1 00:00:10 --> 00:00:13 Are we ready? 2 00:00:13 --> 00:00:19 Okay, ready for me to start. 3 00:00:19 --> 00:00:28 Ready for the taping to start in a minute. 4 00:00:28 --> 00:00:39 He's going to raise his hand and signal when I'm on. 5 00:00:39 --> 00:00:50 Just a minute, though, let them settle.0 6 00:00:50 --> 00:00:51 Okay, guys. 7 00:00:51 --> 00:00:57 Okay, give me the signal, then, when you want me to 8 00:00:57 --> 00:00:57 start. 9 00:00:57 --> 00:01:02 Okay, this is linear algebra, lecture four. 10 00:01:02 --> 00:01:09 And, the first thing I have to do is something that was on the 11 00:01:09 --> 00:01:14 list for last time, but here it is now. 12 00:01:14 --> 00:01:19 What's the inverse of a product? 13 00:01:19 --> 00:01:24 If I multiply two matrices together and I know their 14 00:01:24 --> 00:01:29 inverses, how do I get the inverse of A times B? 15 00:01:29 --> 00:01:35 So I know what inverses mean for a single matrix A and for a 16 00:01:35 --> 00:01:36 matrix B. 17 00:01:36 --> 00:01:42 What matrix do I multiply by to get the identity if I have A B? 18 00:01:42 --> 00:01:47.19 Okay, that'll be simple but so basic. 19 00:01:47.19 --> 00:01:51.87 Then I'm going to use that to -- I will have a product of 20 00:01:51.87 --> 00:01:56 matrices and the product that we'll meet will be these 21 00:01:56 --> 00:02:01 elimination matrices and the net result of today's lectures is 22 00:02:01 --> 00:02:06 the big formula for elimination, so the net result of today's 23 00:02:06 --> 00:02:12 lecture is this great way to look at Gaussian elimination. 24 00:02:12 --> 00:02:16 We know that we get from A to U by elimination. 25 00:02:16 --> 00:02:23 We know the steps -- but now we get the right way to look at it, 26 00:02:23 --> 00:02:24 A equals L U. 27 00:02:24 --> 00:02:27 So that's the high point for today. 28 00:02:27 --> 00:02:28 Okay. 29 00:02:28 --> 00:02:34 Can I take the easy part, the first step first? 30 00:02:34 --> 00:02:39 So, suppose A is invertible -- and of course it's going to be a 31 00:02:39 --> 00:02:43 big question, when is the matrix invertible? 32 00:02:43 --> 00:02:48 But let's say A is invertible and B is invertible, 33 00:02:48 --> 00:02:52 then what matrix gives me the inverse of A B? 34 00:02:52 --> 00:02:56 So that's the direct question. 35 00:02:56 --> 00:02:58 What's the inverse of A B? 36 00:02:58 --> 00:03:01 Do I multiply those separate inverses? 37 00:03:01 --> 00:03:01 Yes. 38 00:03:01 --> 00:03:06 I multiply the two matrices A inverse and B inverse, 39 00:03:06 --> 00:03:08 but what order do I multiply? 40 00:03:08 --> 00:03:10 In reverse order. 41 00:03:10 --> 00:03:11 And you see why. 42 00:03:11 --> 00:03:16 So the right thing to put here is B inverse A inverse. 43 00:03:16 --> 00:03:19 That's the inverse I'm after. 44 00:03:19 --> 00:03:24 We can just check that A B times that matrix gives the 45 00:03:24 --> 00:03:25 identity. 46 00:03:25 --> 00:03:26 Okay. 47 00:03:26 --> 00:03:30 So why -- once again, it's this fact that I can move 48 00:03:30 --> 00:03:32 parentheses around. 49 00:03:32 --> 00:03:38 I can just erase them all and do the multiplications any way I 50 00:03:38 --> 00:03:40 want to. 51 00:03:40 --> 00:03:44.49 So what's the right multiplication to do first? 52 00:03:44.49 --> 00:03:46.1 B times B inverse. 53 00:03:46.1 --> 00:03:49 This product here I is the identity. 54 00:03:49 --> 00:03:54.51 Then A times the identity is the identity and then finally A 55 00:03:54.51 --> 00:03:57 times A inverse gives the identity. 56 00:03:57 --> 00:04:02 So forgive the dumb example in the book. 57 00:04:02 --> 00:04:08.01 Why do you, do the inverse things in reverse order? 58 00:04:08.01 --> 00:04:15 It's just like -- you take off your shoes, you take off your 59 00:04:15 --> 00:04:21 socks, then the good way to invert that process is socks 60 00:04:21 --> 00:04:24 back on first, then shoes. 61 00:04:24 --> 00:04:27 Sorry, okay. 62 00:04:27 --> 00:04:28 I'm sorry that's on the tape. 63 00:04:28 --> 00:04:31 And, of course, on the other side we should 64 00:04:31 --> 00:04:35 really just check -- on the other side I have B inverse, 65 00:04:35 --> 00:04:35 A inverse. 66 00:04:35 --> 00:04:38 That does multiply A B, and this time it's these guys 67 00:04:38 --> 00:04:41 that give the identity, squeeze down, 68 00:04:41 --> 00:04:44 they give the identity, we're in shape. 69 00:04:44 --> 00:04:44 Okay. 70 00:04:44 --> 00:04:47 So there's the inverse. 71 00:04:47 --> 00:04:47 Good. 72 00:04:47 --> 00:04:52 While we're at it, let me do a transpose, 73 00:04:52 --> 00:04:57 because the next lecture has got a lot to -- involves 74 00:04:57 --> 00:04:59 transposes. 75 00:04:59 --> 00:05:04 So how do I -- if I transpose a matrix, 76 00:05:04 --> 00:05:10 I'm talking about square, invertible matrices right now. 77 00:05:10 --> 00:05:14 If I transpose one, what's its inverse? 78 00:05:14 --> 00:05:18 Well, the nice formula is -- let's see. 79 00:05:18 --> 00:05:24 Let me start from A, A inverse equal the identity. 80 00:05:24 --> 00:05:28 And let me transpose both sides. 81 00:05:28 --> 00:05:32 That will bring a transpose into the picture. 82 00:05:32 --> 00:05:38 So if I transpose the identity matrix, what do I have? 83 00:05:38 --> 00:05:39 The identity, right? 84 00:05:39 --> 00:05:45.47 If I exchange rows and columns, the identity is a symmetric 85 00:05:45.47 --> 00:05:46 matrix. 86 00:05:46 --> 00:05:50 It doesn't know the difference. 87 00:05:50 --> 00:05:53 If I transpose these guys, that product, 88 00:05:53 --> 00:05:58 then again it turns out that I have to reverse the order. 89 00:05:58 --> 00:06:03 I can transpose them separately, but when I multiply, 90 00:06:03 --> 00:06:07 those transposes come in the opposite order. 91 00:06:07 --> 00:06:11 So it's A inverse transpose times A transpose giving the 92 00:06:11 --> 00:06:13.68 identity. 93 00:06:13.68 --> 00:06:18 So that's -- this equation is -- just comes directly from that 94 00:06:18 --> 00:06:18 one. 95 00:06:18 --> 00:06:22 But this equation tells me what I wanted to know, 96 00:06:22 --> 00:06:25 namely what is the inverse of this guy A transpose? 97 00:06:25 --> 00:06:29.73 What's the inverse of that -- if I transpose a matrix, 98 00:06:29.73 --> 00:06:33 what'ss the inverse of the result? 99 00:06:33 --> 00:06:38 And this equation tells me that here it is. 100 00:06:38 --> 00:06:42 This is the inverse of A transpose. 101 00:06:42 --> 00:06:45 Inverse of A transpose. 102 00:06:45 --> 00:06:47 Of A transpose. 103 00:06:47 --> 00:06:55 So I'll put a big circle around that, because that's the answer, 104 00:06:55 --> 00:07:01 that's the best answer we could hope for. 105 00:07:01 --> 00:07:07 That if you want to know the inverse of A transpose and you 106 00:07:07 --> 00:07:13 know the inverse of A, then you just transpose that. 107 00:07:13 --> 00:07:19 So in a -- to put it another way, transposing and inversing 108 00:07:19 --> 00:07:25 you can do in either order for a single matrix. 109 00:07:25 --> 00:07:25 Okay. 110 00:07:25 --> 00:07:30 So these are like basic facts that we can now use, 111 00:07:30 --> 00:07:33.54 all right -- so now I put it to use. 112 00:07:33.54 --> 00:07:38.47 I put it to use by thinking -- we're really completing, 113 00:07:38.47 --> 00:07:40 the subject of elimination. 114 00:07:40 --> 00:07:46.14 Actually, -- the thing about elimination is it's the right 115 00:07:46.14 --> 00:07:50 way to understand what the matrix has got. 116 00:07:50 --> 00:08:00 This A equal L U is the most basic factorization of a matrix. 117 00:08:00 --> 00:08:09 I always worry that you will think this course is all 118 00:08:09 --> 00:08:11 elimination. 119 00:08:11 --> 00:08:15 It's just row operations. 120 00:08:15 --> 00:08:19 And please don't. 121 00:08:19 --> 00:08:23 We'll be beyond that, but it's the right algebra to 122 00:08:23 --> 00:08:24 do first. 123 00:08:24 --> 00:08:24 Okay. 124 00:08:24 --> 00:08:30 So, now I'm coming near the end of it, but I want to get it in a 125 00:08:30 --> 00:08:31 decent form. 126 00:08:31 --> 00:08:34.6 So my decent form is matrix form. 127 00:08:34.6 --> 00:08:38 I have a matrix A, let's suppose it's a good 128 00:08:38 --> 00:08:44 matrix, I can do elimination, no row exchanges -- 129 00:08:44 --> 00:08:47 So no row exchanges for now. 130 00:08:47 --> 00:08:52 Pivots all fine, nothing zero in the pivot 131 00:08:52 --> 00:08:53.25 position. 132 00:08:53.25 --> 00:08:57 I get to the very end, which is U. 133 00:08:57 --> 00:08:59 So I get from A to U. 134 00:08:59 --> 00:09:05.35 And I want to know what's the connection? 135 00:09:05.35 --> 00:09:07 How is A related to U? 136 00:09:07 --> 00:09:13.64 And this is going to tell me that there's a matrix L that 137 00:09:13.64 --> 00:09:15 connects them. 138 00:09:15 --> 00:09:15.66 Okay. 139 00:09:15.66 --> 00:09:19 Can I do it for a two by two first? 140 00:09:19 --> 00:09:19 Okay. 141 00:09:19 --> 00:09:22 Two by two, elimination. 142 00:09:22 --> 00:09:25 Okay, so I'll do it under here. 143 00:09:25 --> 00:09:26 Okay. 144 00:09:26 --> 00:09:31 So let my matrix A be -- We'll keep it simple, 145 00:09:31 --> 00:09:35 say two and an eight, so we know that the first pivot 146 00:09:35 --> 00:09:39 is a two, and the multiplier's going to be a four and then let 147 00:09:39 --> 00:09:43 me put a one here and what number do I not want to put 148 00:09:43 --> 00:09:44 there? 149 00:09:44 --> 00:09:44 Four. 150 00:09:44 --> 00:09:48.14 I don't want a four there, because in that case, 151 00:09:48.14 --> 00:09:54 the second pivot would not -- we wouldn't have a second 152 00:09:54 --> 00:09:54.89 pivot. 153 00:09:54.89 --> 00:10:00 The matrix would be singular, general screw-up. 154 00:10:00 --> 00:10:01 Okay. 155 00:10:01 --> 00:10:06 So let me put some other number here like seven. 156 00:10:06 --> 00:10:07 Okay. 157 00:10:07 --> 00:10:07 Okay. 158 00:10:07 --> 00:10:15 Now I want to operate on that with my elementary matrix. 159 00:10:15 --> 00:10:18 So what's the elementary matrix? 160 00:10:18 --> 00:10:22 Strictly speaking, it's E21, because it's the guy 161 00:10:22 --> 00:10:25 that's going to produce a zero in that position. 162 00:10:25 --> 00:10:30 And it's going to produce U in one shot, because it's just a 163 00:10:30 --> 00:10:32 two by two matrix. 164 00:10:32 --> 00:10:37.46 So two one and I'm going to take four of those away from 165 00:10:37.46 --> 00:10:41.39 those, produce that zero and leave a three there. 166 00:10:41.39 --> 00:10:42 And that's U. 167 00:10:42 --> 00:10:45 And what's the matrix that did it? 168 00:10:45 --> 00:10:46 Quick review, then. 169 00:10:46 --> 00:10:50 What's the elimination elementary matrix E21 -- it's 170 00:10:50 --> 00:10:52 one zero, thanks. 171 00:10:52 --> 00:10:55 And -- negative four one, 172 00:10:55 --> 00:10:55 right. 173 00:10:55 --> 00:10:56 Good. 174 00:10:56 --> 00:10:56 Okay. 175 00:10:56 --> 00:11:01.71 So that -- you see the difference between this and what 176 00:11:01.71 --> 00:11:03 I'm shooting for. 177 00:11:03 --> 00:11:08 I'm shooting for A on one side and the other matrices on the 178 00:11:08 --> 00:11:11 other side of the equation. 179 00:11:11 --> 00:11:12 Okay. 180 00:11:12 --> 00:11:15 So I can do that right away. 181 00:11:15 --> 00:11:19 Now here's going to be my A equals L U. 182 00:11:19 --> 00:11:26 And you won't have any trouble telling me what -- so A is still 183 00:11:26 --> 00:11:28 two one eight seven. 184 00:11:28 --> 00:11:34 L is what you're going to tell me and U is still two one zero 185 00:11:34 --> 00:11:35 three. 186 00:11:35 --> 00:11:35 Okay. 187 00:11:35 --> 00:11:39.7 So what's L in this case? 188 00:11:39.7 --> 00:11:43 Well, first -- so how is L related to this E guy? 189 00:11:43 --> 00:11:47 It's the inverse, because I want to multiply 190 00:11:47 --> 00:11:52 through by the inverse of this, which will put the identity 191 00:11:52 --> 00:11:58 here, and the inverse will show up there and I'll call it L. 192 00:11:58 --> 00:12:01 So what is the inverse of this? 193 00:12:01 --> 00:12:10 Remember those elimination matrices are easy to invert. 194 00:12:10 --> 00:12:20 The inverse matrix for this one is 1 0 4 1, it's actually flip 195 00:12:20 --> 00:12:22 sign. 196 00:12:22 --> 00:12:22 Okay. 197 00:12:22 --> 00:12:25.93 Do you want -- if we did the numbers right, 198 00:12:25.93 --> 00:12:28 we must -- this should be correct. 199 00:12:28 --> 00:12:28 Okay. 200 00:12:28 --> 00:12:30 And of course it is. 201 00:12:30 --> 00:12:34 That says the first row's right, four times the first row 202 00:12:34 --> 00:12:37 plus the second row is eight seven. 203 00:12:37 --> 00:12:38.04 Good. 204 00:12:38.04 --> 00:12:38 Okay. 205 00:12:38 --> 00:12:41 That's simple, two by two. 206 00:12:41 --> 00:12:45 But it already shows the form that we're headed for. 207 00:12:45 --> 00:12:48.45 It shows -- so what's the L stand for? 208 00:12:48.45 --> 00:12:49 Why the letter L? 209 00:12:49 --> 00:12:53 If U stood for upper triangular, then of course L 210 00:12:53 --> 00:12:55 stands for lower triangular. 211 00:12:55 --> 00:12:58 And actually, it has ones on the diagonal, 212 00:12:58 --> 00:13:03 where this thing has the pivots on the diagonal. 213 00:13:03 --> 00:13:08 Oh, sometimes we may want to separate out the pivots, 214 00:13:08 --> 00:13:14 so can I just mention that sometimes we could also write 215 00:13:14 --> 00:13:20 this as -- we could have this one zero four one -- I'll just 216 00:13:20 --> 00:13:27 show you how I would divide out this matrix of pivots -- two 217 00:13:27 --> 00:13:28 three. 218 00:13:28 --> 00:13:31 There's a diagonal matrix. 219 00:13:31 --> 00:13:34 And I just -- whatever is left is here. 220 00:13:34 --> 00:13:36 Now what's left? 221 00:13:36 --> 00:13:40 If I divide this first row by two to pull out the two, 222 00:13:40 --> 00:13:43 then I have a one and a one half. 223 00:13:43 --> 00:13:49 And if I divide the second row by three to pull out the three, 224 00:13:49 --> 00:13:52 then I have a one. 225 00:13:52 --> 00:13:57 So if this is L U, this is maybe called L D or 226 00:13:57 --> 00:13:58.05 pivot U. 227 00:13:58.05 --> 00:14:04 And now it's a little more balanced, because we have ones 228 00:14:04 --> 00:14:07 on the diagonal here and here. 229 00:14:07 --> 00:14:12.09 And the diagonal matrix in the middle. 230 00:14:12.09 --> 00:14:14 So both of those... 231 00:14:14 --> 00:14:19 Matlab would produce either one. 232 00:14:19 --> 00:14:21 I'll basically stay with L U. 233 00:14:21 --> 00:14:22 Okay. 234 00:14:22 --> 00:14:26 Now I have to think about bigger than two by two. 235 00:14:26 --> 00:14:30 But right now, this was just like easy 236 00:14:30 --> 00:14:31 exercise. 237 00:14:31 --> 00:14:36 And, to tell the truth, this one was a minus sign and 238 00:14:36 --> 00:14:38 this one was a plus sign. 239 00:14:38 --> 00:14:42 I mean, that's the only difference. 240 00:14:42 --> 00:14:49 But, with three by three, there's a more significant 241 00:14:49 --> 00:14:50 difference. 242 00:14:50 --> 00:14:54 Let me show you how that works. 243 00:14:54 --> 00:15:02 Let me move up to a three by three, let's say some matrix A, 244 00:15:02 --> 00:15:03.03 okay? 245 00:15:03.03 --> 00:15:08 Let's imagine it's three by three. 246 00:15:08 --> 00:15:11 I won't write numbers down for now. 247 00:15:11 --> 00:15:15 So what's the first elimination step that I do, 248 00:15:15 --> 00:15:21 the first matrix I multiply it by, what letter will I use for 249 00:15:21 --> 00:15:21 that? 250 00:15:21 --> 00:15:25 It'll be E two one, because it's -- 251 00:15:25 --> 00:15:30 the first step will be to get a zero in that two one position. 252 00:15:30 --> 00:15:34.99 And then the next step will be to get a zero in the three one 253 00:15:34.99 --> 00:15:35 position. 254 00:15:35 --> 00:15:40.17 And the final step will be to get a zero in the three two 255 00:15:40.17 --> 00:15:40 position. 256 00:15:40 --> 00:15:45 That's what elimination is, and it produced U. 257 00:15:45 --> 00:15:48.04 And again, no row exchanges. 258 00:15:48.04 --> 00:15:52.69 I'm taking the nice case, now, the typical case, 259 00:15:52.69 --> 00:15:57 too -- when I don't have to do any row exchange, 260 00:15:57 --> 00:16:00 all I do is these elimination steps. 261 00:16:00 --> 00:16:01 Okay. 262 00:16:01 --> 00:16:07 Now, suppose I want that stuff over on the right-hand side, 263 00:16:07 --> 00:16:09.61 as I really do. 264 00:16:09.61 --> 00:16:12 That's, like, my point here. 265 00:16:12 --> 00:16:16.7 I can multiply these together to get a matrix E, 266 00:16:16.7 --> 00:16:19 but I want it over on the right. 267 00:16:19 --> 00:16:22 I want its inverse over there. 268 00:16:22 --> 00:16:26 So what's the right expression now? 269 00:16:26 --> 00:16:30.38 If I write A and U, what goes there? 270 00:16:30.38 --> 00:16:30 Okay. 271 00:16:30 --> 00:16:37 So I've got the inverse of this, I've got three matrices in 272 00:16:37 --> 00:16:38 a row now. 273 00:16:38 --> 00:16:43 And it's their inverses that are going to show up, 274 00:16:43 --> 00:16:48 because each one is easy to invert. 275 00:16:48 --> 00:16:50 Question is, what about the whole bunch? 276 00:16:50 --> 00:16:53 How easy is it to invert the whole bunch? 277 00:16:53 --> 00:16:56 So, that's what we know how to do. 278 00:16:56 --> 00:17:00 We know how to invert, we should take the separate 279 00:17:00 --> 00:17:03 inverses, but they go in the opposite order. 280 00:17:03 --> 00:17:05.5 So what goes here? 281 00:17:05.5 --> 00:17:09 E three two inverse, right, because I'll multiply 282 00:17:09 --> 00:17:14 from the left by E three two inverse, then I'll pop it up 283 00:17:14 --> 00:17:15.7 next to U. 284 00:17:15.7 --> 00:17:19.1 And then will come E three one inverse. 285 00:17:19.1 --> 00:17:24.11 And then this'll be the only guy left standing and that's 286 00:17:24.11 --> 00:17:28 gone when Ido an E two one inverse. 287 00:17:28 --> 00:17:29 So there is L. 288 00:17:29 --> 00:17:31 That's L U. 289 00:17:31 --> 00:17:34 L is product of inverses. 290 00:17:34 --> 00:17:41 Now you still can ask why is this guy preferring inverses? 291 00:17:41 --> 00:17:43 And let me explain why. 292 00:17:43 --> 00:17:50 Let me explain why is this product nicer than this one? 293 00:17:50 --> 00:17:57.38 This product turns out to be better than this one. 294 00:17:57.38 --> 00:18:00 Let me take a typical case here. 295 00:18:00 --> 00:18:03 Let me take a typical case. 296 00:18:03 --> 00:18:10 So let me -- I have to do three by three for you to see the 297 00:18:10 --> 00:18:11.89 improvement. 298 00:18:11.89 --> 00:18:17 Two by two, it was just one E, no problem. 299 00:18:17 --> 00:18:21 But let me go up to this case. 300 00:18:21 --> 00:18:29 Suppose my matrices E21 -- suppose E21 has a minus two in 301 00:18:29 --> 00:18:30 there. 302 00:18:30 --> 00:18:39 Suppose that -- and now suppose -- oh, I'll even suppose E31 is 303 00:18:39 --> 00:18:41 the identity. 304 00:18:41 --> 00:18:46 I'm going to make the point with just a couple of these. 305 00:18:46 --> 00:18:47 Okay. 306 00:18:47 --> 00:18:52 Now this guy will have -- do something -- now let's suppose 307 00:18:52 --> 00:18:53 minus five one. 308 00:18:53 --> 00:18:53 Okay. 309 00:18:53 --> 00:18:55 There's typical. 310 00:18:55 --> 00:18:59.77 That's a typical case in which we didn't need an E31. 311 00:18:59.77 --> 00:19:04 Maybe we already had a zero in that three one position. 312 00:19:04 --> 00:19:06 Okay. 313 00:19:06 --> 00:19:09 Let me see -- is that going to be enough to, 314 00:19:09 --> 00:19:10 show my point? 315 00:19:10 --> 00:19:13.18 Let me do that multiplication. 316 00:19:13.18 --> 00:19:13 Okay. 317 00:19:13 --> 00:19:18 So if I do that multiplication it's like good practice to 318 00:19:18 --> 00:19:20 multiply these matrices. 319 00:19:20 --> 00:19:24 Tell me what's above the diagonal when I do this 320 00:19:24 --> 00:19:26.22 multiplication? 321 00:19:26.22 --> 00:19:27 All zeroes. 322 00:19:27 --> 00:19:33 When I do this multiplication, I'm going to get ones on the 323 00:19:33 --> 00:19:35 diagonal and zeroes above. 324 00:19:35 --> 00:19:38 Because -- what does that say? 325 00:19:38 --> 00:19:44 That says that I'm subtracting rows from lower rows. 326 00:19:44 --> 00:19:48 So nothing is moving upwards as it did last time in 327 00:19:48 --> 00:19:49 Gauss-Jordan. 328 00:19:49 --> 00:19:50 Okay. 329 00:19:50 --> 00:19:54 Now -- so really, what I have to do is check this 330 00:19:54 --> 00:19:57 minus two one zero, now this is -- what's that 331 00:19:57 --> 00:19:58.49 number? 332 00:19:58.49 --> 00:20:03 This is the number that I'm really have in mind. 333 00:20:03 --> 00:20:05 That number is ten. 334 00:20:05 --> 00:20:09 And this one is -- what goes here? 335 00:20:09 --> 00:20:16 Row three against column two, it looks like the minus five. 336 00:20:16 --> 00:20:17 It's that ten. 337 00:20:17 --> 00:20:21 How did that ten get in there? 338 00:20:21 --> 00:20:23 I don't like that ten. 339 00:20:23 --> 00:20:28 I mean -- of course, I don't want to erase it, 340 00:20:28 --> 00:20:32 because it's right. 341 00:20:32 --> 00:20:34 But I don't want it there. 342 00:20:34 --> 00:20:40 It's because -- the ten got in there because I subtracted two 343 00:20:40 --> 00:20:46 of row one from row two, and then I subtracted five of 344 00:20:46 --> 00:20:49.56 that new row two from row three. 345 00:20:49.56 --> 00:20:54 So doing it in that order, how did row one effect row 346 00:20:54 --> 00:20:56 three? 347 00:20:56 --> 00:20:59 Well, it did, because two of it got removed 348 00:20:59 --> 00:21:04 from row two and then five of those got removed from row 349 00:21:04 --> 00:21:04.77 three. 350 00:21:04.77 --> 00:21:09 So altogether ten of row one got thrown into row three. 351 00:21:09 --> 00:21:14 Now my point is in the reverse direction -- so now I can do it 352 00:21:14 --> 00:21:17 -- below it I'll do the inverses. 353 00:21:17 --> 00:21:18 Okay. 354 00:21:18 --> 00:21:21 And, of course, opposite order. 355 00:21:21 --> 00:21:22 Reverse order. 356 00:21:22 --> 00:21:24 Reverse order. 357 00:21:24 --> 00:21:24 Okay. 358 00:21:24 --> 00:21:30 So now this is going to -- this is the E that goes on the left 359 00:21:30 --> 00:21:31 side. 360 00:21:31 --> 00:21:32 Left of A. 361 00:21:32 --> 00:21:38 Now I'm going to do the inverses in the opposite order, 362 00:21:38 --> 00:21:43 so what's the -- So the opposite order means I 363 00:21:43 --> 00:21:45 put this inverse first. 364 00:21:45 --> 00:21:47 And what is its inverse? 365 00:21:47 --> 00:21:50 What's the inverse of E21? 366 00:21:50 --> 00:21:53.17 Same thing with a plus sign, right? 367 00:21:53.17 --> 00:21:58 For the individual matrices, instead of taking away two I 368 00:21:58 --> 00:22:03 add back two of row one to row two, so no problem. 369 00:22:03 --> 00:22:09 And now, in reverse order, I want to invert that. 370 00:22:09 --> 00:22:11 Just right? 371 00:22:11 --> 00:22:14 I'm doing just this, this. 372 00:22:14 --> 00:22:23 So now the inverse is again the same thing, but add in the five. 373 00:22:23 --> 00:22:29 And now I'll do that multiplication and I'll get a 374 00:22:29 --> 00:22:31 happy result. 375 00:22:31 --> 00:22:32 I hope. 376 00:22:32 --> 00:22:37 Did I do it right so far? 377 00:22:37 --> 00:22:38 Yes, okay. 378 00:22:38 --> 00:22:40 Let me do the multiplication. 379 00:22:40 --> 00:22:42 I believe this comes out. 380 00:22:42 --> 00:22:46.53 So row one of the answer is one zero zero. 381 00:22:46.53 --> 00:22:50 Oh, I know that all this is going to be left, 382 00:22:50 --> 00:22:50 right? 383 00:22:50 --> 00:22:53 Then I have two one zero. 384 00:22:53 --> 00:22:57 So I get two one zero there, right? 385 00:22:57 --> 00:23:02 And what's the third row? 386 00:23:02 --> 00:23:10 What's the third row in this product? 387 00:23:10 --> 00:23:19 Just read it out to me, the third row? 388 00:23:19 --> 00:23:22 And this is my matrix L. 389 00:23:22 --> 00:23:28 And it's the one that goes on the left of U. 390 00:23:28 --> 00:23:32 It goes into -- what do I mean here? 391 00:23:32 --> 00:23:39 Maybe rather than saying left of A, left of U, 392 00:23:39 --> 00:23:43 let me right down again what I mean. 393 00:23:43 --> 00:23:48 E A is U, whereas A is L U. 394 00:23:48 --> 00:23:49 Okay. 395 00:23:49 --> 00:23:53 Let me make the point now in words. 396 00:23:53 --> 00:24:01 The order that the matrices come for L is the right order. 397 00:24:01 --> 00:24:09 The two and the five don't sort of interfere to produce this ten 398 00:24:09 --> 00:24:16 in the right order, the multipliers just sit in the 399 00:24:16 --> 00:24:18 matrix L. 400 00:24:18 --> 00:24:24 That's the point -- that if I want to know L, 401 00:24:24 --> 00:24:27 I have no work to do. 402 00:24:27 --> 00:24:34 I just keep a record of what those multipliers were, 403 00:24:34 --> 00:24:37 and that gives me L. 404 00:24:37 --> 00:24:42 So I'll draw the -- let me say it. 405 00:24:42 --> 00:24:45 So this is the A=L U. 406 00:24:45 --> 00:24:53 So if no row exchanges, the multipliers that those 407 00:24:53 --> 00:25:01 numbers that we multiplied rows by and subtracted, 408 00:25:01 --> 00:25:11 when we did an elimination step -- the multipliers go directly 409 00:25:11 --> 00:25:13 into L. 410 00:25:13 --> 00:25:14.65 Okay. 411 00:25:14.65 --> 00:25:22 So L is -- this is the way, to look at elimination. 412 00:25:22 --> 00:25:31 You go through the elimination steps, and actually if you do it 413 00:25:31 --> 00:25:39 right, you can throw away A as you create L U. 414 00:25:39 --> 00:25:46 If you think about it, those steps of elimination, 415 00:25:46 --> 00:25:51.22 as when you've finished with row two of A, 416 00:25:51.22 --> 00:25:58 you've created a new row two of U, which you have to save, 417 00:25:58 --> 00:26:05 and you've created the multipliers that you used -- 418 00:26:05 --> 00:26:11 which you have to save, and then you can forget A. 419 00:26:11 --> 00:26:15 So because it's all there in L and U. 420 00:26:15 --> 00:26:21 So that's -- this moment is maybe the new insight in 421 00:26:21 --> 00:26:27 elimination that comes from matrix -- 422 00:26:27 --> 00:26:29 doing it in matrix form. 423 00:26:29 --> 00:26:36.42 So it was -- the product of Es is -- we can't see what that 424 00:26:36.42 --> 00:26:38.32 product of Es is. 425 00:26:38.32 --> 00:26:43 The matrix E is not a particularly attractive one. 426 00:26:43 --> 00:26:50.5 What's great is when we put them on the other side -- 427 00:26:50.5 --> 00:26:56 their inverses in the opposite order, there the L comes out 428 00:26:56 --> 00:26:58.04 just right. 429 00:26:58.04 --> 00:26:58 Okay. 430 00:26:58 --> 00:27:02.52 Now -- oh gosh, so today's a sort of, 431 00:27:02.52 --> 00:27:04 like, practical day. 432 00:27:04 --> 00:27:10.17 Can we think together how expensive is elimination? 433 00:27:10.17 --> 00:27:14 How many operations do we do? 434 00:27:14 --> 00:27:25 So this is now a kind of new topic which I didn't list as -- 435 00:27:25 --> 00:27:31 on the program, but here it came. 436 00:27:31 --> 00:27:39 How many operations on an n by n matrix A. 437 00:27:39 --> 00:27:47 I mean, it's a very practical question. 438 00:27:47 --> 00:27:54 Can we solve systems of order a thousand, in a second or a 439 00:27:54 --> 00:27:56 minute or a week? 440 00:27:56 --> 00:28:03 Can we solve systems of order a million in a second or an hour 441 00:28:03 --> 00:28:04 or a week? 442 00:28:04 --> 00:28:10 I mean, what's the -- if it's n by n, we often want to take n 443 00:28:10 --> 00:28:12 bigger. 444 00:28:12 --> 00:28:16 I mean, we've put in more information. 445 00:28:16 --> 00:28:21 We make the whole thing is more accurate for the bigger matrix. 446 00:28:21 --> 00:28:26.82 But it's more expensive, too, and the question is how 447 00:28:26.82 --> 00:28:29 much more expensive? 448 00:28:29 --> 00:28:32 If I have matrices of order a hundred. 449 00:28:32 --> 00:28:35 Let's say a hundred by a hundred. 450 00:28:35 --> 00:28:38 Let me take n to be a hundred. 451 00:28:38 --> 00:28:40 Say n equal a hundred. 452 00:28:40 --> 00:28:42 How many steps are we doing? 453 00:28:42 --> 00:28:48 How many operations are we actually doing that we -- 454 00:28:48 --> 00:28:54 And let's suppose there aren't any zeroes, because of course if 455 00:28:54 --> 00:28:59.1 a matrix has got a lot of zeroes in good places, 456 00:28:59.1 --> 00:29:03 we don't have to do those operations, and, 457 00:29:03 --> 00:29:05 it'll be much faster. 458 00:29:05 --> 00:29:11 But -- so just think for a moment about the first step. 459 00:29:11 --> 00:29:17 So here's our matrix A, hundred by a hundred. 460 00:29:17 --> 00:29:25 And the first step will be -- that column, is got zeroes down 461 00:29:25 --> 00:29:25 here. 462 00:29:25 --> 00:29:29 So it's down to 99 by 99, right? 463 00:29:29 --> 00:29:37 That's really like the first stage of elimination, 464 00:29:37 --> 00:29:41 to get from this hundred by hundred non-zero matrix to this 465 00:29:41 --> 00:29:46 stage where the first pivot is sitting up here and the first 466 00:29:46 --> 00:29:49 row's okay the first column is okay. 467 00:29:49 --> 00:29:53 So, eventually -- how many steps did that take? 468 00:29:53 --> 00:29:57 You see, I'm trying to get an idea. 469 00:29:57 --> 00:29:59 Is the answer proportional to n? 470 00:29:59 --> 00:30:03 Is the total number of steps in elimination, the total number, 471 00:30:03 --> 00:30:07 is it proportional to n -- in which case if I double n from a 472 00:30:07 --> 00:30:11 hundred to two hundred -- does it take me twice as long? 473 00:30:11 --> 00:30:14 Does it square, so it would take me four times 474 00:30:14 --> 00:30:16 as long? 475 00:30:16 --> 00:30:20 Does it cube so it would take me eight times as long? 476 00:30:20 --> 00:30:24 Or is it n factorial, so it would take me a hundred 477 00:30:24 --> 00:30:25 times as long? 478 00:30:25 --> 00:30:29 I think, you know, from a practical point of view, 479 00:30:29 --> 00:30:33 we have to have some idea of the cost, here. 480 00:30:33 --> 00:30:38 So these are the questions that I'm -- let me ask those 481 00:30:38 --> 00:30:40 questions again. 482 00:30:40 --> 00:30:45 Is it proportional -- does it go like N, like N squared, 483 00:30:45 --> 00:30:50 like N cubed -- or some higher power of N? 484 00:30:50 --> 00:30:55 Like N factorial where every step up multiplies by a hundred 485 00:30:55 --> 00:31:00 and then by a hundred and one and then by a hundred and two -- 486 00:31:00 --> 00:31:01 which is it? 487 00:31:01 --> 00:31:06 Okay, so that's the only way I know to answer that is to think 488 00:31:06 --> 00:31:08 through what we actually had to do. 489 00:31:08 --> 00:31:09 Okay. 490 00:31:09 --> 00:31:11 So what was the cost here? 491 00:31:11 --> 00:31:13 Well, let's see. 492 00:31:13 --> 00:31:16 What do I mean by an operation? 493 00:31:16 --> 00:31:20 I guess I mean, well an addition or -- yeah. 494 00:31:20 --> 00:31:21 No big deal. 495 00:31:21 --> 00:31:26 I guess I mean an addition or a subtraction or a multiplication 496 00:31:26 --> 00:31:27 or a division. 497 00:31:27 --> 00:31:28 Okay. 498 00:31:28 --> 00:31:32 And actually, what operation I doing all the 499 00:31:32 --> 00:31:33 time? 500 00:31:33 --> 00:31:40 When I multiply row one by multiplier L and I subtract from 501 00:31:40 --> 00:31:41.68 row six. 502 00:31:41.68 --> 00:31:45 What's happening there individually? 503 00:31:45 --> 00:31:47 What's going on? 504 00:31:47 --> 00:31:54 If I multiply -- I do a multiplication by L and then a 505 00:31:54 --> 00:31:56 subtraction. 506 00:31:56 --> 00:32:02 So I guess operation -- Can I count that for the moment as, 507 00:32:02 --> 00:32:03 like, one operation? 508 00:32:03 --> 00:32:07 Or you may want to count them separately. 509 00:32:07 --> 00:32:11 The typical operation is multiply plus a subtract. 510 00:32:11 --> 00:32:16 So if I count those together, my answer's going to come out 511 00:32:16 --> 00:32:21 half as many as if -- I mean, if I count them 512 00:32:21 --> 00:32:27 separately, I'd have a certain number of multiplies, 513 00:32:27 --> 00:32:29 certain number of subtracts. 514 00:32:29 --> 00:32:32.44 That's really want to do. 515 00:32:32.44 --> 00:32:32 Okay. 516 00:32:32 --> 00:32:35 How many have I got here? 517 00:32:35 --> 00:32:38 So, I think -- let's see. 518 00:32:38 --> 00:32:42.07 It's about -- well, how many, roughly? 519 00:32:42.07 --> 00:32:45 How many operations to get from here to here? 520 00:32:45 --> 00:32:51 Well, maybe one way to look at it is all these numbers had to 521 00:32:51 --> 00:32:52 get changed. 522 00:32:52 --> 00:32:57 The first row didn't get changed, but all the other rows 523 00:32:57 --> 00:33:00 got changed at this step. 524 00:33:00 --> 00:33:06 So this step -- well, I guess maybe -- shall I say it 525 00:33:06 --> 00:33:09 cost about a hundred squared. 526 00:33:09 --> 00:33:16 I mean, if I had changed the first row, then it would have 527 00:33:16 --> 00:33:21 been exactly hundred squared, because -- 528 00:33:21 --> 00:33:24 because that's how many numbers are here. 529 00:33:24 --> 00:33:29 A hundred squared numbers is the total count of the entry, 530 00:33:29 --> 00:33:33 and all but this insignificant first row got changed. 531 00:33:33 --> 00:33:36 So I would say about a hundred squared. 532 00:33:36 --> 00:33:37 Okay. 533 00:33:37 --> 00:33:40 Now, what about the next step? 534 00:33:40 --> 00:33:43 So now the first row is fine. 535 00:33:43 --> 00:33:46 The second row is fine. 536 00:33:46 --> 00:33:51 And I'm changing these zeroes are all fine, 537 00:33:51 --> 00:33:55.83 so what's up with the second step? 538 00:33:55.83 --> 00:33:58 And then you're with me. 539 00:33:58 --> 00:34:02 Roughly, what's the cost? 540 00:34:02 --> 00:34:06 If this first step cost a hundred squared, 541 00:34:06 --> 00:34:12 about, operations then this one, which is really working on 542 00:34:12 --> 00:34:16 this guy to produce this, costs about what? 543 00:34:16 --> 00:34:19 How many operations to fix? 544 00:34:19 --> 00:34:23 About ninety-nine squared, or ninety-nine times 545 00:34:23 --> 00:34:25 ninety-eight. 546 00:34:25 --> 00:34:26 But less, right? 547 00:34:26 --> 00:34:29 Less, because our problem's getting smaller. 548 00:34:29 --> 00:34:30 About ninety-nine squared. 549 00:34:30 --> 00:34:33 And then I go down and down and the next one will be 550 00:34:33 --> 00:34:36 ninety-eight squared, the next ninety-seven squared 551 00:34:36 --> 00:34:40 and finally I'm down around one squared or -- 552 00:34:40 --> 00:34:43 where it's just like the little numbers. 553 00:34:43 --> 00:34:45 The big numbers are here. 554 00:34:45 --> 00:34:50.31 So the number of operations is about n squared plus that was n, 555 00:34:50.31 --> 00:34:52 right? n was a hundred? 556 00:34:52 --> 00:34:56 n squared for the first step, then n minus one squared, 557 00:34:56 --> 00:35:00 then n minus two squared, finally down to three squared 558 00:35:00 --> 00:35:04 and two squared and even one squared. 559 00:35:04 --> 00:35:13 No way I should have written that -- squeezed that in. 560 00:35:13 --> 00:35:22 Let me try it so the count is N squared plus N minus one squared 561 00:35:22 --> 00:35:28.87 plus -- all the way down to one squared. 562 00:35:28.87 --> 00:35:34 That's a pretty decent count. 563 00:35:34 --> 00:35:41.29 Admittedly, we didn't catch every single tiny operation, 564 00:35:41.29 --> 00:35:46 but we got the right leading term here. 565 00:35:46 --> 00:35:49.52 And what do those add up to? 566 00:35:49.52 --> 00:35:55 Okay, so now we're coming to the punch of this, 567 00:35:55 --> 00:36:00 question, this operation count. 568 00:36:00 --> 00:36:06 So the operations on the left side, on the matrix A to finally 569 00:36:06 --> 00:36:07 get to U. 570 00:36:07 --> 00:36:12 And anybody -- so which of these quantities is the right 571 00:36:12 --> 00:36:15 ballpark for that count? 572 00:36:15 --> 00:36:20.98 If I add a hundred squared to ninety nine squared to ninety 573 00:36:20.98 --> 00:36:25 eight squared -- ninety seven squared, 574 00:36:25 --> 00:36:30 all the way down to two squared then one squared, 575 00:36:30 --> 00:36:33 what have I got, about? 576 00:36:33 --> 00:36:38 It's just one of these -- let's identify it first. 577 00:36:38 --> 00:36:39 Is it N? 578 00:36:39 --> 00:36:42 Certainly not. 579 00:36:42 --> 00:36:43 Is it n factorial? 580 00:36:43 --> 00:36:44.11 No. 581 00:36:44.11 --> 00:36:49 If it was n factorial, we would -- with determinants, 582 00:36:49 --> 00:36:51 it is n factorial. 583 00:36:51 --> 00:36:55 I'll put in a bad mark against determinants, 584 00:36:55 --> 00:36:58 because that -- okay, so what is it? 585 00:36:58 --> 00:37:03.51 It's n -- well, this is the answer. 586 00:37:03.51 --> 00:37:06 It's this order -- n cubed. 587 00:37:06 --> 00:37:10 It's like I have n terms, right? 588 00:37:10 --> 00:37:13 I've got n terms in this sum. 589 00:37:13 --> 00:37:16 And the biggest one is N squared. 590 00:37:16 --> 00:37:23 So the worst it could be would be n cubed, but it's not as bad 591 00:37:23 --> 00:37:30 as -- it's N cubed times -- it's about one third of n 592 00:37:30 --> 00:37:31.11 cubed. 593 00:37:31.11 --> 00:37:34 That's the magic operation count. 594 00:37:34 --> 00:37:40 Somehow that one third takes account of the fact that the 595 00:37:40 --> 00:37:43 numbers are getting smaller. 596 00:37:43 --> 00:37:48 If they weren't getting smaller, we would have n terms 597 00:37:48 --> 00:37:53 times n squared, but it would be exactly n 598 00:37:53 --> 00:37:54 cubed. 599 00:37:54 --> 00:37:58 But our numbers are getting smaller -- actually, 600 00:37:58 --> 00:38:02 do you remember where does one third come in this -- I'll even 601 00:38:02 --> 00:38:05 allow a mention of calculus. 602 00:38:05 --> 00:38:09 So calculus can be mentioned, integration can be mentioned 603 00:38:09 --> 00:38:13 now in the next minute and not again for weeks. 604 00:38:13 --> 00:38:21 It's not that I don't like 18.01, but18.06 is better. 605 00:38:21 --> 00:38:21 Okay. 606 00:38:21 --> 00:38:30 So, -- so what's -- what's the calculus formula that looks 607 00:38:30 --> 00:38:32 like? 608 00:38:32 --> 00:38:37.31 It looks like -- if we were in calculus instead of summing 609 00:38:37.31 --> 00:38:39 stuff, we would integrate. 610 00:38:39 --> 00:38:45 So I would integrate x squared and I would get one third x 611 00:38:45 --> 00:38:45 cubed. 612 00:38:45 --> 00:38:49 So if that was like an integral from one to N, 613 00:38:49 --> 00:38:53 of x squared b x, if the answer would be one 614 00:38:53 --> 00:38:58 third n cubed -- and it's correct for the sum 615 00:38:58 --> 00:39:01 also, because that's, like, the whole point of 616 00:39:01 --> 00:39:02 calculus. 617 00:39:02 --> 00:39:06 The whole point of calculus is -- oh, I don't want to tell you 618 00:39:06 --> 00:39:10 the whole -- I mean, you know the whole point of 619 00:39:10 --> 00:39:11 calculus. 620 00:39:11 --> 00:39:15 Calculus is like sums except it's continuous. 621 00:39:15 --> 00:39:15 Okay. 622 00:39:15 --> 00:39:17 And algebra is discreet. 623 00:39:17 --> 00:39:18 Okay. 624 00:39:18 --> 00:39:21 So the answer is one third n cubed. 625 00:39:21 --> 00:39:27 Now I'll just -- let me say one more thing about operations. 626 00:39:27 --> 00:39:30 What about the right-hand side? 627 00:39:30 --> 00:39:34 This was what it cost on the left side. 628 00:39:34 --> 00:39:35 This is on A. 629 00:39:35 --> 00:39:40 Because this is A that we're working with. 630 00:39:40 --> 00:39:46 But what's the cost on the extra column vector b that we're 631 00:39:46 --> 00:39:48 hanging around here? 632 00:39:48 --> 00:39:53 So b costs a lot less, obviously, because it's just 633 00:39:53 --> 00:39:55 one column. 634 00:39:55 --> 00:40:01 We carry it through elimination and then actually we do back 635 00:40:01 --> 00:40:03 substitution. 636 00:40:03 --> 00:40:07 Let me just tell you the answer there. 637 00:40:07 --> 00:40:08 It's n squared. 638 00:40:08 --> 00:40:14 So the cost for every right hand side is n squared. 639 00:40:14 --> 00:40:19.69 So let me -- I'll just fit that in here -- 640 00:40:19.69 --> 00:40:24 for the the cost of b turns out to be n squared. 641 00:40:24 --> 00:40:27 So you see if we have, as we often have, 642 00:40:27 --> 00:40:31 a a matrix A and several right-hand sides, 643 00:40:31 --> 00:40:37.42 then we pay the price on A, the higher price on A to get it 644 00:40:37.42 --> 00:40:41 split up into L and U to do elimination on A, 645 00:40:41 --> 00:40:48.25 but then we can process every right-hand side at low cost. 646 00:40:48.25 --> 00:40:49 Okay. 647 00:40:49 --> 00:41:00 So the -- We really have discussed the most fundamental 648 00:41:00 --> 00:41:08 algorithm for a system of equations. 649 00:41:08 --> 00:41:09 Okay. 650 00:41:09 --> 00:41:18 So, I'm ready to allow row exchanges. 651 00:41:18 --> 00:41:23.47 I'm ready to allow -- now what happens to this whole -- today's 652 00:41:23.47 --> 00:41:26 lecture if there are row exchanges? 653 00:41:26 --> 00:41:28.53 When would there be row exchanges? 654 00:41:28.53 --> 00:41:33 There are row -- we need to do row exchanges if a zero shows up 655 00:41:33 --> 00:41:34 in the pivot position. 656 00:41:34 --> 00:41:38 So moving then into the final section of this chapter, 657 00:41:38 --> 00:41:46 which is about transposes -- well, we've already seen some 658 00:41:46 --> 00:41:53 transposes, and -- the title of this section is, 659 00:41:53 --> 00:41:58 "Transposes and Permutations." 660 00:41:58 --> 00:41:58 Okay. 661 00:41:58 --> 00:42:05 So can I say, now, where does a permutation 662 00:42:05 --> 00:42:07 come in? 663 00:42:07 --> 00:42:11 Let me talk a little about permutations. 664 00:42:11 --> 00:42:15 So that'll be up here, permutations. 665 00:42:15 --> 00:42:21 So these are the matrices that I need to do row exchanges. 666 00:42:21 --> 00:42:26 And I may have to do two row exchanges. 667 00:42:26 --> 00:42:32 Can you invent a matrix where I would have to do two row 668 00:42:32 --> 00:42:37 exchanges and then would come out fine? 669 00:42:37 --> 00:42:44 Yeah let's just, for the heck of it -- so I'll 670 00:42:44 --> 00:42:45 put it here. 671 00:42:45 --> 00:42:49 Let me do three by threes. 672 00:42:49 --> 00:42:56 Actually, why don't I just plain list all the three by 673 00:42:56 --> 00:43:01 three permutation matrices. 674 00:43:01 --> 00:43:05 There're a nice little group of them. 675 00:43:05 --> 00:43:10 What are all the matrices that exchange no rows at all? 676 00:43:10 --> 00:43:14 Well, I'll include the identity. 677 00:43:14 --> 00:43:19 So that's a permutation matrix that doesn't do anything. 678 00:43:19 --> 00:43:25 Now what's the permutation matrix that exchanges -- what is 679 00:43:25 --> 00:43:27 P12? 680 00:43:27 --> 00:43:33 The permutation matrix that exchanges rows one and two would 681 00:43:33 --> 00:43:36 be -- 0 1 0 -- 1 0 0, right. 682 00:43:36 --> 00:43:42 I just exchanged those rows of the identity and I've got it. 683 00:43:42 --> 00:43:42 Okay. 684 00:43:42 --> 00:43:45 Actually, I'll -- yes. 685 00:43:45 --> 00:43:48 Let me clutter this up. 686 00:43:48 --> 00:43:48 Okay. 687 00:43:48 --> 00:43:53 Give me a complete list of all the row exchange matrices. 688 00:43:53 --> 00:43:55 So what are they? 689 00:43:55 --> 00:44:00 They're all the ways I can take the identity matrix and 690 00:44:00 --> 00:44:01 rearrange its rows. 691 00:44:01 --> 00:44:04 How many will there be? 692 00:44:04 --> 00:44:08 How many three by three permutation matrices? 693 00:44:08 --> 00:44:13 Shall we keep going and get the answer? 694 00:44:13 --> 00:44:16 So tell me some more. 695 00:44:16 --> 00:44:20 What one are you going to do now? 696 00:44:20 --> 00:44:26 I'm going to switch rows one and -- 697 00:44:26 --> 00:44:28 One and three, okay. 698 00:44:28 --> 00:44:31.31 One and three, leaving two alone. 699 00:44:31.31 --> 00:44:31 Okay. 700 00:44:31 --> 00:44:33 Now what else? 701 00:44:33 --> 00:44:38 Switch -- what would be the next easy one -- is switch two 702 00:44:38 --> 00:44:40.53 and three, good. 703 00:44:40.53 --> 00:44:46 So I'll leave one zero zero alone and I'll switch -- 704 00:44:46 --> 00:44:50 I'll move number three up and number two down. 705 00:44:50 --> 00:44:50 Okay. 706 00:44:50 --> 00:44:55 Those are the ones that just exchange single -- a pair of 707 00:44:55 --> 00:44:55 rows. 708 00:44:55 --> 00:45:00 This guy, this guy and this guy exchanges a pair of rows, 709 00:45:00 --> 00:45:04.25 but now there are more possibilities. 710 00:45:04.25 --> 00:41:52 What's left? 711 00:41:52 --> 00:32:15 So tell -- there is another one here. 712 00:32:15 --> 00:29:03 What's that? 713 00:29:03 --> 00:15:26 It's going to move -- it's going to change all rows, 714 00:15:26 --> 00:13:50 right? 715 00:13:50 --> 00:05:18 Where shall we put them?4 716 00:05:18 --> 00:09:01.4 So -- give me a first row. 717 00:09:01.4 --> 00:12:19 STUDENT: Zero one zero? 718 00:12:19 --> 00:15:28 STRANG: Zero one zero. 719 00:15:28 --> 00:24:03 Okay, now a second row -- say zero zero one and the third guy 720 00:24:03 --> 00:26:03 one zero zero. 721 00:26:03 --> 00:29:30 So that is like a cycle. 722 00:29:30 --> 00:37:57 That puts row two moves up to row one, row three moves up to 723 00:37:57 --> 00:45:41 row two and row one moves down to row three. 724 00:45:41 --> 00:45:45 And there's one more, which is -- let's see. 725 00:45:45 --> 00:45:47 What's left? 726 00:45:47 --> 00:45:48 I'm lost. 727 00:45:48 --> 00:45:51 STUDENT: Is it zero zero one? 728 00:45:51 --> 00:45:52 STRANG: Okay. 729 00:45:52 --> 00:45:55 STUDENT: One zero zero. 730 00:45:55 --> 00:45:59 STRANG: One zero zero, okay. 731 00:45:59 --> 00:46:00.95 Zero one zero. 732 00:46:00.95 --> 00:46:01 Okay. 733 00:46:01 --> 00:46:02 Great. 734 00:46:02 --> 00:46:02 Six. 735 00:46:02 --> 00:46:04 Six of them. 736 00:46:04 --> 00:46:04 Six P. 737 00:46:04 --> 00:46:11 And they're sort of nice, because what happens if I 738 00:46:11 --> 00:46:15 write, multiply two of them together? 739 00:46:15 --> 00:46:20 If I multiply two of these matrices together, 740 00:46:20 --> 00:46:27 what can you tell me about the answer? 741 00:46:27 --> 00:46:28 It's on the list. 742 00:46:28 --> 00:46:34 If I do some row exchanges and then I do some more row 743 00:46:34 --> 00:46:39.94 exchanges, then all together I've done row exchanges. 744 00:46:39.94 --> 00:46:44.57 So if I multiply -- but, I don't know. 745 00:46:44.57 --> 00:46:47 And if I invert, then I'm just doing row 746 00:46:47 --> 00:46:50 exchanges to get back again. 747 00:46:50 --> 00:46:52.9 So the inverses are all there. 748 00:46:52.9 --> 00:46:57 It's a little family of matrices that -- they've got 749 00:46:57 --> 00:47:02 their own -- if I multiply, I'm still inside this group. 750 00:47:02 --> 00:47:05 If I invert I'm inside this group -- 751 00:47:05 --> 00:47:09 actually, group is the right name for this subject. 752 00:47:09 --> 00:47:14 It's a group of six matrices, and what about the inverses? 753 00:47:14 --> 00:47:17 What's the inverse of this guy, for example? 754 00:47:17 --> 00:47:21 What's the inverse -- if I exchange rows one and two, 755 00:47:21 --> 00:47:24 what's the inverse matrix? 756 00:47:24 --> 00:47:26 Just tell me fast. 757 00:47:26 --> 00:47:31 The inverse of that matrix is -- if I exchange rows one and 758 00:47:31 --> 00:47:37 two, then what I should do to get back to where I started is 759 00:47:37 --> 00:47:38 the same thing. 760 00:47:38 --> 00:47:41 So this thing is its own inverse. 761 00:47:41 --> 00:47:44 That's probably its own inverse. 762 00:47:44 --> 00:47:50 This is probably not -- actually, I think these are 763 00:47:50 --> 00:47:52 inverses of each other. 764 00:47:52 --> 00:47:56 Oh, yeah, actually -- the inverse is the transpose. 765 00:47:56 --> 00:48:01 There's a curious fact about permutations matrices, 766 00:48:01 --> 00:48:04 that the inverses are the transposes. 767 00:48:04 --> 00:48:09 And final moment -- how many are there if I -- 768 00:48:09 --> 00:48:12 how many four by four permutations? 769 00:48:12 --> 00:48:16.68 So let me take four by four -- how many Ps? 770 00:48:16.68 --> 00:48:17 Well, okay. 771 00:48:17 --> 00:48:19 Make a good guess. 772 00:48:19 --> 00:48:21 Twenty four, right. 773 00:48:21 --> 00:48:22 Twenty four Ps. 774 00:48:22 --> 00:48:22 Okay. 775 00:48:22 --> 00:48:28 So, we've got these permutation matrices, and in the next 776 00:48:28 --> 00:48:31 lecture, we'll use them.