1 00:00:00 --> 00:00:01 2 00:00:01 --> 00:00:02 The following content is provided under a Creative 3 00:00:03 --> 00:00:03 Commons license. 4 00:00:03 --> 00:00:06 Your support will help MIT OpenCourseWare continue to 5 00:00:06 --> 00:00:10 offer high-quality educational resources for free. 6 00:00:10 --> 00:00:12 To make a donation, or to view additional materials from 7 00:00:12 --> 00:00:15 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:15 --> 00:00:20 at ocw.mit.edu. 9 00:00:20 --> 00:00:26 PROFESSOR STRANG: OK, so review session on the first part of 10 00:00:26 --> 00:00:31 the Fourier chapter, these two topics that we've done and that 11 00:00:31 --> 00:00:34 homework is now coming on. 12 00:00:34 --> 00:00:37 Fourier series, the classical facts. 13 00:00:37 --> 00:00:44 Plus, paying attention to the rate of decay of the Fourier 14 00:00:44 --> 00:00:48 coefficients, it's an aspect not always mentioned. 15 00:00:48 --> 00:00:52 And the energy equality is important. 16 00:00:52 --> 00:00:56 So it's not just here's the function, find the coefficient. 17 00:00:56 --> 00:01:02 That's part of it but not all of it. 18 00:01:02 --> 00:01:07 And then the discrete series, we're doing today. 19 00:01:07 --> 00:01:11 OK, so those are the two topics for today, and then the next 20 00:01:11 --> 00:01:15 review session, which would be two weeks from now, would focus 21 00:01:15 --> 00:01:20 especially on convolution and Fourier integrals. 22 00:01:20 --> 00:01:25 OK, so I'm open to questions on the homework, or 23 00:01:25 --> 00:01:27 off the homework. 24 00:01:27 --> 00:01:32 Always fine. 25 00:01:32 --> 00:01:34 I didn't know how many questions to ask you 26 00:01:34 --> 00:01:35 on the homework. 27 00:01:35 --> 00:01:41 I wanted you to have enough practice doing this stuff 28 00:01:41 --> 00:01:46 because the time for this Fourier part of the course 29 00:01:46 --> 00:01:47 is a little shorter. 30 00:01:47 --> 00:01:54 Thanksgiving comes into it, so needed to do some exercise. 31 00:01:54 --> 00:01:56 And you've got a good question. 32 00:01:56 --> 00:01:56 Thanks. 33 00:01:56 --> 00:02:00 AUDIENCE: [INAUDIBLE] 34 00:02:00 --> 00:02:05 PROFESSOR STRANG: Number 18 on the homework, OK. 35 00:02:05 --> 00:02:08 Ah yes, OK. 36 00:02:08 --> 00:02:10 Right, alright. 37 00:02:10 --> 00:02:17 So the idea of that problem, I'm really asking you to read 38 00:02:17 --> 00:02:22 the two pages, the last two pages of the section. 39 00:02:22 --> 00:02:26 That use Fourier series to solve the heat equation. 40 00:02:26 --> 00:02:30 So we've used, briefly, Fourier series to solve 41 00:02:30 --> 00:02:32 Laplace's equation. 42 00:02:32 --> 00:02:34 So that was just to recall. 43 00:02:34 --> 00:02:37 So Fourier series to solve Laplace's equation was when 44 00:02:37 --> 00:02:40 the region was a circle. 45 00:02:40 --> 00:02:44 The function was given, the boundary values were given. 46 00:02:44 --> 00:02:47 It's 2pi periodic because it is a circle. 47 00:02:47 --> 00:02:53 And we solved Laplace inside. 48 00:02:53 --> 00:02:58 Because on the boundary, the perfect thing we needed was 49 00:02:58 --> 00:03:00 the Fourier series to match the path. 50 00:03:00 --> 00:03:04 Now, I'm taking another, classical, classical 51 00:03:04 --> 00:03:06 application too. 52 00:03:06 --> 00:03:07 The heat equation. 53 00:03:07 --> 00:03:13 So I made it heat equation, so this direction is time. 54 00:03:13 --> 00:03:15 This direction is space. 55 00:03:15 --> 00:03:23 The heat equation is u_t=u_xx, the coefficient, everybody here 56 00:03:23 --> 00:03:26 knows there would be a c in there. 57 00:03:26 --> 00:03:29 But let's take it to be one. 58 00:03:29 --> 00:03:34 Then what are the solutions, and how does a Fourier series 59 00:03:34 --> 00:03:37 help help you to match the initial functions? 60 00:03:37 --> 00:03:43 So I'm matching, I'm given u(x,0) here. 61 00:03:43 --> 00:03:44 OK. 62 00:03:44 --> 00:03:47 Along this is at time zero. 63 00:03:47 --> 00:03:50 So that says at a time zero, so I have a bar. 64 00:03:50 --> 00:03:53 I have a conducting bar. 65 00:03:53 --> 00:03:59 And this is such a classical example that I didn't feel you 66 00:03:59 --> 00:04:02 could miss it completely. 67 00:04:02 --> 00:04:06 Even though we look beyond formulas. 68 00:04:06 --> 00:04:09 But here's one where the formula shows us 69 00:04:09 --> 00:04:10 something important. 70 00:04:10 --> 00:04:15 OK, so what are solutions to this equation? 71 00:04:15 --> 00:04:18 You look for solutions, so the classical idea 72 00:04:18 --> 00:04:22 of separate variables. 73 00:04:22 --> 00:04:29 Look for solutions that are of the form, some function of t, 74 00:04:29 --> 00:04:32 maybe I'll try to use the same letters as the text. 75 00:04:32 --> 00:04:38 Some function of t, times some function of x, OK? 76 00:04:38 --> 00:04:43 And the text uses, look for solutions. u(x,t) that 77 00:04:43 --> 00:04:44 are of this form. 78 00:04:44 --> 00:04:50 Some function of t, and I didn't remember. 79 00:04:50 --> 00:04:51 Ah yes, it's A(x)B(t). 80 00:04:51 --> 00:04:56 81 00:04:56 --> 00:04:59 OK, that's what I mean by separating. 82 00:04:59 --> 00:05:02 So those will be especially simple solutions. 83 00:05:02 --> 00:05:06 And when we go to match the initial condition, I'll just 84 00:05:06 --> 00:05:08 plug in t=0 and I'll see the A(x)'s. 85 00:05:09 --> 00:05:12 Well, what are they? 86 00:05:12 --> 00:05:15 In this case, so their eigenfunction - oh, I have to 87 00:05:15 --> 00:05:18 tell you about the remaining boundary conditions, don't I? 88 00:05:18 --> 00:05:25 Because that will decide what the A(x) has to satisfy, and 89 00:05:25 --> 00:05:28 will decide what those eigenfunctions are. 90 00:05:28 --> 00:05:29 So, let's see. 91 00:05:29 --> 00:05:35 In this problem I think I picked free conditions. 92 00:05:35 --> 00:05:40 So I made the interval minus pi to pi, that's a change. 93 00:05:40 --> 00:05:43 Minus pi to pi just so we have nice Fourier series. 94 00:05:43 --> 00:05:49 And here I have this boundary is free, du/dx, u'. 95 00:05:51 --> 00:05:52 At x=-pi. 96 00:05:54 --> 00:05:59 For all time, so up this line is zero. 97 00:05:59 --> 00:06:05 And also u' at plus pi, and all time is zero. 98 00:06:05 --> 00:06:09 So up those lines, heat's going out. 99 00:06:09 --> 00:06:11 That's what that means. 100 00:06:11 --> 00:06:13 Is that what that means, or does that mean 101 00:06:13 --> 00:06:14 heat can't go out? 102 00:06:14 --> 00:06:18 No, so what's happening? 103 00:06:18 --> 00:06:19 Heat's not going out. 104 00:06:19 --> 00:06:21 Is that right? 105 00:06:21 --> 00:06:23 The slope is zero, right? 106 00:06:23 --> 00:06:26 The slope is the temperature gradient, we're requiring 107 00:06:26 --> 00:06:27 the temperature gradient. 108 00:06:27 --> 00:06:30 So the ends of the bar are insulated. 109 00:06:30 --> 00:06:32 So this is insulated. 110 00:06:32 --> 00:06:35 No passage through, right? 111 00:06:35 --> 00:06:36 Is that what that means? 112 00:06:36 --> 00:06:37 Yeah. 113 00:06:37 --> 00:06:43 It's like there's nobody, it's cut off there. 114 00:06:43 --> 00:06:46 The rod isn't extended for heat to go further. 115 00:06:46 --> 00:06:55 OK, so that tells us what the x and the t, what 116 00:06:55 --> 00:06:57 the A(x) and B(t) are. 117 00:06:57 --> 00:06:58 So here's the point. 118 00:06:58 --> 00:07:01 You plug that hoped-for solution into the 119 00:07:01 --> 00:07:04 equation, right? 120 00:07:04 --> 00:07:07 So I plug it in here. 121 00:07:07 --> 00:07:10 What do I have on the left side, just the time derivative? 122 00:07:10 --> 00:07:16 So that's A times A(x), B'(t), you see taking the time 123 00:07:16 --> 00:07:17 derivative doesn't touch A(x). 124 00:07:18 --> 00:07:22 On the right-hand side, I don't touch B(t), but I 125 00:07:22 --> 00:07:25 have a second derivative. 126 00:07:25 --> 00:07:27 Of the x part. 127 00:07:27 --> 00:07:30 So far, so good. 128 00:07:30 --> 00:07:35 Now, a little trick. 129 00:07:35 --> 00:07:39 If I divide both sides by A and by B, I get 130 00:07:39 --> 00:07:43 B'/B equaling A''/A. 131 00:07:43 --> 00:07:47 132 00:07:47 --> 00:07:48 Right? 133 00:07:48 --> 00:07:52 Just, put the A under here, put the B under here. 134 00:07:52 --> 00:07:53 Now what? 135 00:07:53 --> 00:07:58 This is neat, because that function is only 136 00:07:58 --> 00:08:00 depending on time. 137 00:08:00 --> 00:08:05 This function depends only on x, so that the both sides 138 00:08:05 --> 00:08:06 have to be constant. 139 00:08:06 --> 00:08:11 One can't actually change with time, because this side is 140 00:08:11 --> 00:08:13 not changing with time. 141 00:08:13 --> 00:08:15 And this couldn't actually change with x, because 142 00:08:15 --> 00:08:17 that's not changing with x. 143 00:08:17 --> 00:08:20 So those are both constants. 144 00:08:20 --> 00:08:23 So both constants. 145 00:08:23 --> 00:08:27 Let's see, I'll maybe just put a constant. 146 00:08:27 --> 00:08:30 And various constants are possible. 147 00:08:30 --> 00:08:32 OK, so now you see the point here. 148 00:08:32 --> 00:08:36 Now I have two separate equations, I have an equation 149 00:08:36 --> 00:08:39 for the B part, dB/dt, B'. 150 00:08:40 --> 00:08:46 If I bring the B up there, I have equals, the 151 00:08:46 --> 00:08:48 constant times B. 152 00:08:48 --> 00:08:50 And I know the solution to that. 153 00:08:50 --> 00:08:56 B(t) is, as everybody knows, what's the solution to 154 00:08:56 --> 00:08:59 a first order constant coefficient equation? 155 00:08:59 --> 00:08:59 Just e^(ct)*B(0). 156 00:08:59 --> 00:09:03 157 00:09:03 --> 00:09:06 Good, we've got B. 158 00:09:06 --> 00:09:09 We've got a B(t) that works, and now what's 159 00:09:09 --> 00:09:11 the A that also works? 160 00:09:11 --> 00:09:12 That has A''. 161 00:09:14 --> 00:09:20 And I bring the A up, so now I have A''=cA, so the A 162 00:09:20 --> 00:09:25 that goes with it is? 163 00:09:25 --> 00:09:32 Oh, OK, I've used a c there, so what's the good? 164 00:09:32 --> 00:09:35 I want two derivatives should bring down a c. 165 00:09:35 --> 00:09:42 Let me change c to minus lambda squared. 166 00:09:42 --> 00:09:46 How about if I look ahead, change this constant to minus 167 00:09:46 --> 00:09:50 lambda squared, because I want something where two derivatives 168 00:09:50 --> 00:09:55 bring down minus lambda squared, and what will do that? 169 00:09:55 --> 00:10:00 Any amount of cos(lambda*x), right? 170 00:10:00 --> 00:10:02 Because two derivatives will bring down a 171 00:10:02 --> 00:10:04 minus lambda squared. 172 00:10:04 --> 00:10:06 And any amount of sin(lambda*x). 173 00:10:07 --> 00:10:18 And this is now e to the minus lambda squared t. 174 00:10:18 --> 00:10:21 I'm doing this fast, but actually it's totally simple. 175 00:10:21 --> 00:10:25 The conclusion is that I now have a bunch of solutions of 176 00:10:25 --> 00:10:28 this special separated form. 177 00:10:28 --> 00:10:33 Where B(t) could be that and A(x) could be either of those 178 00:10:33 --> 00:10:35 or any combination of those. 179 00:10:35 --> 00:10:40 And I have to use the same lambda for each, so that the 180 00:10:40 --> 00:10:46 two equations will work in the original problem. 181 00:10:46 --> 00:10:47 Good. 182 00:10:47 --> 00:10:51 Now, so far, no boundary conditions. 183 00:10:51 --> 00:10:54 What I've got so far is just a lot of solutions. 184 00:10:54 --> 00:10:56 These times that. 185 00:10:56 --> 00:10:58 With any lambda. 186 00:10:58 --> 00:11:01 But of course the boundary conditions will tell me the 187 00:11:01 --> 00:11:03 lambdas, first of all. 188 00:11:03 --> 00:11:04 And how do they tell me? 189 00:11:04 --> 00:11:08 The only boundary condition I have is this free stuff. 190 00:11:08 --> 00:11:14 So it's, free the slope should be zero at pi, and zero at? 191 00:11:14 --> 00:11:16 So that's the x direction. 192 00:11:16 --> 00:11:19 So that's going to tell me - I've forgotten. 193 00:11:19 --> 00:11:23 Do I want cosines or sines? 194 00:11:23 --> 00:11:24 Cosines. 195 00:11:24 --> 00:11:29 I want the derivative to be zero at pi. 196 00:11:29 --> 00:11:32 Yeah, so I think I want cosines, good. 197 00:11:32 --> 00:11:35 And then lambdas can't be anything at all. 198 00:11:35 --> 00:11:41 Because, should lambda be an integer or something like that? 199 00:11:41 --> 00:11:44 I think maybe lambda should be an integer, because 200 00:11:44 --> 00:11:47 I want to plug in pi. 201 00:11:47 --> 00:11:50 So let me take the second derivative. 202 00:11:50 --> 00:11:53 Is minus lambda squared cos(lambda*x). 203 00:11:54 --> 00:11:56 And then I want to plug in lambda=x=pi. 204 00:11:58 --> 00:12:04 And I want this to be zero. 205 00:12:04 --> 00:12:10 So lambda should be an integer, is that right? 206 00:12:10 --> 00:12:13 At multiples of pi, the cosine is zero, yes. 207 00:12:13 --> 00:12:16 Is it? no. 208 00:12:16 --> 00:12:19 Did I want sine? 209 00:12:19 --> 00:12:23 Maybe I wanted sine. 210 00:12:23 --> 00:12:26 Oh, it's the first derivative I'm looking at, thank you. 211 00:12:26 --> 00:12:27 Thank you. 212 00:12:27 --> 00:12:31 OK, good. 213 00:12:31 --> 00:12:33 OK, now I've got it. 214 00:12:33 --> 00:12:34 Thank you. 215 00:12:34 --> 00:12:37 And now I see lambda should be an integer. 216 00:12:37 --> 00:12:41 Lambda should be, zero is, yeah zero's alright. 217 00:12:41 --> 00:12:43 That's the constant term, yeah, we need that. zero, 218 00:12:43 --> 00:12:46 one, two, and so on. 219 00:12:46 --> 00:12:52 So do you see that I've now got, now I can take, I've 220 00:12:52 --> 00:12:53 got a lot of solutions. 221 00:12:53 --> 00:12:55 And I have a linear problem. 222 00:12:55 --> 00:12:57 So I can take any combination. 223 00:12:57 --> 00:13:05 So finally I have final solution is that u(x,t) is any 224 00:13:05 --> 00:13:12 combination with coefficients, I'm free to choose, of A(x), 225 00:13:12 --> 00:13:18 which is cos(nx), because lambda had to be an n. 226 00:13:18 --> 00:13:22 And n could be anywhere from zero on up. 227 00:13:22 --> 00:13:31 Times e to the minus, lambda's n, so that's n squared t. 228 00:13:31 --> 00:13:32 Did I get that? 229 00:13:32 --> 00:13:34 Let me draw a circle and step back. 230 00:13:34 --> 00:13:35 What's up? 231 00:13:35 --> 00:13:37 AUDIENCE: [INAUDIBLE] 232 00:13:37 --> 00:13:40 PROFESSOR STRANG: I could have negative n's, they wouldn't 233 00:13:40 --> 00:13:42 give me anything new, right? 234 00:13:42 --> 00:13:48 I mean cos(nx) and cos(-nx) are just, one's just the 235 00:13:48 --> 00:13:51 negative of the other. 236 00:13:51 --> 00:13:53 So these are the good guys. 237 00:13:53 --> 00:13:56 I've got a cosine series because I've got 238 00:13:56 --> 00:13:58 free n's, right. 239 00:13:58 --> 00:14:00 Because of the boundary conditions. 240 00:14:00 --> 00:14:01 Do you say that? 241 00:14:01 --> 00:14:04 That's pretty nice. 242 00:14:04 --> 00:14:08 There's my A(x), there's my B(t), I can take 243 00:14:08 --> 00:14:10 any combination. 244 00:14:10 --> 00:14:11 Usual stuff. 245 00:14:11 --> 00:14:14 You get to that solution. 246 00:14:14 --> 00:14:18 OK, and now I have to meet the initial conditions, right? 247 00:14:18 --> 00:14:21 Boundary conditions are now built-in because 248 00:14:21 --> 00:14:23 I chose cosine. 249 00:14:23 --> 00:14:24 Or you did. 250 00:14:24 --> 00:14:30 Now, this will tell me what the c's are. 251 00:14:30 --> 00:14:31 I'm going to set t=0. 252 00:14:33 --> 00:14:38 At t=0, I'm given the initial condition, u(x,0), and I 253 00:14:38 --> 00:14:40 have the same sum of c_n*cos(nx), e^0. 254 00:14:40 --> 00:14:45 255 00:14:45 --> 00:14:49 So this will tell me the c's are the cosine coefficients of 256 00:14:49 --> 00:14:51 the given initial conditions. 257 00:14:51 --> 00:14:56 So I expand, so here's where Fourier series is paid off. 258 00:14:56 --> 00:15:01 Expand the initial function in a cosine series. 259 00:15:01 --> 00:15:04 And then go forward in time. 260 00:15:04 --> 00:15:06 This is just the old e^(lambda*t). 261 00:15:06 --> 00:15:10 262 00:15:10 --> 00:15:14 Only the lambda we're talking about here is minus n squared. 263 00:15:14 --> 00:15:18 And you see what's happening here? 264 00:15:18 --> 00:15:21 What's going to happen for large time? 265 00:15:21 --> 00:15:23 So this is a very physical problem. 266 00:15:23 --> 00:15:27 That I think you cannot take 18.085 without 267 00:15:27 --> 00:15:29 seeing this problem. 268 00:15:29 --> 00:15:32 You can't learn about Fourier series without using it 269 00:15:32 --> 00:15:36 for the initial value. 270 00:15:36 --> 00:15:41 And then propagating in time with the usual exponentials. 271 00:15:41 --> 00:15:46 For each one, and now as n increases what do I see? 272 00:15:46 --> 00:15:48 Faster and faster decay. 273 00:15:48 --> 00:15:54 For large n, these are going to zero extremely fast. 274 00:15:54 --> 00:16:00 So that what you see with a solid bar, which starts with 275 00:16:00 --> 00:16:08 the temperature u in some probably not negative unless 276 00:16:08 --> 00:16:10 it's a really cold bar. 277 00:16:10 --> 00:16:16 But, anyway, it starts with some initial temperature. 278 00:16:16 --> 00:16:18 That flattens out fast. 279 00:16:18 --> 00:16:21 The heat flows, to equilibrate. 280 00:16:21 --> 00:16:24 What I approach is the constant terms, c_0. 281 00:16:26 --> 00:16:27 This approaches c_0. 282 00:16:28 --> 00:16:33 Because all these other n positives, they go to zero. 283 00:16:33 --> 00:16:38 So the heat distributes itself equally. 284 00:16:38 --> 00:16:42 OK, and now I guess the particular u(0) in the 285 00:16:42 --> 00:16:46 problem was a delta. 286 00:16:46 --> 00:16:53 OK, and so the particular u(0) was all, was from 287 00:16:53 --> 00:16:58 that really hot point. 288 00:16:58 --> 00:17:00 So we know the coefficients. 289 00:17:00 --> 00:17:05 We know the cosine coefficients for the delta function, we know 290 00:17:05 --> 00:17:08 these c's, and what were they? 291 00:17:08 --> 00:17:12 1/2pi was c_0. 292 00:17:12 --> 00:17:15 And the other c's were 1/pi, I think. 293 00:17:15 --> 00:17:18 Is that right? 294 00:17:18 --> 00:17:19 Maybe they're all 1/2pi. 295 00:17:21 --> 00:17:22 Maybe. 296 00:17:22 --> 00:17:22 Yeah. 297 00:17:22 --> 00:17:24 Whatever. 298 00:17:24 --> 00:17:25 They disappear fast. 299 00:17:25 --> 00:17:28 And this is what we approach. 300 00:17:28 --> 00:17:31 So the heat from the delta function is, yeah. 301 00:17:31 --> 00:17:37 So is that everything the problem wanted? 302 00:17:37 --> 00:17:37 Yeah. 303 00:17:37 --> 00:17:38 Yeah. 304 00:17:38 --> 00:17:40 I think we've done it. 305 00:17:40 --> 00:17:44 We'll put in the c_n's to complete that picture. 306 00:17:44 --> 00:17:46 Into here. 307 00:17:46 --> 00:17:51 And then c_0 is the one that survives over time. 308 00:17:51 --> 00:17:54 Yeah. 309 00:17:54 --> 00:17:56 I guess you've, once I got rolling I couldn't 310 00:17:56 --> 00:18:01 stop and that's u. 311 00:18:01 --> 00:18:06 For investing time this afternoon you get a fast look 312 00:18:06 --> 00:18:11 at this classical, classical problem of separating the 313 00:18:11 --> 00:18:14 variables using the Fourier series for 314 00:18:14 --> 00:18:16 the initial function. 315 00:18:16 --> 00:18:20 And recognizing that we're doing this on 316 00:18:20 --> 00:18:24 a finite interval. 317 00:18:24 --> 00:18:32 If the bar was infinitely long, then we would be talking 318 00:18:32 --> 00:18:34 about Fourier integrals. 319 00:18:34 --> 00:18:37 And that's what's coming up a bit later. 320 00:18:37 --> 00:18:40 We would integrate instead of sum. 321 00:18:40 --> 00:18:40 Yeah. 322 00:18:40 --> 00:18:45 But the idea would not be different, if we had infinite 323 00:18:45 --> 00:18:48 bar then we would not be restricted to n equals zero, 324 00:18:48 --> 00:18:52 one, two, three, any n, any cosine, wouldn't have to 325 00:18:52 --> 00:18:54 be an integer at all. 326 00:18:54 --> 00:18:57 Any number, any frequency would be allowed. 327 00:18:57 --> 00:19:02 And so we would have to integrate that, yeah. 328 00:19:02 --> 00:19:08 And that's a classical problem too, again. 329 00:19:08 --> 00:19:11 It's come up in a modern way, that the famous 330 00:19:11 --> 00:19:15 Black-Scholes equation. 331 00:19:15 --> 00:19:17 So. 332 00:19:17 --> 00:19:21 The heat equation is for 18.086. 333 00:19:21 --> 00:19:26 Here, we brought it up because we could solve it fast. 334 00:19:26 --> 00:19:30 But the actual, yeah. 335 00:19:30 --> 00:19:33 The most important solution I could give you to the heat 336 00:19:33 --> 00:19:36 equation would be the one that starts from that 337 00:19:36 --> 00:19:39 point source of heat. 338 00:19:39 --> 00:19:42 But on the whole line. 339 00:19:42 --> 00:19:46 The one that would be integrals instead of sums. 340 00:19:46 --> 00:19:47 Yeah. 341 00:19:47 --> 00:19:51 So we came pretty close to solving the most important 342 00:19:51 --> 00:19:53 heat equation problem. 343 00:19:53 --> 00:20:00 But doing the periodic case, with just cosines, and the 344 00:20:00 --> 00:20:05 infinite line case would be the most famous of all. 345 00:20:05 --> 00:20:05 Yeah. 346 00:20:05 --> 00:20:09 And it has a beautiful form, and I was going to say that 347 00:20:09 --> 00:20:12 the heat equation's pretty classical. 348 00:20:12 --> 00:20:19 But let's see, where can I write the magic 349 00:20:19 --> 00:20:22 words, Black-Scholes. 350 00:20:22 --> 00:20:27 Next to the heat equation, so that's the heat equation but 351 00:20:27 --> 00:20:36 it's also - do you know these names, Black and Scholes? 352 00:20:36 --> 00:20:40 Anybody in Mathematics of Finance? 353 00:20:40 --> 00:20:47 So these much-despised option, derivative options, so people 354 00:20:47 --> 00:20:53 on Wall Street, traders, will carry around a little 355 00:20:53 --> 00:20:58 calculator that solves the Black-Scholes equation so they 356 00:20:58 --> 00:21:02 can price the options that they're bidding to buy and 357 00:21:02 --> 00:21:04 sell, so they can price them fast. 358 00:21:04 --> 00:21:10 And that little calculator does a finite differences, or a 359 00:21:10 --> 00:21:15 Fourier series solution to this Black-Scholes equation, which 360 00:21:15 --> 00:21:18 actually, if you change variables on it, is 361 00:21:18 --> 00:21:19 the heat equation. 362 00:21:19 --> 00:21:25 So what you see here as is actually important on Wall 363 00:21:25 --> 00:21:28 Street except, it's probably not the right moment 364 00:21:28 --> 00:21:29 to mention it. 365 00:21:29 --> 00:21:34 Right? 366 00:21:34 --> 00:21:38 So you can blame the whole meltdown on mathematicians. 367 00:21:38 --> 00:21:41 But that wouldn't be entirely fair. 368 00:21:41 --> 00:21:43 They didn't mean it, anyway. 369 00:21:43 --> 00:21:50 But that's been the biggest source of employment, I 370 00:21:50 --> 00:21:55 guess, apart from teaching, in the last ten years. 371 00:21:55 --> 00:21:59 People who could work out the partial differential equations, 372 00:21:59 --> 00:22:01 and they get more complicated than the heat equation, 373 00:22:01 --> 00:22:03 you can be sure. 374 00:22:03 --> 00:22:11 And so the classical one, these guys are economists at MIT 375 00:22:11 --> 00:22:13 and Harvard, or they were. 376 00:22:13 --> 00:22:18 And I guess maybe the Nobel Prize in Economics came 377 00:22:18 --> 00:22:20 to part of that group. 378 00:22:20 --> 00:22:23 And also to Merton. 379 00:22:23 --> 00:22:28 Maybe Black, possibly Black died before the time 380 00:22:28 --> 00:22:31 of the Nobel Prize. 381 00:22:31 --> 00:22:32 Anyway, they were the first. 382 00:22:32 --> 00:22:36 And it's a beautiful paper, beautiful paper too. 383 00:22:36 --> 00:22:41 Just to figure out how do you price the, what's the value 384 00:22:41 --> 00:22:47 of an option to buy that allows you to buy or sell 385 00:22:47 --> 00:22:49 a stock at a later time? 386 00:22:49 --> 00:22:50 Yeah. 387 00:22:50 --> 00:22:54 So it's, well of course you have to make assumptions on 388 00:22:54 --> 00:22:58 what's going to happen over that time and that's where Wall 389 00:22:58 --> 00:23:01 Street came to grief, I guess. 390 00:23:01 --> 00:23:10 If you had to put it in a nutshell, I mean, the options, 391 00:23:10 --> 00:23:13 the standard straightforward options, those are fine. 392 00:23:13 --> 00:23:18 Using Black-Scholes, and then what's happened is they now 393 00:23:18 --> 00:23:21 price more and more complicated things. 394 00:23:21 --> 00:23:27 To the point that the banks were buying and selling credit 395 00:23:27 --> 00:23:32 default swaps, insurance swaps, that practically nobody 396 00:23:32 --> 00:23:33 understood what they were. 397 00:23:33 --> 00:23:37 They just assumed that if there was always a market, for them 398 00:23:37 --> 00:23:39 like insurance, somehow it wouldn't happen. 399 00:23:39 --> 00:23:41 And you get on. 400 00:23:41 --> 00:23:42 But it happened. 401 00:23:42 --> 00:23:46 So, now we're in trouble. 402 00:23:46 --> 00:23:50 OK, that's not 18.085, fortunately. 403 00:23:50 --> 00:23:55 Or math, but. 404 00:23:55 --> 00:23:58 Anyway, a lot of people got involved with things they 405 00:23:58 --> 00:24:01 didn't really know about. 406 00:24:01 --> 00:24:06 And then were selling them as well as of course 407 00:24:06 --> 00:24:07 the mortgage problems. 408 00:24:07 --> 00:24:11 Anyway. 409 00:24:11 --> 00:24:20 Ready for other questions on our homework, or these topics. 410 00:24:20 --> 00:24:21 Yeah. 411 00:24:21 --> 00:24:22 OK. 412 00:24:22 --> 00:24:24 AUDIENCE: [INAUDIBLE] 413 00:24:24 --> 00:24:24 PROFESSOR STRANG: OK 414 00:24:24 --> 00:24:28 AUDIENCE: [INAUDIBLE] 415 00:24:28 --> 00:24:35 PROFESSOR STRANG: OK, right, yeah. 416 00:24:35 --> 00:24:37 Have an image of waves, so -- 417 00:24:37 --> 00:24:38 AUDIENCE: [INAUDIBLE] 418 00:24:38 --> 00:24:41 PROFESSOR STRANG: Yeah. 419 00:24:41 --> 00:24:45 I suppose if I had to have a picture of the discrete thing, 420 00:24:45 --> 00:24:52 if my picture of the function case was a bunch of sines, and 421 00:24:52 --> 00:24:57 cosines, somehow adding up to my function. 422 00:24:57 --> 00:25:03 And if time, I'm solving the heat equation then those 423 00:25:03 --> 00:25:09 separate waves are maybe decaying in time. 424 00:25:09 --> 00:25:09 Here. 425 00:25:09 --> 00:25:12 So that when I add them up at a later time I get 426 00:25:12 --> 00:25:13 something different. 427 00:25:13 --> 00:25:18 Or if it was the wave equation, which is probably your image, 428 00:25:18 --> 00:25:20 they're moving in time. 429 00:25:20 --> 00:25:23 So they add up to different things at different times, 430 00:25:23 --> 00:25:26 because they can move at different speeds. 431 00:25:26 --> 00:25:31 Yes, so a function is a sum of waves, right? 432 00:25:31 --> 00:25:34 Then what would the discrete guy be? 433 00:25:34 --> 00:25:38 I guess I would just have to imagine the function as only 434 00:25:38 --> 00:25:41 having those n values. 435 00:25:41 --> 00:25:45 And my wave would just be, a wave might be 436 00:25:45 --> 00:25:51 just n values there. 437 00:25:51 --> 00:25:58 But still, if I have a time- dependent problem maybe 438 00:25:58 --> 00:26:01 that thing is pulsing thing up and down. 439 00:26:01 --> 00:26:05 It's just that it's only got a fixed number of points. 440 00:26:05 --> 00:26:09 And I'm not looking at the whole wave on a, yeah. 441 00:26:09 --> 00:26:15 But I don't think it's essentially different. 442 00:26:15 --> 00:26:19 And of course, the fast Fourier transform and the discrete case 443 00:26:19 --> 00:26:23 is used to approximate the continuous one. 444 00:26:23 --> 00:26:24 Yeah. 445 00:26:24 --> 00:26:29 You can look in numerical recipes for a discussion of 446 00:26:29 --> 00:26:32 approximating the Fourier series by discrete 447 00:26:32 --> 00:26:33 Fourier series. 448 00:26:33 --> 00:26:34 I mean, that's an important question. 449 00:26:34 --> 00:26:37 Because of course, Fourier series has got all 450 00:26:37 --> 00:26:39 these integrals. 451 00:26:39 --> 00:26:46 The coefficients come from an integral formula. 452 00:26:46 --> 00:26:50 We're not going to do those integrals exactly, so we 453 00:26:50 --> 00:26:53 have to do them some approximate way. 454 00:26:53 --> 00:26:57 And one way would be to use equally spaced 455 00:26:57 --> 00:26:58 points and do the DFT. 456 00:27:00 --> 00:27:04 Can you just remind me what that integral formula is? 457 00:27:04 --> 00:27:07 I don't want you to, it was on the board today. 458 00:27:07 --> 00:27:11 What's the formula for the coefficient c_k in 459 00:27:11 --> 00:27:15 the Fourier series. 460 00:27:15 --> 00:27:20 I'm really just asking you this because I think you should have 461 00:27:20 --> 00:27:26 it in some memory cache, you know in fast memory, rather 462 00:27:26 --> 00:27:28 than in the textbook. 463 00:27:28 --> 00:27:32 OK, so what's the formula for c_k, in the 464 00:27:32 --> 00:27:34 Fourier series case? 465 00:27:34 --> 00:27:36 Everybody think about it. 466 00:27:36 --> 00:27:39 It's going to be an integral, right? 467 00:27:39 --> 00:27:42 And I'll take it over zero to 2pi, I don't mind. 468 00:27:42 --> 00:27:44 Or minus pi, pi. 469 00:27:44 --> 00:27:46 And what do I integrate? 470 00:27:46 --> 00:27:48 As the Fourier coefficients of the function f(x)? 471 00:27:49 --> 00:27:55 So I take f(x), I remember to divide by 2pi, I'm doing the 472 00:27:55 --> 00:28:00 continuous one, what do I multiply by to get the 473 00:28:00 --> 00:28:03 coefficients? e^(-ikx). 474 00:28:03 --> 00:28:12 475 00:28:12 --> 00:28:16 So I've forgotten whether I signed some of these 476 00:28:16 --> 00:28:19 very early questions. 477 00:28:19 --> 00:28:20 It just gave you the function and said 478 00:28:20 --> 00:28:22 what's the coefficient. 479 00:28:22 --> 00:28:24 If I just look at one or two. 480 00:28:24 --> 00:28:26 Suppose my function is f(x)=x. 481 00:28:27 --> 00:28:31 I guess in that problem, in Problem 1, I made 482 00:28:31 --> 00:28:33 it minus pi to pi. 483 00:28:33 --> 00:28:34 Suppose f(x)=x. 484 00:28:34 --> 00:28:40 485 00:28:40 --> 00:28:43 So you have to integrate x times e^(-ikx). 486 00:28:43 --> 00:28:48 487 00:28:48 --> 00:28:50 Well, you got an integral to do. 488 00:28:50 --> 00:28:55 OK, it's doable but not instantly doable. 489 00:28:55 --> 00:28:58 Let me ask you some questions and you tell me about it. 490 00:28:58 --> 00:29:01 So I draw the function. 491 00:29:01 --> 00:29:05 The function is x from minus pi to pi. 492 00:29:05 --> 00:29:08 And tell me about the coefficients, how 493 00:29:08 --> 00:29:13 quickly do they decay? 494 00:29:13 --> 00:29:18 This is like some constant over k to some power 495 00:29:18 --> 00:29:23 p, and what's p? 496 00:29:23 --> 00:29:27 What rate of decay are you expecting for the coefficients? 497 00:29:27 --> 00:29:29 Well you say to yourself, it looks like a pretty nice 498 00:29:29 --> 00:29:32 function, smooth as can be. 499 00:29:32 --> 00:29:37 But, what's the answer here? 500 00:29:37 --> 00:29:43 The rate of decay will be, what will that power be? 501 00:29:43 --> 00:29:44 One. 502 00:29:44 --> 00:29:49 Because the function jumps. 503 00:29:49 --> 00:29:53 The function has a jump there, and the Fourier coefficients 504 00:29:53 --> 00:29:55 have got to deal with it. 505 00:29:55 --> 00:29:59 So the Fourier series for this is going to be, if I took 100 506 00:29:59 --> 00:30:01 terms it would be really close. 507 00:30:01 --> 00:30:09 And then it'll go down here to the, but it's got to get there. 508 00:30:09 --> 00:30:13 And got to start, and pick up below there. 509 00:30:13 --> 00:30:18 So it's got the same issue that the Gibbs phenomenon, 510 00:30:18 --> 00:30:21 that the square wave had. 511 00:30:21 --> 00:30:22 It's got that jump. 512 00:30:22 --> 00:30:25 So it'll go like 1/k. 513 00:30:26 --> 00:30:29 OK, at coming back. 514 00:30:29 --> 00:30:31 Could you actually find those numbers? 515 00:30:31 --> 00:30:34 And do you remember how to do integral like that? 516 00:30:34 --> 00:30:38 Well, look it up, I guess is the best answer. 517 00:30:38 --> 00:30:44 But whoever did it the first time, well, it's integration 518 00:30:44 --> 00:30:46 by parts, somehow. 519 00:30:46 --> 00:30:50 The derivative of this makes it real simple, and this we can 520 00:30:50 --> 00:30:52 integrate really easily, right? 521 00:30:52 --> 00:30:55 So we integrate that, take the derivative of that. 522 00:30:55 --> 00:31:02 We get a boundary term, so I don't exactly 523 00:31:02 --> 00:31:03 remember the formula. 524 00:31:03 --> 00:31:09 But it'll have a couple of terms, but not bad. 525 00:31:09 --> 00:31:12 And we'll see this 1/k. 526 00:31:13 --> 00:31:16 OK, and that's the formula. 527 00:31:16 --> 00:31:21 If I changed x to something else, let's see. 528 00:31:21 --> 00:31:24 As long as I'm looking at number one, what if I took e^x? 529 00:31:24 --> 00:31:25 Oh, easy. 530 00:31:25 --> 00:31:31 Right? e^x, yeah, let's do e^x, because that's an easy one. 531 00:31:31 --> 00:31:35 Now, e^x looks like, so what's my graph of e^x? 532 00:31:36 --> 00:31:41 It's quite small here at minus pi, and pretty large at e^pi. 533 00:31:42 --> 00:31:45 What's the rate of decay of the Fourier coefficients 534 00:31:45 --> 00:31:48 for that guy? 535 00:31:48 --> 00:31:50 Same. 536 00:31:50 --> 00:31:51 Got to jump again. 537 00:31:51 --> 00:31:52 Drops down here. 538 00:31:52 --> 00:31:55 So let's find it. 539 00:31:55 --> 00:31:58 Let's find those Fourier, that integral. 540 00:31:58 --> 00:31:59 That's e^(x-ik). 541 00:32:02 --> 00:32:06 Sorry, let's put down what I'm integrating here. 542 00:32:06 --> 00:32:08 I'm integrating e^x(1-ik). 543 00:32:08 --> 00:32:13 544 00:32:13 --> 00:32:16 Right? 545 00:32:16 --> 00:32:18 That's what I've got to integrate. 546 00:32:18 --> 00:32:22 The integral is the same thing divided by 1-ik. 547 00:32:22 --> 00:32:24 548 00:32:24 --> 00:32:28 I plug in the endpoints, minus pi and pi, and 549 00:32:28 --> 00:32:29 I've got the answer. 550 00:32:29 --> 00:32:30 And I divide by 2pi. 551 00:32:30 --> 00:32:33 552 00:32:33 --> 00:32:34 Yeah. 553 00:32:34 --> 00:32:36 So that's a totally doable one. 554 00:32:36 --> 00:32:37 Let's plug it in. 555 00:32:37 --> 00:32:37 1/2pi. 556 00:32:37 --> 00:32:40 557 00:32:40 --> 00:32:45 And then the denominator is this 1-ik, yeah, well there 558 00:32:45 --> 00:32:47 you see it already, right? 559 00:32:47 --> 00:32:51 You see already the 1/k in the denominator. 560 00:32:51 --> 00:32:54 That's giving us the slow decay rate. 561 00:32:54 --> 00:32:56 And now I'm plugging in x=e^pi(1-ik)-e^-pi(1-ik). 562 00:32:56 --> 00:33:10 563 00:33:10 --> 00:33:21 So as k gets big, this is slow decay. 564 00:33:21 --> 00:33:24 Now, what's happening here as k gets large? 565 00:33:24 --> 00:33:27 Oh, k is multiplied by i, there. 566 00:33:27 --> 00:33:33 So this e^(pi*ik) is just sitting, it may even be 567 00:33:33 --> 00:33:36 just one or something, or minus one, or whatever. 568 00:33:36 --> 00:33:37 Is it? 569 00:33:37 --> 00:33:41 Yeah. k is just an integer, this is k*pi*i, that's just 570 00:33:41 --> 00:33:43 minus one to something. 571 00:33:43 --> 00:33:49 So all that numerator is minus 1one to the to the k'th 572 00:33:49 --> 00:33:52 power or something. 573 00:33:52 --> 00:33:53 Times e^pi. 574 00:33:53 --> 00:33:56 575 00:33:56 --> 00:34:05 So e^pi pi there. e^-ik, at e^(ii*pi*k) is just minus one. 576 00:34:05 --> 00:34:08 And this is maybe e to the minus pi times 577 00:34:08 --> 00:34:09 the same e^(+i*pi*k). 578 00:34:09 --> 00:34:14 579 00:34:14 --> 00:34:16 I think that's minus one to the k. 580 00:34:16 --> 00:34:17 Well, wait a minute. 581 00:34:17 --> 00:34:20 Maybe they're not both, whatever. 582 00:34:20 --> 00:34:22 It's a number. 583 00:34:22 --> 00:34:24 It's a number. 584 00:34:24 --> 00:34:28 That's just of this size. 585 00:34:28 --> 00:34:31 There's the big number, there's the small one. 586 00:34:31 --> 00:34:37 And divided by that, that's the thing that shows is 587 00:34:37 --> 00:34:39 yes, there is this jump. 588 00:34:39 --> 00:34:41 Right? 589 00:34:41 --> 00:34:45 OK, that's a couple of sets of Fourier coefficients. 590 00:34:45 --> 00:34:48 You could ask yourself, because on the quiz there'll probably 591 00:34:48 --> 00:34:54 be one, and I'll try to pick a function that's interesting. 592 00:34:54 --> 00:35:00 And I mean, I don't plan to pick xe^x or anything. 593 00:35:00 --> 00:35:03 Yeah. 594 00:35:03 --> 00:35:05 OK, good. 595 00:35:05 --> 00:35:05 Yes, thanks. 596 00:35:05 --> 00:35:11 AUDIENCE: [INAUDIBLE] 597 00:35:11 --> 00:35:13 Fourier series has twice the energy as another, 598 00:35:13 --> 00:35:13 what's that mean? 599 00:35:13 --> 00:35:13 PROFESSOR STRANG: I don't know. 600 00:35:13 --> 00:35:14 I guess. 601 00:35:14 --> 00:35:17 Hm. 602 00:35:17 --> 00:35:22 Maybe we're talking about power, and things like if we 603 00:35:22 --> 00:35:31 were dealing with electronics, I guess I would interpret that 604 00:35:31 --> 00:35:33 energy in terms of power. 605 00:35:33 --> 00:35:37 So that's what I'd be seeing. 606 00:35:37 --> 00:35:42 I'm not thinking of a really good answer to say well why 607 00:35:42 --> 00:35:46 is that energy equality, but it's really useful. 608 00:35:46 --> 00:35:50 You know, so much of signal processing, and we'll do a bit 609 00:35:50 --> 00:35:57 of signal processing, is simply based on that energy or 610 00:35:57 --> 00:36:04 equality there, and the moving into frequency space. 611 00:36:04 --> 00:36:05 And convolution. 612 00:36:05 --> 00:36:09 Actually, they would call it filtering. 613 00:36:09 --> 00:36:13 So we'll call it filtering when we use convolution. 614 00:36:13 --> 00:36:15 But it's pure convolution. 615 00:36:15 --> 00:36:22 Pure linear algebra, for these special bases. 616 00:36:22 --> 00:36:24 So, OK. 617 00:36:24 --> 00:36:29 I could try to come up with a better answer, or more focused 618 00:36:29 --> 00:36:42 answer than just to say power or, to use an electric power 619 00:36:42 --> 00:36:45 word is just one step. 620 00:36:45 --> 00:36:48 Yeah, thanks. 621 00:36:48 --> 00:36:48 Yes. 622 00:36:48 --> 00:36:51 AUDIENCE: [INAUDIBLE] 623 00:36:51 --> 00:36:53 PROFESSOR STRANG: 4.3, eight or nine. 624 00:36:53 --> 00:37:00 Let's just have a look and see what they're about. 625 00:37:00 --> 00:37:02 OK, just some regular guys. 626 00:37:02 --> 00:37:08 Yeah so that, OK, let me look at, you want 627 00:37:08 --> 00:37:12 me to do 4.3 eight? 628 00:37:12 --> 00:37:19 AUDIENCE: [INAUDIBLE] 629 00:37:19 --> 00:37:20 PROFESSOR STRANG: Ah, yes. 630 00:37:20 --> 00:37:22 Good, good question. 631 00:37:22 --> 00:37:26 OK, so 4.3 eight gave a couple of vectors, a little bit like 632 00:37:26 --> 00:37:28 the ones I did this morning. 633 00:37:28 --> 00:37:32 I mean here the c and 4.3 eight has a couple of ones, and this 634 00:37:32 --> 00:37:35 morning I just had a . 635 00:37:35 --> 00:37:37 But no big deal. 636 00:37:37 --> 00:37:39 Now, yeah. 637 00:37:39 --> 00:37:45 So if two vectors are orthogonal, are their 638 00:37:45 --> 00:37:49 transforms orthogonal? 639 00:37:49 --> 00:37:51 I think, yes. 640 00:37:51 --> 00:37:52 Yes. 641 00:37:52 --> 00:37:56 The Fourier, so the, yeah. 642 00:37:56 --> 00:38:00 So maybe this is worth a moment, here. 643 00:38:00 --> 00:38:01 Yeah. 644 00:38:01 --> 00:38:07 Because what do we, let me write down the letters for 645 00:38:07 --> 00:38:09 my question, and then answer the question. 646 00:38:09 --> 00:38:18 OK, so suppose I have two vectors, c and d. 647 00:38:18 --> 00:38:22 And they're orthogonal. 648 00:38:22 --> 00:38:27 And then I want to ask about their, if I multiply by the 649 00:38:27 --> 00:38:33 Fourier matrices, are those orthogonal? 650 00:38:33 --> 00:38:36 That's sort of the question. 651 00:38:36 --> 00:38:39 So here the vectors in frequency space, here they 652 00:38:39 --> 00:38:41 are in physical space. 653 00:38:41 --> 00:38:43 I don't mind if you started in physical space and 654 00:38:43 --> 00:38:47 went to frequency with F inverse, same question. 655 00:38:47 --> 00:38:53 Does the Fourier matrix preserve angles? 656 00:38:53 --> 00:38:55 Does it preserve angles? 657 00:38:55 --> 00:39:02 Do matrices, and F wouldn't be the only one with this 658 00:39:02 --> 00:39:05 property, do they preserve length. 659 00:39:05 --> 00:39:06 Right? 660 00:39:06 --> 00:39:09 If you preserve length, do you preserve angles too? 661 00:39:09 --> 00:39:13 Preserve length, you're just looking at one vector. 662 00:39:13 --> 00:39:17 We know that we preserve length, and how 663 00:39:17 --> 00:39:18 do we know that? 664 00:39:18 --> 00:39:20 Let's just remember that. 665 00:39:20 --> 00:39:22 So here's the length question. 666 00:39:22 --> 00:39:24 And this'll be the angle question. 667 00:39:24 --> 00:39:27 So this is the energy inequality, coming back 668 00:39:27 --> 00:39:28 to that key thing. 669 00:39:28 --> 00:39:37 This is c bar transpose c, and over here we have (Fc,Fc) and 670 00:39:37 --> 00:39:39 this is what I did this morning. 671 00:39:39 --> 00:39:46 So that's c bar transpose F bar transpose F c, right? 672 00:39:46 --> 00:39:51 That's why the bar as well as the transpose because 673 00:39:51 --> 00:39:53 we're doing complex. 674 00:39:53 --> 00:39:59 OK, let me make a little more space. 675 00:39:59 --> 00:40:01 And what did we do this morning? 676 00:40:01 --> 00:40:07 We replaced that by, well I wish it were the identity. 677 00:40:07 --> 00:40:11 But it's a multiple of the identity, that's what matters. 678 00:40:11 --> 00:40:16 So this was N, this is the identity with an 679 00:40:16 --> 00:40:19 N, c bar transpose c. 680 00:40:19 --> 00:40:24 So the conclusion was that this thing is just 681 00:40:24 --> 00:40:27 N times this thing. 682 00:40:27 --> 00:40:31 Well, and times this one. 683 00:40:31 --> 00:40:34 I'm expecting oh, but over here we got a zero. 684 00:40:34 --> 00:40:36 So my N is going to wash out. 685 00:40:36 --> 00:40:39 OK, let's just do the same thing for angle. 686 00:40:39 --> 00:40:46 This is the key energy equality for length. 687 00:40:46 --> 00:40:52 And all I want to say is, this Fourier matrix, like other 688 00:40:52 --> 00:40:57 orthogonal matrices, is just rotating the space. 689 00:40:57 --> 00:41:01 It's sort of amazing to think you have one space, physical 690 00:41:01 --> 00:41:05 space, and you rotate it. 691 00:41:05 --> 00:41:07 I'll use that word. 692 00:41:07 --> 00:41:10 Because somehow that's what an orthogonal matrix does. 693 00:41:10 --> 00:41:13 Well it's complex N-dimensional space. 694 00:41:13 --> 00:41:17 Sorry, so it's not so easy to visualize rotations in 695 00:41:17 --> 00:41:21 C^N, but it's a rotation. 696 00:41:21 --> 00:41:24 Angles don't change, and let's just see why? 697 00:41:24 --> 00:41:28 The inner product of this with this is c bar transpose, F 698 00:41:28 --> 00:41:31 bar transpose, because I have to transpose that. 699 00:41:31 --> 00:41:36 Times Fd, what do I do now? 700 00:41:36 --> 00:41:43 This one was c bar transpose d, with zero. 701 00:41:43 --> 00:41:45 You see you have it? 702 00:41:45 --> 00:41:50 Again, this fact that this rotation and inverse, and the 703 00:41:50 --> 00:41:55 transpose is the identity, apart from an N. 704 00:41:55 --> 00:41:59 So again, inner products are multiplied by N. 705 00:41:59 --> 00:42:04 Not only the length squared, which is inner product with 706 00:42:04 --> 00:42:08 itself, but all inner products are just multiplied by N. 707 00:42:08 --> 00:42:11 So if they start zero they end up zero. 708 00:42:11 --> 00:42:14 Yeah. 709 00:42:14 --> 00:42:15 That was your question, right? 710 00:42:15 --> 00:42:17 Yep, right. 711 00:42:17 --> 00:42:20 So if I have a couple of vectors, as I guess that 712 00:42:20 --> 00:42:24 problem proposed, it happened to propose two inputs that 713 00:42:24 --> 00:42:28 happen to be orthogonal, then you should be able to see that 714 00:42:28 --> 00:42:30 the transforms are, too. 715 00:42:30 --> 00:42:31 Yeah. 716 00:42:31 --> 00:42:33 Yep. 717 00:42:33 --> 00:42:37 Can I ask you about question two, was that on the homework? 718 00:42:37 --> 00:42:41 Problem two in Section 4.3? 719 00:42:41 --> 00:42:43 Was it? 720 00:42:43 --> 00:42:49 All I want to say is that's a simple fact. 721 00:42:49 --> 00:42:53 Let me just write that fact down so we're looking 722 00:42:53 --> 00:42:56 at that fact. 723 00:42:56 --> 00:43:02 If I look at this F matrix, it's so simple but it leads 724 00:43:02 --> 00:43:05 to lots of good, it's just a key fact. 725 00:43:05 --> 00:43:12 But if I look at my F matrix, am I looking at rows here? 726 00:43:12 --> 00:43:15 Yeah, I happened to look at rows, columns would be 727 00:43:15 --> 00:43:16 the same, it's symmetric. 728 00:43:16 --> 00:43:23 So here's row one, here's row two, I'm sorry that's row zero. 729 00:43:23 --> 00:43:26 This is row one. 730 00:43:26 --> 00:43:28 And let me look at row N-1. 731 00:43:29 --> 00:43:35 This is w, w^(N-1), this was w^2, this is w^2(N-1). 732 00:43:35 --> 00:43:38 733 00:43:38 --> 00:43:40 And so on. 734 00:43:40 --> 00:43:42 Everybody's got the idea of the, and then all 735 00:43:42 --> 00:43:45 these in between rows. 736 00:43:45 --> 00:43:48 Two, three, et cetera. 737 00:43:48 --> 00:43:55 And the question asks, show that this and this are 738 00:43:55 --> 00:43:57 complex conjugates. 739 00:43:57 --> 00:44:03 That that row and that row are the same, except conjugates. 740 00:44:03 --> 00:44:09 So if we look at the row number one in F, we'll be looking 741 00:44:09 --> 00:44:14 at row number N-1 in F bar. 742 00:44:14 --> 00:44:16 Now, why is that row the conjugate of that row? 743 00:44:16 --> 00:44:19 Why is this the conjugate of this? 744 00:44:19 --> 00:44:23 Why are those two conjugates? 745 00:44:23 --> 00:44:30 So I'm asking you to explain to me why w bar is the, why is 746 00:44:30 --> 00:44:40 the conjugate of this, this? 747 00:44:40 --> 00:44:45 So it's just one more neat fact about these important, 748 00:44:45 --> 00:44:48 crucial, numbers w. 749 00:44:48 --> 00:44:50 And so how do I see this? 750 00:44:50 --> 00:44:54 Well, this is w, this is one way to do it. 751 00:44:54 --> 00:44:56 This is w^N times w^-1. 752 00:44:57 --> 00:44:58 And what's w^N? 753 00:45:00 --> 00:45:01 One. 754 00:45:01 --> 00:45:04 Everybody knows, w^N is one. 755 00:45:04 --> 00:45:07 So I have w, so I'm trying to show that. 756 00:45:07 --> 00:45:12 Well, we know that this is true. 757 00:45:12 --> 00:45:18 So we know that the conjugate of w, right? 758 00:45:18 --> 00:45:20 There's w. 759 00:45:20 --> 00:45:25 Here's its conjugate, and it's also the inverse. 760 00:45:25 --> 00:45:30 This w is some e^(i*theta), this is some e to the 761 00:45:30 --> 00:45:32 the i some angle. 762 00:45:32 --> 00:45:35 Then always, it's sitting on the unit circle. 763 00:45:35 --> 00:45:40 So its reciprocal is also on the unit circle. 764 00:45:40 --> 00:45:45 The reciprocal has e^(-i*theta), so it's 765 00:45:45 --> 00:45:46 just the conjugate. 766 00:45:46 --> 00:45:50 That's a great fact. 767 00:45:50 --> 00:45:54 That's a beautiful fact about all these w's. 768 00:45:54 --> 00:45:55 And their powers. 769 00:45:55 --> 00:45:59 Conjugate and inverse the same. 770 00:45:59 --> 00:46:01 Right. 771 00:46:01 --> 00:46:04 So that was the key to problem two. 772 00:46:04 --> 00:46:09 I don't think I asked you to go through all the steps of 773 00:46:09 --> 00:46:13 problem three, but just in case you didn't read problem three, 774 00:46:13 --> 00:46:18 let me tell you in one tiny space what happens. 775 00:46:18 --> 00:46:21 In problem three you discover that the fourth power 776 00:46:21 --> 00:46:25 of F is the identity. 777 00:46:25 --> 00:46:34 Except there is an N squared, because from two F's we got an 778 00:46:34 --> 00:46:38 N, so from four F's, so that's another fantastic fact 779 00:46:38 --> 00:46:41 about the Fourier matrix. 780 00:46:41 --> 00:46:47 That its fourth power, it must be somehow, the Fourier 781 00:46:47 --> 00:46:50 matrix is rotating, yeah. 782 00:46:50 --> 00:46:54 Somehow, I don't know, how do you understand that the 783 00:46:54 --> 00:46:58 Fourier matrix is, its fourth power brings you back 784 00:46:58 --> 00:46:59 to the identity. 785 00:46:59 --> 00:47:03 Apart from, we just didn't normalize it. 786 00:47:03 --> 00:47:07 So that we wouldn't, to avoid that N squared, 787 00:47:07 --> 00:47:09 so we had to use it. 788 00:47:09 --> 00:47:10 Yeah. 789 00:47:10 --> 00:47:13 That's pretty amazing. 790 00:47:13 --> 00:47:15 Pretty amazing. 791 00:47:15 --> 00:47:21 So if I had normalized it right, if I'd took F over the 792 00:47:21 --> 00:47:26 square root of N, that's the exact normalization to give me 793 00:47:26 --> 00:47:32 a to put the N's where they belong, you could say. 794 00:47:32 --> 00:47:35 Then the fourth power of that matrix is the identity. 795 00:47:35 --> 00:47:37 Yeah. 796 00:47:37 --> 00:47:43 New math and new applications keep coming for these things. 797 00:47:43 --> 00:47:45 I'll tell you, actually. 798 00:47:45 --> 00:47:48 You could tell me eigenvalues of this matrix. 799 00:47:48 --> 00:47:50 Yeah, this is a good question. 800 00:47:50 --> 00:47:53 What are the eigenvalues of a matrix whose fourth 801 00:47:53 --> 00:47:57 power is the identity? 802 00:47:57 --> 00:47:58 AUDIENCE: [INAUDIBLE] 803 00:47:58 --> 00:47:59 PROFESSOR STRANG: Yeah, one. 804 00:47:59 --> 00:48:01 What else could the eigenvalue be? 805 00:48:01 --> 00:48:05 If the fourth power of the matrix is i, what are the 806 00:48:05 --> 00:48:07 possible eigenvalues? 807 00:48:07 --> 00:48:16 So let me, so this matrix is, can I call it M for the, or 808 00:48:16 --> 00:48:20 maybe u for the matrix F? 809 00:48:20 --> 00:48:32 So if u^4 is the identity, what are the eigenvalues? 810 00:48:32 --> 00:48:34 What could, well of course u could be the 811 00:48:34 --> 00:48:36 identity, but it's not. 812 00:48:36 --> 00:48:41 It's a Fourier matrix. 813 00:48:41 --> 00:48:46 So the eigenvalues could be one, what else could they be? 814 00:48:46 --> 00:48:48 Minus one is possible. 815 00:48:48 --> 00:48:55 Because minus the identity, that'd be fine. i, and minus i. 816 00:48:55 --> 00:48:57 Four possible eigenvalues. 817 00:48:57 --> 00:49:01 And when the matrix reaches, I think if you put the 818 00:49:01 --> 00:49:03 four by four Fourier 819 00:49:03 --> 00:49:06 - I don't know. 820 00:49:06 --> 00:49:10 If you, say, put the four by four Fourier matrix into 821 00:49:10 --> 00:49:14 MATLAB, and see what you get for eigenvalues, I've 822 00:49:14 --> 00:49:16 forgotten whether you get one of each of these. 823 00:49:16 --> 00:49:21 Or whether somebody's repeated at the level four, but then go 824 00:49:21 --> 00:49:25 up to five or six you'll see these guys start showing up. 825 00:49:25 --> 00:49:28 Different multiplicity. 826 00:49:28 --> 00:49:28 Right? 827 00:49:28 --> 00:49:34 The 1,024 matrix has got these guys. 828 00:49:34 --> 00:49:37 Some number of times, adding up to a 1,024. 829 00:49:37 --> 00:49:38 Right, yeah. 830 00:49:38 --> 00:49:41 So it's quite neat. 831 00:49:41 --> 00:49:45 Now, here's a question, which I actually just learned 832 00:49:45 --> 00:49:46 a good answer to. 833 00:49:46 --> 00:49:49 What are the eigenvectors? 834 00:49:49 --> 00:49:51 What are the eigenvectors? 835 00:49:51 --> 00:49:55 You could give a sort of half-baked description. 836 00:49:55 --> 00:49:57 Because once you know the eigenvalues. 837 00:49:57 --> 00:50:01 But to really get a handle on the eigenvectors, that's has 838 00:50:01 --> 00:50:08 been a problem that was studied in IEEE transactions papers. 839 00:50:08 --> 00:50:12 But not really the right, not a nice specific description 840 00:50:12 --> 00:50:13 of the eigenvectors. 841 00:50:13 --> 00:50:19 And somebody was in my office, this Fall, a guy, post-doc at 842 00:50:19 --> 00:50:25 Berkeley who's seen the right way to look at that problem and 843 00:50:25 --> 00:50:28 describe the eigenvectors of the Fourier matrix. 844 00:50:28 --> 00:50:32 So that's, like, amazing to me that such a fundamental 845 00:50:32 --> 00:50:34 question was waiting. 846 00:50:34 --> 00:50:40 And turned out to involve quite important, deep ideas. 847 00:50:40 --> 00:50:42 OK, ready for one more for today? 848 00:50:42 --> 00:50:45 Anything? 849 00:50:45 --> 00:50:48 Or not. 850 00:50:48 --> 00:50:52 OK, so I hope you're getting the good stuff now on 851 00:50:52 --> 00:50:54 the Fourier series. 852 00:50:54 --> 00:50:57 Fourier discrete transform. 853 00:50:57 --> 00:51:00 Please come on Friday, because Friday will be the big 854 00:51:00 --> 00:51:02 day for convolution. 855 00:51:02 --> 00:51:09 And that's the essential thing that we still have to do. 856 00:51:09 --> 00:51:11 OK, see you Friday.