non-zero pivots. 580 00:32:15,160 --> 00:32:20,260 And here they are, two, 3/2, 4/3 and 5/4. 581 00:32:20,260 --> 00:32:23,270 Worth knowing because this matrix K is so important. 582 00:32:23,270 --> 00:32:24,750 We'll see it over and over again. 583 00:32:24,750 --> 00:32:30,950 Part of my purpose today is to give some matrices 584 00:32:30,950 --> 00:32:35,380 a name because we'll see them again and you'll know them 585 00:32:35,380 --> 00:32:38,640 and you'll recognize them. 586 00:32:38,640 --> 00:32:43,900 While I'm on this invertible or not invertible business 587 00:32:43,900 --> 00:32:52,020 I want to ask you to change K. To make it not invertible. 588 00:32:52,020 --> 00:32:54,650 Change that matrix. 589 00:32:54,650 --> 00:32:56,710 How could I change that matrix? 590 00:32:56,710 --> 00:32:58,430 Well, of course, many ways. 591 00:32:58,430 --> 00:33:00,950 But I'm interested in another matrix 592 00:33:00,950 --> 00:33:04,880 and this'll be among my special matrices. 593 00:33:04,880 --> 00:33:07,660 And it will start out the same. 594 00:33:07,660 --> 00:33:14,560 It'll have these same diagonals. 595 00:33:14,560 --> 00:33:16,900 It'll be Toeplitz. 596 00:33:16,900 --> 00:33:20,600 I'm going to call it C and I want 597 00:33:20,600 --> 00:33:25,600 to say the reason I'm talking about it now is that it's not 598 00:33:25,600 --> 00:33:29,200 going to be invertible. 599 00:33:29,200 --> 00:33:38,150 And I'm going to tell you a C and see if you can tell me 600 00:33:38,150 --> 00:33:40,320 why it is not invertible. 601 00:33:40,320 --> 00:33:43,270 So here's the difference: I'm going to put minus one 602 00:33:43,270 --> 00:33:45,230 in the corners. 603 00:33:45,230 --> 00:33:49,880 Still zeros there. 604 00:33:49,880 --> 00:33:56,180 So that matrix C still has that pattern. 605 00:33:56,180 --> 00:33:58,740 It's still a Toeplitz matrix, actually. 606 00:33:58,740 --> 00:34:07,970 That would still be the matrix Toeplitz of 2, -1, 0, -1. 607 00:34:07,970 --> 00:34:11,440 I claim that matrix is not invertible 608 00:34:11,440 --> 00:34:18,700 and I claim that we can see that without computing determinants, 609 00:34:18,700 --> 00:34:22,290 we can see it without doing elimination, too. 610 00:34:22,290 --> 00:34:24,380 MATLAB would see it by doing elimination. 611 00:34:24,380 --> 00:34:30,720 We can see it by just human intelligence. 612 00:34:30,720 --> 00:34:33,720 Now why? 613 00:34:33,720 --> 00:34:39,110 How do I recognize a matrix that's not invertible? 614 00:34:39,110 --> 00:34:44,450 And then, by converse, how a matrix that is invertible. 615 00:34:44,450 --> 00:34:48,690 I claim-- and let may say first, let 616 00:34:48,690 --> 00:34:56,830 me say why that letter C. That letter C stands for circulant. 617 00:34:56,830 --> 00:35:02,010 It's because-- This word circulant, why circulant, 618 00:35:02,010 --> 00:35:05,520 it's because that diagonal which only had three guys 619 00:35:05,520 --> 00:35:09,060 circled around to the fourth. 620 00:35:09,060 --> 00:35:11,370 This diagonal that only had three entries 621 00:35:11,370 --> 00:35:14,070 circled around to the fourth entry. 622 00:35:14,070 --> 00:35:16,740 This diagonal with two zeros circled around 623 00:35:16,740 --> 00:35:17,840 to the other two zeros. 624 00:35:17,840 --> 00:35:22,550 The diagonal are not only constant, they loop around. 625 00:35:22,550 --> 00:35:24,790 And you use the word periodic. 626 00:35:24,790 --> 00:35:29,460 Now for me, that's the periodic matrix. 627 00:35:29,460 --> 00:35:35,140 See, a circulant matrix comes from a periodic problem. 628 00:35:35,140 --> 00:35:38,180 Because it loops around. 629 00:35:38,180 --> 00:35:41,970 It brings numbers, zero is the same 630 00:35:41,970 --> 00:35:45,400 as number four or something. 631 00:35:45,400 --> 00:35:51,320 And why is that not invertible? 632 00:35:51,320 --> 00:35:55,570 The thing is can you find a vector? 633 00:35:55,570 --> 00:35:57,100 Because matrices multiply vectors, 634 00:35:57,100 --> 00:35:59,140 that's their whole point. 635 00:35:59,140 --> 00:36:03,060 Can you see a vector that it takes to zero? 636 00:36:03,060 --> 00:36:06,520 Can you see a solution to Cu=0? 637 00:36:06,520 --> 00:36:10,990 I'm looking for a u with four entries 638 00:36:10,990 --> 00:36:18,680 so that I get four zeros. 639 00:36:18,680 --> 00:36:20,810 Do you see it? 640 00:36:20,810 --> 00:36:21,990 All ones. 641 00:36:21,990 --> 00:36:23,710 All ones. 642 00:36:23,710 --> 00:36:25,460 That will do it. 643 00:36:25,460 --> 00:36:33,230 So that's a nice, natural entry, a constant. 644 00:36:33,230 --> 00:36:37,360 And do you see why when I-- we haven't 645 00:36:37,360 --> 00:36:42,170 spoken about multiplying matrices times vectors. 646 00:36:42,170 --> 00:36:44,010 And most people will do it this way. 647 00:36:44,010 --> 00:36:46,280 And let's do this one this way. 648 00:36:46,280 --> 00:36:49,142 You take row one times that, you get two, minus one, zero, 649 00:36:49,142 --> 00:36:51,020 minus one. 650 00:36:51,020 --> 00:36:53,730 You get the zero because of that new number. 651 00:36:53,730 --> 00:36:58,920 Here we always got zero from the all ones vector and now 652 00:36:58,920 --> 00:37:04,190 over here that minus one, you see it's just right. 653 00:37:04,190 --> 00:37:09,290 If all the rows add to zero then this vector of all ones 654 00:37:09,290 --> 00:37:14,110 will be, I would use the word "in the null space" 655 00:37:14,110 --> 00:37:18,280 if you wanted a fancy word, a linear algebra word. 656 00:37:18,280 --> 00:37:19,390 What does that mean? 657 00:37:19,390 --> 00:37:21,330 It solves Cu=0. 658 00:37:21,330 --> 00:37:24,770 659 00:37:24,770 --> 00:37:29,080 And why does that show that the matrix isn't invertible? 660 00:37:29,080 --> 00:37:31,340 Because that's our point here. 661 00:37:31,340 --> 00:37:35,210 I have a solution to Cu=0. 662 00:37:35,210 --> 00:37:39,050 I claim that the existence of such a solution 663 00:37:39,050 --> 00:37:45,210 has wiped out the possibility that the matrix is invertible 664 00:37:45,210 --> 00:37:49,120 because if it was invertible, what would this lead to? 665 00:37:49,120 --> 00:37:55,850 If invertible, if C inverse exists what would 666 00:37:55,850 --> 00:38:01,760 I do to that equation that would show me 667 00:38:01,760 --> 00:38:04,360 that C inverse can't exist? 668 00:38:04,360 --> 00:38:08,480 Multiply both sides by C inverse. 669 00:38:08,480 --> 00:38:11,230 So you're seeing, just this first day you're 670 00:38:11,230 --> 00:38:14,330 seeing some of the natural steps of linear algebra. 671 00:38:14,330 --> 00:38:17,140 Row reduction, multiply-- when you 672 00:38:17,140 --> 00:38:20,530 want to see what's happening, multiply both sides 673 00:38:20,530 --> 00:38:21,460 by C inverse. 674 00:38:21,460 --> 00:38:24,330 That's the same as in ordinary language, 675 00:38:24,330 --> 00:38:27,330 do the same thing to all the equations. 676 00:38:27,330 --> 00:38:30,100 So I multiply both sides by the same matrix. 677 00:38:30,100 --> 00:38:32,670 And here I would get C^(-1) C u = C^(-1) 0. 678 00:38:32,670 --> 00:38:36,050 679 00:38:36,050 --> 00:38:40,190 So what does that tell me? 680 00:38:40,190 --> 00:38:43,970 I made it long, I threw in this extra step. 681 00:38:43,970 --> 00:38:51,350 You were going to jump immediately to C^(-1) C is I, 682 00:38:51,350 --> 00:38:54,540 is the identity matrix and when the identity matrix multiplies 683 00:38:54,540 --> 00:38:57,780 a vector u, you get u. 684 00:38:57,780 --> 00:38:59,700 And on the right side, C inverse, 685 00:38:59,700 --> 00:39:02,290 whatever it is, if it existed, times zero 686 00:39:02,290 --> 00:39:05,160 would have to be zero. 687 00:39:05,160 --> 00:39:08,380 So this would say that if C inverse exists, 688 00:39:08,380 --> 00:39:13,020 then the only solution is u equals zero. 689 00:39:13,020 --> 00:39:15,660 That's a good way to recognize invertible matrices. 690 00:39:15,660 --> 00:39:21,770 If it is invertible then the only solution to Cu=0 is u=0. 691 00:39:21,770 --> 00:39:24,620 And that wasn't true here. 692 00:39:24,620 --> 00:39:28,160 So we conclude C is not invertible. 693 00:39:28,160 --> 00:39:32,400 C is therefore not invertible. 694 00:39:32,400 --> 00:39:36,040 Now can I even jump in. 695 00:39:36,040 --> 00:39:37,800 I've got two more matrices that I 696 00:39:37,800 --> 00:39:43,250 want to tell you about that are also close cousins of K and C. 697 00:39:43,250 --> 00:39:50,250 But let me just explain physically a little bit 698 00:39:50,250 --> 00:39:54,050 about where these matrices are coming from. 699 00:39:54,050 --> 00:39:59,760 So maybe next to K-- so I'm not going to put periodic there. 700 00:39:59,760 --> 00:40:01,570 Right? 701 00:40:01,570 --> 00:40:03,770 That's the one that I would call periodic. 702 00:40:03,770 --> 00:40:08,450 This one is fixed at the ends. 703 00:40:08,450 --> 00:40:13,760 Can I draw a little picture that aims to show that? 704 00:40:13,760 --> 00:40:18,730 Aims to show where this is coming from. 705 00:40:18,730 --> 00:40:22,610 It's coming from I think of this as controlling 706 00:40:22,610 --> 00:40:23,990 like four masses. 707 00:40:23,990 --> 00:40:27,660 Mass one, mass two, mass three and mass four 708 00:40:27,660 --> 00:40:40,390 with springs attached and with endpoints fixed. 709 00:40:40,390 --> 00:40:47,290 So if I put some weights on those masses-- we'll do this; 710 00:40:47,290 --> 00:40:51,750 masses and springs is going to be the very first application 711 00:40:51,750 --> 00:40:55,370 and it will connect to all these matrices. 712 00:40:55,370 --> 00:41:05,050 And all I'm doing now is just asking to draw the system. 713 00:41:05,050 --> 00:41:06,490 Draw the mechanical system. 714 00:41:06,490 --> 00:41:09,950 Actually I'll usually draw it vertically. 715 00:41:09,950 --> 00:41:14,580 But anyway, it's got four masses and the fact 716 00:41:14,580 --> 00:41:17,250 that this minus one here got chopped off, 717 00:41:17,250 --> 00:41:19,820 what would I call that end? 718 00:41:19,820 --> 00:41:21,830 I'd call that a fixed end. 719 00:41:21,830 --> 00:41:25,910 So this is a fixed, fixed matrix. 720 00:41:25,910 --> 00:41:28,560 Both ends are fixed. 721 00:41:28,560 --> 00:41:30,620 And it's the matrix that would govern-- 722 00:41:30,620 --> 00:41:34,270 and the springs and masses all the same 723 00:41:34,270 --> 00:41:38,530 is what tells me that the thing is Toeplitz. 724 00:41:38,530 --> 00:41:42,970 Now what's the picture that goes with C? 725 00:41:42,970 --> 00:41:46,680 What's the picture with C? 726 00:41:46,680 --> 00:41:49,520 Do you have an instinct of that? 727 00:41:49,520 --> 00:41:52,080 So C is periodic. 728 00:41:52,080 --> 00:41:57,440 So again we've got four masses connected by springs. 729 00:41:57,440 --> 00:42:03,210 But what's up with those masses to make the problem cyclic, 730 00:42:03,210 --> 00:42:07,140 periodic, circular, whatever word you like. 731 00:42:07,140 --> 00:42:13,190 They're arranged in a ring. 732 00:42:13,190 --> 00:42:16,450 The fourth guy comes back to the first one. 733 00:42:16,450 --> 00:42:22,120 So the four masses would be, so in some kind of a ring, 734 00:42:22,120 --> 00:42:27,500 the springs would connect them. 735 00:42:27,500 --> 00:42:31,940 I don't know if that's suggestive, but I hope so. 736 00:42:31,940 --> 00:42:36,990 And what's the point of, can we just 737 00:42:36,990 --> 00:42:39,530 speak about mechanics one moment? 738 00:42:39,530 --> 00:42:46,080 How does that system differ from this fixed system? 739 00:42:46,080 --> 00:42:53,340 Here the whole system can't move, right? 740 00:42:53,340 --> 00:42:55,880 If there no force, then nothing can happen. 741 00:42:55,880 --> 00:43:00,560 Here the whole system can turn. 742 00:43:00,560 --> 00:43:02,750 They can all displace the same amount 743 00:43:02,750 --> 00:43:05,760 and just turn without any compression of the springs, 744 00:43:05,760 --> 00:43:08,890 without any force having to do anything. 745 00:43:08,890 --> 00:43:13,280 And that's why the solution that kills this matrix is 746 00:43:13,280 --> 00:43:15,050 [1, 1, 1, 1]. 747 00:43:15,050 --> 00:43:19,360 So [1, 1, 1, 1] would describe a case where all 748 00:43:19,360 --> 00:43:21,840 the displacements were equal. 749 00:43:21,840 --> 00:43:25,940 In a way it's like the arbitrary constant in calculus. 750 00:43:25,940 --> 00:43:30,250 You're always adding plus C. So here we've 751 00:43:30,250 --> 00:43:36,350 got a solution of all ones that produces zero the way 752 00:43:36,350 --> 00:43:41,610 the derivative of a constant function is the zero function. 753 00:43:41,610 --> 00:43:49,550 So this is just like an indication. 754 00:43:49,550 --> 00:43:51,090 Yes, perfect. 755 00:43:51,090 --> 00:43:52,750 I've got two more matrices. 756 00:43:52,750 --> 00:43:58,770 Are you okay for two more? 757 00:43:58,770 --> 00:44:03,810 Yes okay, what are they? 758 00:44:03,810 --> 00:44:10,110 Okay a different blackboard for the last two. 759 00:44:10,110 --> 00:44:17,640 So one of them is going to come by freeing up this end. 760 00:44:17,640 --> 00:44:24,580 So I'm going to take that support away. 761 00:44:24,580 --> 00:44:29,980 And you might imagine like a tower oscillating up and down 762 00:44:29,980 --> 00:44:34,020 or you might turn it upside down and like a hanging spring, 763 00:44:34,020 --> 00:44:39,160 or rather four springs with four masses hanging onto them. 764 00:44:39,160 --> 00:44:43,470 But this end is fixed and this is not fixed anymore, 765 00:44:43,470 --> 00:44:46,160 this is now free. 766 00:44:46,160 --> 00:44:50,590 And can I tell you the matrix, the free-fixed matrix. 767 00:44:50,590 --> 00:44:53,590 Free-fixed. 768 00:44:53,590 --> 00:44:56,370 Because it's the top end that I changed, 769 00:44:56,370 --> 00:44:59,680 I'm going to call it T. So all the other guys 770 00:44:59,680 --> 00:45:11,340 are going to be the same but the top one, the top row, 771 00:45:11,340 --> 00:45:13,230 the boundary row, boundary conditions 772 00:45:13,230 --> 00:45:17,870 are always the tough part, the tricky part, 773 00:45:17,870 --> 00:45:20,950 the key part of a model, and here 774 00:45:20,950 --> 00:45:25,120 the natural boundary condition is to have a 1 there. 775 00:45:25,120 --> 00:45:34,610 That two changed to a one. 776 00:45:34,610 --> 00:45:37,900 Now if I asked you for the properties of that matrix-- 777 00:45:37,900 --> 00:45:41,010 so that's the third. shall I do the fourth one? 778 00:45:41,010 --> 00:45:44,480 So you have them all, you'll have the whole picture. 779 00:45:44,480 --> 00:45:45,940 The fourth one, well you can guess. 780 00:45:45,940 --> 00:45:48,950 What's the fourth? 781 00:45:48,950 --> 00:45:51,470 What am I going to do? 782 00:45:51,470 --> 00:45:53,380 Free up the other end. 783 00:45:53,380 --> 00:45:59,420 So this guy had one free end and the other guy 784 00:45:59,420 --> 00:46:01,570 has B for both ends. 785 00:46:01,570 --> 00:46:04,050 B for both ends are going to be free. 786 00:46:04,050 --> 00:46:06,840 So this is free-fixed. 787 00:46:06,840 --> 00:46:08,930 This'll be free-free. 788 00:46:08,930 --> 00:46:12,240 So that means I have this free end, 789 00:46:12,240 --> 00:46:17,010 the usual stuff in the middle, no change, 790 00:46:17,010 --> 00:46:23,510 and the last row is what? 791 00:46:23,510 --> 00:46:27,560 What am I going to put in the last row? -1, 1. 792 00:46:27,560 --> 00:46:29,160 -1, 1. 793 00:46:29,160 --> 00:46:34,740 So I've changed the diagonal. 794 00:46:34,740 --> 00:46:38,150 There I put a single one in because I freed up one end. 795 00:46:38,150 --> 00:46:41,900 With B I freed both ends and I got two minus ones. 796 00:46:41,900 --> 00:46:44,420 Now what do you think? 797 00:46:44,420 --> 00:46:52,860 So we've drawn the free-fixed one and what's your guess? 798 00:46:52,860 --> 00:46:55,320 They're all symmetric. 799 00:46:55,320 --> 00:46:57,200 That's no accident. 800 00:46:57,200 --> 00:47:00,690 They're all tridiagonal, no accident again. 801 00:47:00,690 --> 00:47:02,210 Why are they tridiagonal? 802 00:47:02,210 --> 00:47:06,020 Physically they're tridiagonal because that mass is only 803 00:47:06,020 --> 00:47:07,890 connected to its two neighbors, it's 804 00:47:07,890 --> 00:47:10,940 not connected to that mass. 805 00:47:10,940 --> 00:47:16,390 That's why we get a zero in the two, four position. 806 00:47:16,390 --> 00:47:19,180 Because two is not connected to four. 807 00:47:19,180 --> 00:47:21,920 So it's tridiagonal. 808 00:47:21,920 --> 00:47:25,030 And it's not Toeplitz anymore, right? 809 00:47:25,030 --> 00:47:27,720 Toeplitz says constant diagonals and these are not 810 00:47:27,720 --> 00:47:29,310 quite constant. 811 00:47:29,310 --> 00:47:33,990 I would create K, I would take T equal K, 812 00:47:33,990 --> 00:47:36,830 if I was going to create this matrix and then I would say 813 00:47:36,830 --> 00:47:37,330 T(1, 1) = 1. 814 00:47:37,330 --> 00:47:40,060 815 00:47:40,060 --> 00:47:49,020 That command would fix up the first entry. 816 00:47:49,020 --> 00:47:50,580 Yeah, that's a serious question. 817 00:47:50,580 --> 00:47:53,730 Maybe, can I hang on until Friday, and even 818 00:47:53,730 --> 00:47:54,710 maybe next week. 819 00:47:54,710 --> 00:47:56,310 Because it's very important. 820 00:47:56,310 --> 00:48:00,980 When I said boundary conditions are the key to problems, 821 00:48:00,980 --> 00:48:02,760 I'm serious. 822 00:48:02,760 --> 00:48:06,790 If I had to think okay, what do people come in my office 823 00:48:06,790 --> 00:48:08,780 ask about questions, I say right away, 824 00:48:08,780 --> 00:48:10,030 what's the boundary condition? 825 00:48:10,030 --> 00:48:12,910 Because I know that's where the problem is. 826 00:48:12,910 --> 00:48:16,910 And so here we'll see these guys clearly. 827 00:48:16,910 --> 00:48:23,360 Fixed and free, very important. 828 00:48:23,360 --> 00:48:26,330 But also let me say two more words, I never can resist. 829 00:48:26,330 --> 00:48:30,510 So fixed means the displacement is zero. 830 00:48:30,510 --> 00:48:32,790 Something was set to zero. 831 00:48:32,790 --> 00:48:36,540 The fifth guy, the fifth over here, that fifth column was 832 00:48:36,540 --> 00:48:39,270 knocked out. 833 00:48:39,270 --> 00:48:47,300 Free means that in here it could mean that the fifth guy is 834 00:48:47,300 --> 00:48:49,450 the same as the fourth. 835 00:48:49,450 --> 00:48:52,080 The slope is zero. 836 00:48:52,080 --> 00:48:55,760 Fixed is u is zero. 837 00:48:55,760 --> 00:48:59,010 Free is slope is zero. 838 00:48:59,010 --> 00:49:04,710 So here I have a slope of zero at that end, 839 00:49:04,710 --> 00:49:05,960 here I have it at both ends. 840 00:49:05,960 --> 00:49:09,440 So maybe that's a sort of part answer. 841 00:49:09,440 --> 00:49:12,640 Now I wanted to get to the difference between these two 842 00:49:12,640 --> 00:49:15,140 matrices. 843 00:49:15,140 --> 00:49:19,030 And the main properties. 844 00:49:19,030 --> 00:49:19,960 So what are we see? 845 00:49:19,960 --> 00:49:22,210 Symmetric again, tridiagonal again, 846 00:49:22,210 --> 00:49:27,760 not quite Toeplitz, but almost, sort of morally Toeplitz. 847 00:49:27,760 --> 00:49:32,590 But then the key question was invertible or not. 848 00:49:32,590 --> 00:49:34,590 Key question was invertible or not. 849 00:49:34,590 --> 00:49:35,090 Right. 850 00:49:35,090 --> 00:49:37,680 And what's your guess on these two? 851 00:49:37,680 --> 00:49:41,030 Do you think that one's invertible or not? 852 00:49:41,030 --> 00:49:41,770 Make a guess. 853 00:49:41,770 --> 00:49:46,140 You're allowed to guess. 854 00:49:46,140 --> 00:49:47,550 Yeah it is. 855 00:49:47,550 --> 00:49:48,460 Why's that? 856 00:49:48,460 --> 00:49:52,740 Because this thing has still got a support. 857 00:49:52,740 --> 00:49:56,260 It's not free to shift forever. 858 00:49:56,260 --> 00:49:57,860 It's held in there. 859 00:49:57,860 --> 00:50:01,020 So that gives you a hint about this guy. 860 00:50:01,020 --> 00:50:04,890 Invertible or not for B? 861 00:50:04,890 --> 00:50:06,210 No. 862 00:50:06,210 --> 00:50:09,780 And now prove that it's not. 863 00:50:09,780 --> 00:50:12,800 Physically you were saying, well this free guy 864 00:50:12,800 --> 00:50:19,640 with this thing gone now, this is now free-free. 865 00:50:19,640 --> 00:50:21,710 Physically we're saying the whole thing can move, 866 00:50:21,710 --> 00:50:24,130 there's nothing holding it. 867 00:50:24,130 --> 00:50:27,680 But now, for linear algebra, that's not the proper language. 868 00:50:27,680 --> 00:50:31,040 You have to say something about that matrix. 869 00:50:31,040 --> 00:50:33,010 Maybe tell me something about Bu=0. 870 00:50:33,010 --> 00:50:36,820 871 00:50:36,820 --> 00:50:38,950 What are you going to take for u? 872 00:50:38,950 --> 00:50:39,770 Yeah. 873 00:50:39,770 --> 00:50:41,420 Same u. 874 00:50:41,420 --> 00:50:46,740 We're lucky in this course, u = [1, 1, 1, 1] is the guilty main 875 00:50:46,740 --> 00:50:48,800 vector many times. 876 00:50:48,800 --> 00:50:55,090 Because again the rows are all adding to zero and the all ones 877 00:50:55,090 --> 00:51:02,360 vector is in the null space. 878 00:51:02,360 --> 00:51:06,200 If I could just close with one more word. 879 00:51:06,200 --> 00:51:07,810 Because it's the most important. 880 00:51:07,810 --> 00:51:10,160 Two words, two words. 881 00:51:10,160 --> 00:51:11,890 Because they're the most important words, 882 00:51:11,890 --> 00:51:15,200 they're the words that we're leading to in this chapter. 883 00:51:15,200 --> 00:51:18,770 And I'm assuming that for most people they will be new words, 884 00:51:18,770 --> 00:51:21,480 but not for all. 885 00:51:21,480 --> 00:51:24,290 It's a further property of this matrix. 886 00:51:24,290 --> 00:51:25,380 So we've got, how many? 887 00:51:25,380 --> 00:51:27,630 Four properties, or five? 888 00:51:27,630 --> 00:51:29,900 I'm going to go for one more. 889 00:51:29,900 --> 00:51:33,680 And I'm just going to say that name first 890 00:51:33,680 --> 00:51:36,400 so you know it's coming. 891 00:51:36,400 --> 00:51:38,110 And then I'll say, I can't resist 892 00:51:38,110 --> 00:51:41,110 saying a tiny bit about it. 893 00:51:41,110 --> 00:51:45,980 I'll use a whole blackboard for this. 894 00:51:45,980 --> 00:51:54,530 So I'm going to say that K and T are -- here it comes, 895 00:51:54,530 --> 00:52:07,690 take a breath -- positive definite matrices. 896 00:52:07,690 --> 00:52:10,390 So if you don't know what that means, I'm happy. 897 00:52:10,390 --> 00:52:10,890 Right? 898 00:52:10,890 --> 00:52:13,420 Because well, I can tell you one way 899 00:52:13,420 --> 00:52:16,830 to recognize a positive definite matrix. 900 00:52:16,830 --> 00:52:21,190 And while we're at it, let me tell you about C and B. 901 00:52:21,190 --> 00:52:32,160 Those are positive semi-definite because they hit zero somehow. 902 00:52:32,160 --> 00:52:35,830 Positive means up there, greater than zero. 903 00:52:35,830 --> 00:52:40,270 And what is greater than zero that we've already seen? 904 00:52:40,270 --> 00:52:42,900 And we'll say more. 905 00:52:42,900 --> 00:52:44,550 The pivots were. 906 00:52:44,550 --> 00:52:50,620 So if I have a symmetric matrix and the pivots are all positive 907 00:52:50,620 --> 00:52:54,700 then that matrix is not only invertible, because I'm 908 00:52:54,700 --> 00:52:56,920 in good shape, the determinant isn't zero, 909 00:52:56,920 --> 00:53:00,690 I can go backwards and do everything, 910 00:53:00,690 --> 00:53:04,540 those positive numbers are telling me that more than that, 911 00:53:04,540 --> 00:53:07,810 the matrix is positive definite. 912 00:53:07,810 --> 00:53:11,240 So that's a test. 913 00:53:11,240 --> 00:53:13,500 We'll say more about positive definite, 914 00:53:13,500 --> 00:53:17,760 but one way to recognize it is compute the pivots 915 00:53:17,760 --> 00:53:18,930 by elimination. 916 00:53:18,930 --> 00:53:20,630 Are they positive? 917 00:53:20,630 --> 00:53:23,980 We'll see that all the eigenvalues are positive. 918 00:53:23,980 --> 00:53:27,520 The word positive definite just brings the whole 919 00:53:27,520 --> 00:53:29,740 of linear algebra together. 920 00:53:29,740 --> 00:53:33,440 It connects to pivots, it connects to eigenvalues, 921 00:53:33,440 --> 00:53:36,530 it connects to least squares, it's all over the place. 922 00:53:36,530 --> 00:53:39,790 Determinants too. 923 00:53:39,790 --> 00:53:42,050 Questions or discussion. 924 00:53:42,050 --> 00:53:44,970 It's a big class and we're just meeting for the first time 925 00:53:44,970 --> 00:53:49,740 but there's lots of time to, chance to ask me. 926 00:53:49,740 --> 00:53:52,590 I'll always be here after class. 927 00:53:52,590 --> 00:53:53,760 So shall we stop today? 928 00:53:53,760 --> 00:53:58,890 I'll see you Friday or this afternoon. 929 00:53:58,890 --> 00:54:03,710 If this wasn't familiar, this afternoon would be a good idea. 930 00:54:03,710 --> 00:54:05,220 Thank you.