1 00:00:00,000 --> 00:00:11,240 -- two, one and -- okay. 2 00:00:11,240 --> 00:00:18,910 Here is a lecture on the applications of eigenvalues 3 00:00:18,910 --> 00:00:24,070 and, if I can -- so that will be Markov matrices. 4 00:00:24,070 --> 00:00:27,260 I'll tell you what a Markov matrix is, 5 00:00:27,260 --> 00:00:32,070 so this matrix A will be a Markov matrix 6 00:00:32,070 --> 00:00:38,890 and I'll explain how they come in applications. 7 00:00:38,890 --> 00:00:42,060 And -- and then if I have time, I would like to say a little 8 00:00:42,060 --> 00:00:46,060 bit about Fourier series, which is a fantastic application 9 00:00:46,060 --> 00:00:48,540 of the projection chapter. 10 00:00:48,540 --> 00:00:49,100 Okay. 11 00:00:49,100 --> 00:00:50,930 What's a Markov matrix? 12 00:00:50,930 --> 00:01:01,940 Can I just write down a typical Markov matrix, say .1, .2, .7, 13 00:01:01,940 --> 00:01:08,820 .01, .99 0, let's say, .3, .3, .4. 14 00:01:08,820 --> 00:01:09,970 Okay. 15 00:01:09,970 --> 00:01:13,922 There's a -- a totally just invented Markov matrix. 16 00:01:16,640 --> 00:01:19,540 What makes it a Markov matrix? 17 00:01:19,540 --> 00:01:22,510 Two properties that this -- this matrix has. 18 00:01:22,510 --> 00:01:24,970 So two properties are -- 19 00:01:24,970 --> 00:01:30,430 one, every entry is greater equal zero. 20 00:01:30,430 --> 00:01:38,720 All entries greater than or equal to zero. 21 00:01:38,720 --> 00:01:43,100 And, of course, when I square the matrix, 22 00:01:43,100 --> 00:01:45,660 the entries will still be greater/equal zero. 23 00:01:45,660 --> 00:01:50,150 I'm going to be interested in the powers of this matrix. 24 00:01:50,150 --> 00:01:54,250 And this property, of course, is going to -- stay there. 25 00:01:54,250 --> 00:01:58,870 It -- really Markov matrices you'll see are connected 26 00:01:58,870 --> 00:02:04,230 to probability ideas and probabilities are never 27 00:02:04,230 --> 00:02:05,120 negative. 28 00:02:05,120 --> 00:02:09,669 The other property -- do you see the other property in there? 29 00:02:09,669 --> 00:02:13,590 If I add down the columns, what answer do I get? 30 00:02:13,590 --> 00:02:14,490 One. 31 00:02:14,490 --> 00:02:17,640 So all columns add to one. 32 00:02:17,640 --> 00:02:25,380 All columns add to one. 33 00:02:28,900 --> 00:02:31,580 And actually when I square the matrix, 34 00:02:31,580 --> 00:02:33,820 that will be true again. 35 00:02:33,820 --> 00:02:38,920 So that the powers of my matrix are all 36 00:02:38,920 --> 00:02:42,330 Markov matrices, and I'm interested 37 00:02:42,330 --> 00:02:46,520 in, always, the eigenvalues and the eigenvectors. 38 00:02:46,520 --> 00:02:50,780 And this question of steady state will come up. 39 00:02:50,780 --> 00:02:53,860 You remember we had steady state for differential equations 40 00:02:53,860 --> 00:02:54,940 last time? 41 00:02:54,940 --> 00:02:57,080 When -- what was the steady state -- 42 00:02:57,080 --> 00:02:59,100 what was the eigenvalue? 43 00:02:59,100 --> 00:03:03,240 What was the eigenvalue in the differential equation case 44 00:03:03,240 --> 00:03:05,620 that led to a steady state? 45 00:03:05,620 --> 00:03:08,360 It was lambda equals zero. 46 00:03:08,360 --> 00:03:11,210 When -- you remember that we did an example and one 47 00:03:11,210 --> 00:03:15,180 of the eigenvalues was lambda equals zero, and that -- 48 00:03:15,180 --> 00:03:20,140 so then we had an E to the zero T, a constant one -- 49 00:03:20,140 --> 00:03:24,430 as time went on, there that thing stayed steady. 50 00:03:24,430 --> 00:03:30,760 Now what -- in the powers case, it's not a zero eigenvalue. 51 00:03:30,760 --> 00:03:33,740 Actually with powers of a matrix, a zero eigenvalue, 52 00:03:33,740 --> 00:03:36,570 that part is going to die right away. 53 00:03:36,570 --> 00:03:41,710 It's an eigenvalue of one that's all important. 54 00:03:41,710 --> 00:03:44,540 So this steady state will correspond -- 55 00:03:44,540 --> 00:03:49,000 will be totally connected with an eigenvalue of one and its 56 00:03:49,000 --> 00:03:49,910 eigenvector. 57 00:03:49,910 --> 00:03:53,510 In fact, the steady state will be the eigenvector 58 00:03:53,510 --> 00:03:55,260 for that eigenvalue. 59 00:03:55,260 --> 00:03:56,670 Okay. 60 00:03:56,670 --> 00:03:58,580 So that's what's coming. 61 00:03:58,580 --> 00:04:04,390 Now, for some reason then that we have to see, 62 00:04:04,390 --> 00:04:08,210 this matrix has an eigenvalue of one. 63 00:04:08,210 --> 00:04:12,040 This property, that the columns all add to one -- 64 00:04:12,040 --> 00:04:16,680 turns out -- guarantees that one is an eigenvalue, 65 00:04:16,680 --> 00:04:19,600 so that you can actually find the eigenvalue -- 66 00:04:19,600 --> 00:04:24,330 find that eigenvalue of a Markov matrix without computing any 67 00:04:24,330 --> 00:04:27,640 determinants of A minus lambda I -- 68 00:04:27,640 --> 00:04:30,310 that matrix will have an eigenvalue of one, 69 00:04:30,310 --> 00:04:32,270 and we want to see why. 70 00:04:32,270 --> 00:04:35,850 And then the other thing is -- 71 00:04:35,850 --> 00:04:39,190 so the key points -- let me -- let me write these underneath. 72 00:04:39,190 --> 00:04:45,837 The key points are -- the key points are lambda equal one is 73 00:04:45,837 --> 00:04:46,420 an eigenvalue. 74 00:04:53,760 --> 00:04:56,410 I'll add in a little -- an additional -- well, 75 00:04:56,410 --> 00:04:59,030 a thing about eigenvalues -- 76 00:04:59,030 --> 00:05:02,400 key point two, the other eigenval- values -- 77 00:05:02,400 --> 00:05:12,340 all other eigenvalues are, in magnitude, smaller than one -- 78 00:05:12,340 --> 00:05:14,630 in absolute value, smaller than one. 79 00:05:14,630 --> 00:05:18,390 Well, there could be some exceptional case when -- 80 00:05:18,390 --> 00:05:22,380 when an eigen -- another eigenvalue might have magnitude 81 00:05:22,380 --> 00:05:23,300 equal one. 82 00:05:23,300 --> 00:05:26,690 It never has an eigenvalue larger than one. 83 00:05:26,690 --> 00:05:29,230 So these two facts -- somehow we ought to -- 84 00:05:29,230 --> 00:05:32,600 linear algebra ought to tell us. 85 00:05:32,600 --> 00:05:35,440 And then, of course, linear algebra is going to tell us 86 00:05:35,440 --> 00:05:40,040 what the -- what's -- what happens if I take -- if -- 87 00:05:40,040 --> 00:05:42,720 you remember when I solve -- 88 00:05:42,720 --> 00:05:47,980 when I multiply by A time after time the K-th thing is A 89 00:05:47,980 --> 00:05:54,940 to the K u0 and I'm asking what's special about this -- 90 00:05:54,940 --> 00:06:01,240 these powers of A, and very likely the quiz will have 91 00:06:01,240 --> 00:06:06,440 a problem to computer s- to computer some powers of A or -- 92 00:06:06,440 --> 00:06:08,570 or applied to an initial vector. 93 00:06:08,570 --> 00:06:11,040 So, you remember the general form? 94 00:06:11,040 --> 00:06:14,180 The general form is that there's some amount 95 00:06:14,180 --> 00:06:17,660 of the first eigenvalue to the K-th power 96 00:06:17,660 --> 00:06:22,790 times the first eigenvector, and another amount 97 00:06:22,790 --> 00:06:25,340 of the second eigenvalue to the K-th power 98 00:06:25,340 --> 00:06:27,430 times the second eigenvector and so on. 99 00:06:27,430 --> 00:06:30,700 A -- just -- 100 00:06:30,700 --> 00:06:34,380 my conscience always makes me say at least once per lecture 101 00:06:34,380 --> 00:06:40,060 that this requires a complete set of eigenvectors, 102 00:06:40,060 --> 00:06:43,080 otherwise we might not be able to expand u0 103 00:06:43,080 --> 00:06:45,840 in the eigenvectors and we couldn't get started. 104 00:06:45,840 --> 00:06:49,410 But once we're started with u0 when K is zero, 105 00:06:49,410 --> 00:06:53,880 then every A brings in these lambdas. 106 00:06:53,880 --> 00:06:56,830 And now you can see what the steady state is going to be. 107 00:06:56,830 --> 00:07:01,630 If lambda one is one -- so lambda one equals one 108 00:07:01,630 --> 00:07:07,900 to the K-th power and these other eigenvalues are smaller 109 00:07:07,900 --> 00:07:10,280 than one -- 110 00:07:10,280 --> 00:07:14,560 so I've sort of scratched over the equation there to -- 111 00:07:14,560 --> 00:07:18,260 we had this term, but what happens to this term -- 112 00:07:18,260 --> 00:07:21,180 if the lambda's smaller than one, then the -- when -- 113 00:07:21,180 --> 00:07:25,160 as we take powers, as we iterate as we -- 114 00:07:25,160 --> 00:07:30,662 as we go forward in time, this goes to zero, Can I just -- 115 00:07:30,662 --> 00:07:32,370 having scratched over it, I might as well 116 00:07:32,370 --> 00:07:34,050 scratch right? further. 117 00:07:34,050 --> 00:07:36,880 That term and all the other terms are going to zero 118 00:07:36,880 --> 00:07:39,750 because all the other eigenvalues are smaller than 119 00:07:39,750 --> 00:07:45,440 one and the steady state that we're approaching is just -- 120 00:07:45,440 --> 00:07:48,780 whatever there was -- this was -- this was the -- 121 00:07:48,780 --> 00:07:56,990 this is the x1 part of un- of the initial condition u0 -- 122 00:07:56,990 --> 00:07:59,150 is the steady state. 123 00:08:05,120 --> 00:08:08,350 This much we know from general -- from -- you know, 124 00:08:08,350 --> 00:08:11,390 what we've already done. 125 00:08:11,390 --> 00:08:15,740 So I want to see why -- let's at least see number one, 126 00:08:15,740 --> 00:08:18,160 why one is an eigenvalue. 127 00:08:18,160 --> 00:08:20,580 And then there's actually -- 128 00:08:20,580 --> 00:08:23,370 in this chapter we're interested not only in eigenvalues, 129 00:08:23,370 --> 00:08:25,390 but also eigenvectors. 130 00:08:25,390 --> 00:08:28,760 And there's something special about the eigenvector. 131 00:08:28,760 --> 00:08:30,800 Let me write down what that is. 132 00:08:30,800 --> 00:08:36,700 The eigenvector x1 -- 133 00:08:36,700 --> 00:08:41,909 x1 is the eigenvector and all its components are positive, 134 00:08:41,909 --> 00:08:49,100 so the steady state is positive, if the start was. 135 00:08:49,100 --> 00:08:51,480 If the start was -- so -- 136 00:08:51,480 --> 00:08:54,780 well, actually, in general, I -- 137 00:08:54,780 --> 00:09:00,710 this might have a -- might have some component zero always, 138 00:09:00,710 --> 00:09:03,880 but no negative components in that eigenvector. 139 00:09:03,880 --> 00:09:04,410 Okay. 140 00:09:04,410 --> 00:09:08,470 Can I come to that point? 141 00:09:08,470 --> 00:09:11,370 How can I look at that matrix -- 142 00:09:11,370 --> 00:09:15,280 so that was just an example. 143 00:09:15,280 --> 00:09:21,000 How could I be sure -- how can I see that a matrix -- 144 00:09:21,000 --> 00:09:24,750 if the columns add to zero -- add to one, sorry -- 145 00:09:24,750 --> 00:09:30,930 if the columns add to one, this property means that lambda 146 00:09:30,930 --> 00:09:32,780 equal one is an eigenvalue. 147 00:09:32,780 --> 00:09:33,540 Okay. 148 00:09:33,540 --> 00:09:37,080 So let's just think that through. 149 00:09:37,080 --> 00:09:41,760 What I saying about -- let me ca- let me look at A, 150 00:09:41,760 --> 00:09:44,940 and if I believe that one is an eigenvalue, 151 00:09:44,940 --> 00:09:47,810 then I should be able to subtract off one times 152 00:09:47,810 --> 00:09:56,150 the identity and then I would get a matrix that's, what, -.9, 153 00:09:56,150 --> 00:09:59,160 -.01 and -.6 -- 154 00:09:59,160 --> 00:10:02,700 wh- I took the ones away and the other parts, of course, 155 00:10:02,700 --> 00:10:11,430 are still what they were, and this is still .2 and .7 and -- 156 00:10:11,430 --> 00:10:14,200 okay, what's -- 157 00:10:14,200 --> 00:10:15,700 what's up with this matrix now? 158 00:10:15,700 --> 00:10:19,380 I've shifted the matrix, this Markov matrix by one, 159 00:10:19,380 --> 00:10:24,350 by the identity, and what do I want to prove? 160 00:10:24,350 --> 00:10:28,940 I -- what is it that I believe this matrix -- 161 00:10:28,940 --> 00:10:30,760 about this matrix? 162 00:10:30,760 --> 00:10:33,190 I believe it's singular. 163 00:10:33,190 --> 00:10:36,580 Singular will -- if A minus I is singular, 164 00:10:36,580 --> 00:10:40,020 that tells me that one is an eigenvalue, right? 165 00:10:40,020 --> 00:10:43,420 The eigenvalues are the numbers that I subtract off -- 166 00:10:43,420 --> 00:10:45,010 the shifts -- 167 00:10:45,010 --> 00:10:47,880 the numbers that I subtract from the diagonal -- 168 00:10:47,880 --> 00:10:48,810 to make it singular. 169 00:10:48,810 --> 00:10:50,610 Now why is that matrix singular? 170 00:10:50,610 --> 00:10:55,570 I -- we could compute its determinant, 171 00:10:55,570 --> 00:11:00,000 but we want to see a reason that would work for every Markov 172 00:11:00,000 --> 00:11:05,550 matrix not just this particular random example. 173 00:11:05,550 --> 00:11:07,780 So what is it about that matrix? 174 00:11:07,780 --> 00:11:11,040 Well, I guess you could look at its columns now -- 175 00:11:11,040 --> 00:11:12,890 what do they add up to? 176 00:11:12,890 --> 00:11:13,390 Zero. 177 00:11:13,390 --> 00:11:21,660 The columns add to zero, so all columns -- 178 00:11:21,660 --> 00:11:25,240 let me put all columns now of -- of -- 179 00:11:25,240 --> 00:11:32,450 of A minus I add to zero, and then I 180 00:11:32,450 --> 00:11:40,305 want to realize that this means A minus I is singular. 181 00:11:45,400 --> 00:11:47,790 Okay. 182 00:11:47,790 --> 00:11:49,560 Why? 183 00:11:49,560 --> 00:11:52,210 So I could I -- you know, that could be a quiz question, 184 00:11:52,210 --> 00:11:54,280 a sort of theoretical quiz question. 185 00:11:54,280 --> 00:11:56,070 If I give you a matrix and I tell you 186 00:11:56,070 --> 00:12:00,740 all its columns add to zero, give me a reason, 187 00:12:00,740 --> 00:12:06,270 because it is true, that the matrix is singular. 188 00:12:06,270 --> 00:12:06,940 Okay. 189 00:12:06,940 --> 00:12:09,290 I guess actually -- now what -- 190 00:12:09,290 --> 00:12:11,680 I think of -- you know, I'm thinking of two or three ways 191 00:12:11,680 --> 00:12:12,480 to see that. 192 00:12:17,370 --> 00:12:18,410 How would you do it? 193 00:12:18,410 --> 00:12:21,880 We don't want to take its determinant somehow. 194 00:12:21,880 --> 00:12:24,520 For the matrix to be singular, well, 195 00:12:24,520 --> 00:12:31,180 it means that these three columns are dependent, right? 196 00:12:31,180 --> 00:12:33,530 The determinant will be zero when those three columns 197 00:12:33,530 --> 00:12:34,260 are dependent. 198 00:12:34,260 --> 00:12:36,650 You see, we're -- we're at a point in this course, now, 199 00:12:36,650 --> 00:12:40,100 where we have several ways to look at an idea. 200 00:12:40,100 --> 00:12:42,390 We can take the determinant -- here we don't want to. 201 00:12:42,390 --> 00:12:45,810 B- but we met singular before that -- 202 00:12:45,810 --> 00:12:47,830 those columns are dependent. 203 00:12:47,830 --> 00:12:50,670 So how do I see that those columns are dependent? 204 00:12:50,670 --> 00:12:51,925 They all add to zero. 205 00:12:57,020 --> 00:13:00,440 Let's see, whew -- 206 00:13:00,440 --> 00:13:05,070 well, oh, actually, what -- 207 00:13:05,070 --> 00:13:07,010 another thing I know is that the -- 208 00:13:07,010 --> 00:13:11,410 I would like to be able to show is that the rows are dependent. 209 00:13:11,410 --> 00:13:14,250 Maybe that's easier. 210 00:13:14,250 --> 00:13:15,940 If I know that all the columns add 211 00:13:15,940 --> 00:13:18,480 to zero, that's my information, how 212 00:13:18,480 --> 00:13:25,120 do I see that those three rows are linearly dependent? 213 00:13:25,120 --> 00:13:30,210 What -- what combination of those rows gives the zero row? 214 00:13:30,210 --> 00:13:32,920 How -- how could I combine those three rows -- 215 00:13:32,920 --> 00:13:38,930 those three row vectors to produce the zero row vector? 216 00:13:38,930 --> 00:13:42,360 And that would tell me those rows are dependent, 217 00:13:42,360 --> 00:13:43,910 therefore the columns are dependent, 218 00:13:43,910 --> 00:13:46,940 the matrix is singular, the determinant is zero -- well, 219 00:13:46,940 --> 00:13:47,890 you see it. 220 00:13:47,890 --> 00:13:49,010 I just add the rows. 221 00:13:49,010 --> 00:13:52,610 One times that row plus one times that row plus one times 222 00:13:52,610 --> 00:13:53,770 that row -- 223 00:13:53,770 --> 00:13:56,990 it's the zero row. 224 00:13:56,990 --> 00:13:58,960 The rows are dependent. 225 00:13:58,960 --> 00:14:03,470 In a way, that one one one, because it's multiplying 226 00:14:03,470 --> 00:14:07,810 the rows, is like an eigenvector in the -- 227 00:14:07,810 --> 00:14:10,700 it's in the left null space, right? 228 00:14:10,700 --> 00:14:12,870 One one one is in the left null space. 229 00:14:12,870 --> 00:14:23,190 It's singular because the rows are dependent -- 230 00:14:23,190 --> 00:14:25,380 and can I just keep the reasoning going? 231 00:14:25,380 --> 00:14:32,790 Because this vector one one one is -- 232 00:14:32,790 --> 00:14:34,940 it's not in the null space of the matrix, 233 00:14:34,940 --> 00:14:37,410 but it's in the null space of the transpose -- 234 00:14:37,410 --> 00:14:43,160 is in the null space of the transpose. 235 00:14:43,160 --> 00:14:45,600 And that's good enough. 236 00:14:45,600 --> 00:14:48,890 If we have a square matrix -- if we have a square matrix 237 00:14:48,890 --> 00:14:52,930 and the rows are dependent, that matrix is singular. 238 00:14:52,930 --> 00:14:58,630 So it turned out that the immediate guy we could identify 239 00:14:58,630 --> 00:15:00,470 was one one one. 240 00:15:00,470 --> 00:15:04,650 Of course, the -- 241 00:15:04,650 --> 00:15:06,480 there will be somebody in the null space, 242 00:15:06,480 --> 00:15:07,300 too. 243 00:15:07,300 --> 00:15:10,280 And actually, who will it be? 244 00:15:10,280 --> 00:15:13,110 So what's -- so -- 245 00:15:13,110 --> 00:15:15,830 so now I want to ask about the null space of -- 246 00:15:15,830 --> 00:15:17,400 of the matrix itself. 247 00:15:17,400 --> 00:15:20,890 What combination of the columns gives zero? 248 00:15:20,890 --> 00:15:23,450 I -- I don't want to compute it because I just made up this 249 00:15:23,450 --> 00:15:24,830 matrix and -- 250 00:15:24,830 --> 00:15:28,020 it will -- it would take me a while -- 251 00:15:28,020 --> 00:15:31,390 it looks sort of doable because it's three by three but wh- 252 00:15:31,390 --> 00:15:34,720 my point is, what -- 253 00:15:34,720 --> 00:15:37,350 what vector is it if we -- once we've found it, 254 00:15:37,350 --> 00:15:41,360 what have we got that's in the -- in the null space of A? 255 00:15:41,360 --> 00:15:44,060 It's the eigenvector, right? 256 00:15:44,060 --> 00:15:46,060 That's where we find X one. 257 00:15:46,060 --> 00:15:54,350 Then X one, the eigenvector, is in the null space of A. 258 00:15:54,350 --> 00:15:57,690 That's the eigenvector corresponding to the eigenvalue 259 00:15:57,690 --> 00:15:58,470 one. 260 00:15:58,470 --> 00:15:58,980 Right? 261 00:15:58,980 --> 00:16:01,330 That's how we find eigenvectors. 262 00:16:01,330 --> 00:16:05,280 So those three columns must be dependent -- 263 00:16:05,280 --> 00:16:08,090 some combination of columns -- of those three columns is 264 00:16:08,090 --> 00:16:11,930 the zero column and that -- the three components in that 265 00:16:11,930 --> 00:16:14,020 combination are the eigenvector. 266 00:16:14,020 --> 00:16:17,250 And that guy is the steady state. 267 00:16:17,250 --> 00:16:18,280 Okay. 268 00:16:18,280 --> 00:16:21,110 So I'm happy about the -- 269 00:16:21,110 --> 00:16:26,010 the thinking here, but I haven't given -- 270 00:16:26,010 --> 00:16:29,620 I haven't completed it because I haven't found x1. 271 00:16:29,620 --> 00:16:32,560 But it's there. 272 00:16:32,560 --> 00:16:36,330 Can I -- another thought came to me as I was doing this, 273 00:16:36,330 --> 00:16:39,280 another little comment that -- you -- 274 00:16:39,280 --> 00:16:41,500 about eigenvalues and eigenvectors, 275 00:16:41,500 --> 00:16:45,330 because of A and A transpose. 276 00:16:45,330 --> 00:16:50,800 What can you tell me about eigenvalues of A -- 277 00:16:50,800 --> 00:16:56,275 of A and eigenvalues of A transpose? 278 00:17:01,160 --> 00:17:01,660 Whoops. 279 00:17:04,952 --> 00:17:05,660 They're the same. 280 00:17:05,660 --> 00:17:11,079 They're -- so this is a little comment -- we -- it's useful, 281 00:17:11,079 --> 00:17:14,550 since eigenvalues are generally not easy to find -- 282 00:17:14,550 --> 00:17:19,180 it's always useful to know some cases where you've got them, 283 00:17:19,180 --> 00:17:20,210 where -- 284 00:17:20,210 --> 00:17:23,089 and this is -- if you know the eigenvalues of A, 285 00:17:23,089 --> 00:17:25,260 then you know the eigenvalues of A transpose. 286 00:17:25,260 --> 00:17:27,500 eigenvalues of A transpose are the same. 287 00:17:32,560 --> 00:17:37,890 And can I just, like, review why that is? 288 00:17:37,890 --> 00:17:44,280 So to find the eigenvalues of A, this would be determinate of A 289 00:17:44,280 --> 00:17:53,490 minus lambda I equals zero, that gives me an eigenvalue of A -- 290 00:17:53,490 --> 00:17:58,520 now how can I get A transpose into the picture here? 291 00:17:58,520 --> 00:18:00,690 I'll use the fact that the determinant 292 00:18:00,690 --> 00:18:06,690 of a matrix and the determinant of its transpose are the same. 293 00:18:06,690 --> 00:18:09,550 The determinant of a matrix equals the determinant of a -- 294 00:18:09,550 --> 00:18:10,520 of the transpose. 295 00:18:10,520 --> 00:18:13,950 That was property ten, the very last guy 296 00:18:13,950 --> 00:18:15,860 in our determinant list. 297 00:18:15,860 --> 00:18:18,280 So I'll transpose that matrix. 298 00:18:18,280 --> 00:18:21,680 This leads to -- 299 00:18:21,680 --> 00:18:24,690 I just take the matrix and transpose it, 300 00:18:24,690 --> 00:18:29,520 but now what do I get when I transpose lambda I? 301 00:18:29,520 --> 00:18:33,920 I just get lambda I. 302 00:18:33,920 --> 00:18:37,800 So that's -- that's all there was to the reasoning. 303 00:18:37,800 --> 00:18:40,000 The reasoning is that the eigenvalues of A 304 00:18:40,000 --> 00:18:42,370 solved that equation. 305 00:18:42,370 --> 00:18:44,530 The determinant of a matrix is the determinant 306 00:18:44,530 --> 00:18:47,480 of its transpose, so that gives me this equation 307 00:18:47,480 --> 00:18:50,160 and that tells me that the same lambdas 308 00:18:50,160 --> 00:18:53,240 are eigenvalues of A transpose. 309 00:18:53,240 --> 00:18:56,660 So that, backing up to the Markov case, 310 00:18:56,660 --> 00:19:00,880 one is an eigenvalue of A transpose and we actually found 311 00:19:00,880 --> 00:19:05,480 its eigenvector, one one one, and that tell us that one is 312 00:19:05,480 --> 00:19:08,090 also an eigenvalue of A -- but, of course, 313 00:19:08,090 --> 00:19:10,360 it has a different eigenvector, the -- 314 00:19:10,360 --> 00:19:13,530 the left null space isn't the same as the null space and we 315 00:19:13,530 --> 00:19:14,750 would have to find it. 316 00:19:14,750 --> 00:19:20,960 So there's some vector here which is x1 that produces zero 317 00:19:20,960 --> 00:19:22,460 zero zero. 318 00:19:22,460 --> 00:19:24,900 Actually, it wouldn't be that hard to find, you know, I -- 319 00:19:24,900 --> 00:19:26,520 as I'm talking I'm thinking, okay, 320 00:19:26,520 --> 00:19:29,440 I going to follow through and actually find it? 321 00:19:29,440 --> 00:19:32,750 Well, I can tell from this one -- look, 322 00:19:32,750 --> 00:19:37,650 if I put a point six there and a point seven there, 323 00:19:37,650 --> 00:19:43,510 that's what -- then I'll be okay in the last row, right? 324 00:19:43,510 --> 00:19:47,840 Now I only -- remains to find one guy. 325 00:19:47,840 --> 00:19:49,860 And let me take the first row, then. 326 00:19:49,860 --> 00:19:53,500 Minus point 54 plus point 21 -- 327 00:19:53,500 --> 00:19:56,790 there's some big number going in there, right? 328 00:19:56,790 --> 00:20:00,320 So I have -- just to make the first row come out zero, 329 00:20:00,320 --> 00:20:03,980 I'm getting minus point 54 plus point 21, 330 00:20:03,980 --> 00:20:09,420 so that was minus point 33 and what -- what do I want? 331 00:20:09,420 --> 00:20:11,350 Like thirty three hundred? 332 00:20:11,350 --> 00:20:13,980 This is the first time in the history of linear algebra 333 00:20:13,980 --> 00:20:16,980 that an eigenvector has every had a component 334 00:20:16,980 --> 00:20:18,980 thirty three hundred. 335 00:20:18,980 --> 00:20:21,030 But I guess it's true. 336 00:20:21,030 --> 00:20:24,760 Because then I multiply by minus one over a hundred -- oh no, 337 00:20:24,760 --> 00:20:26,830 it was point 33. 338 00:20:26,830 --> 00:20:29,180 So is this just -- oh, shoot. 339 00:20:29,180 --> 00:20:30,800 Only 33. 340 00:20:30,800 --> 00:20:31,880 Okay. 341 00:20:31,880 --> 00:20:33,030 Only 33. 342 00:20:33,030 --> 00:20:35,610 Okay, so there's the eigenvector. 343 00:20:35,610 --> 00:20:39,150 Oh, and notice that it -- that it turned -- did turn out, 344 00:20:39,150 --> 00:20:42,940 at least, to be all positive. 345 00:20:42,940 --> 00:20:45,990 So that was, like, the theory -- predicts that part, too. 346 00:20:45,990 --> 00:20:48,500 I won't give the proof of that part. 347 00:20:48,500 --> 00:20:50,600 So 30 -- 33 -- 348 00:20:50,600 --> 00:20:52,770 point six 33 point seven. 349 00:20:52,770 --> 00:20:53,420 Okay. 350 00:20:53,420 --> 00:20:58,750 Now those are the ma- that's the linear algebra part. 351 00:20:58,750 --> 00:21:00,980 Can I get to the applications? 352 00:21:00,980 --> 00:21:03,120 Where do these Markov matrices come from? 353 00:21:03,120 --> 00:21:06,360 Because that's -- that's part of this course and absolutely part 354 00:21:06,360 --> 00:21:07,570 of this lecture. 355 00:21:07,570 --> 00:21:08,080 Okay. 356 00:21:08,080 --> 00:21:12,270 So where's -- what's an application of Markov matrices? 357 00:21:12,270 --> 00:21:12,770 Okay. 358 00:21:17,600 --> 00:21:21,180 Markov matrices -- so, my equation, then, 359 00:21:21,180 --> 00:21:24,842 that I'm solving and studying is this equation u(k+1)=Auk. 360 00:21:28,290 --> 00:21:31,280 And now A is a Markov matrix. 361 00:21:31,280 --> 00:21:31,930 A is Markov. 362 00:21:36,060 --> 00:21:39,110 And I want to give an example. 363 00:21:39,110 --> 00:21:41,190 Can I just create an example? 364 00:21:41,190 --> 00:21:44,120 It'll be two by two. 365 00:21:44,120 --> 00:21:48,870 And it's one I've used before because it seems 366 00:21:48,870 --> 00:21:50,990 to me to bring out the idea. 367 00:21:50,990 --> 00:21:55,740 It's -- because we have two by two, we have two states, 368 00:21:55,740 --> 00:22:01,510 let's say California and Massachusetts. 369 00:22:01,510 --> 00:22:05,310 And I'm looking at the populations in those two 370 00:22:05,310 --> 00:22:08,720 states, the people in those two states, California 371 00:22:08,720 --> 00:22:10,900 and Massachusetts. 372 00:22:10,900 --> 00:22:17,040 And my matrix A is going to tell me in a -- in a year, 373 00:22:17,040 --> 00:22:19,230 some movement has happened. 374 00:22:19,230 --> 00:22:21,180 Some people stayed in Massachusetts, 375 00:22:21,180 --> 00:22:24,050 some people moved to California, some smart people 376 00:22:24,050 --> 00:22:26,300 moved from California to Massachusetts, 377 00:22:26,300 --> 00:22:29,240 some people stayed in California and made a billion. 378 00:22:29,240 --> 00:22:29,740 Okay. 379 00:22:29,740 --> 00:22:36,150 So that -- there's a matrix there with four entries 380 00:22:36,150 --> 00:22:41,044 and those tell me the fractions of my population -- 381 00:22:41,044 --> 00:22:41,710 so I'm making -- 382 00:22:41,710 --> 00:22:45,140 I'm going to use fractions, so they won't be negative, 383 00:22:45,140 --> 00:22:48,500 of course, because -- because only positive people are 384 00:22:48,500 --> 00:22:51,550 in- involved here -- and they'll add up to one, 385 00:22:51,550 --> 00:22:54,620 because I'm accounting for all people. 386 00:22:54,620 --> 00:22:57,830 So that's why I have these two key properties. 387 00:22:57,830 --> 00:22:59,650 The entries are greater equal zero 388 00:22:59,650 --> 00:23:02,890 because I'm looking at probabilities. 389 00:23:02,890 --> 00:23:06,050 Do they move, do they stay? 390 00:23:06,050 --> 00:23:09,130 Those probabilities are all between zero and one. 391 00:23:09,130 --> 00:23:12,290 And the probabilities add to one because everybody's 392 00:23:12,290 --> 00:23:12,990 accounted for. 393 00:23:12,990 --> 00:23:17,970 I'm not losing anybody, gaining anybody in this Markov chain. 394 00:23:17,970 --> 00:23:22,490 It's -- it conserves the total population. 395 00:23:22,490 --> 00:23:22,990 Okay. 396 00:23:22,990 --> 00:23:25,450 So what would be a typical matrix, then? 397 00:23:25,450 --> 00:23:34,820 So this would be u, California and u Massachusetts at time t 398 00:23:34,820 --> 00:23:36,740 equal k+1. 399 00:23:36,740 --> 00:23:40,680 And it's some matrix, which we'll 400 00:23:40,680 --> 00:23:48,730 think of, times u California and u Massachusetts at time k. 401 00:23:51,280 --> 00:23:54,460 And notice this matrix is going to stay the same, 402 00:23:54,460 --> 00:23:57,480 you know, forever. 403 00:23:57,480 --> 00:24:01,940 So that's a severe limitation on the example. 404 00:24:01,940 --> 00:24:05,090 The example has a -- the same Markov matrix, 405 00:24:05,090 --> 00:24:08,510 the same probabilities act at every time. 406 00:24:08,510 --> 00:24:09,010 Okay. 407 00:24:09,010 --> 00:24:11,030 So what's a reasonable, say -- 408 00:24:11,030 --> 00:24:16,750 say point nine of the people in California at time k 409 00:24:16,750 --> 00:24:19,410 stay there. 410 00:24:19,410 --> 00:24:24,440 And point one of the people in California 411 00:24:24,440 --> 00:24:26,930 move to Massachusetts. 412 00:24:26,930 --> 00:24:28,900 Notice why that column added to one, 413 00:24:28,900 --> 00:24:32,170 because we've now accounted for all the people in California 414 00:24:32,170 --> 00:24:33,890 at time k. 415 00:24:33,890 --> 00:24:36,250 Nine tenths of them are still in California, 416 00:24:36,250 --> 00:24:41,220 one tenth are here at time k+1. 417 00:24:41,220 --> 00:24:42,130 Okay. 418 00:24:42,130 --> 00:24:45,120 What about the people who are in Massachusetts? 419 00:24:45,120 --> 00:24:47,530 This is going to multiply column two, right, 420 00:24:47,530 --> 00:24:52,370 by our fundamental rule of multiplying matrix by vector, 421 00:24:52,370 --> 00:24:57,050 it's the -- it's the population in Massachusetts. 422 00:24:57,050 --> 00:25:06,350 Shall we say that -- that after the Red Sox, fail again, 423 00:25:06,350 --> 00:25:11,630 eight -- only 80 percent of the people in Massachusetts stay 424 00:25:11,630 --> 00:25:15,120 and 20 percent move to California. 425 00:25:15,120 --> 00:25:15,870 Okay. 426 00:25:15,870 --> 00:25:18,960 So again, this adds to one, which 427 00:25:18,960 --> 00:25:23,070 accounts for all people in Massachusetts where they are. 428 00:25:23,070 --> 00:25:26,770 So there is a Markov matrix. 429 00:25:26,770 --> 00:25:28,400 Non-negative entries adding to one. 430 00:25:28,400 --> 00:25:29,980 What's the steady state? 431 00:25:29,980 --> 00:25:33,314 If everybody started in Massachusetts, say, at -- 432 00:25:33,314 --> 00:25:34,980 you know, when the Pilgrims showed up or 433 00:25:34,980 --> 00:25:35,600 something. 434 00:25:35,600 --> 00:25:40,280 Then where are they now? 435 00:25:40,280 --> 00:25:45,211 Where are they at time 100, let's say, or maybe -- 436 00:25:45,211 --> 00:25:47,210 I don't know, how many years since the Pilgrims? 437 00:25:47,210 --> 00:25:49,090 300 and something. 438 00:25:49,090 --> 00:25:51,520 Or -- and actually where will they be, like, 439 00:25:51,520 --> 00:25:54,820 way out a million years from now? 440 00:25:54,820 --> 00:26:00,980 I -- I could multiply -- 441 00:26:00,980 --> 00:26:03,360 take the powers of this matrix. 442 00:26:03,360 --> 00:26:07,040 In fact, you'll -- you would -- ought to be able to figure out 443 00:26:07,040 --> 00:26:10,490 what is the hundredth power of that matrix? 444 00:26:10,490 --> 00:26:12,520 Why don't we do that? 445 00:26:12,520 --> 00:26:15,050 But let me follow the steady state. 446 00:26:15,050 --> 00:26:17,760 So what -- what's my starting -- 447 00:26:17,760 --> 00:26:28,560 my starting u Cal, u Mass at time zero is, shall we say -- 448 00:26:28,560 --> 00:26:30,400 shall we put anybody in California? 449 00:26:30,400 --> 00:26:32,920 Let's make -- let's make zero there, 450 00:26:32,920 --> 00:26:36,400 and say the population of Massachusetts is -- 451 00:26:36,400 --> 00:26:38,780 let's say a thousand just to -- okay. 452 00:26:43,330 --> 00:26:46,030 So the population is -- 453 00:26:46,030 --> 00:26:49,400 so the populations are zero and a thousand at the start. 454 00:26:49,400 --> 00:26:52,790 What can you tell me about this population after -- 455 00:26:52,790 --> 00:26:55,990 after k steps? 456 00:26:55,990 --> 00:27:01,420 What will u Cal plus u Mass add to? 457 00:27:01,420 --> 00:27:02,790 A thousand. 458 00:27:02,790 --> 00:27:06,260 Those thousand people are always accounted for. 459 00:27:06,260 --> 00:27:10,590 But -- so u Mass will start dropping from a thousand and u 460 00:27:10,590 --> 00:27:11,930 Cal will start growing. 461 00:27:11,930 --> 00:27:15,060 Actually, we could see -- why don't we figure out what it is 462 00:27:15,060 --> 00:27:16,420 after one? 463 00:27:16,420 --> 00:27:22,100 After one time step, what are the populations at time one? 464 00:27:24,890 --> 00:27:26,530 So what happens in one step? 465 00:27:26,530 --> 00:27:30,230 You multiply once by that matrix and, let's see, 466 00:27:30,230 --> 00:27:33,460 zero times this column -- so it's just a thousand times this 467 00:27:33,460 --> 00:27:39,940 column, so I think we're getting 200 and 800. 468 00:27:39,940 --> 00:27:42,860 So after the first step, 200 people have -- 469 00:27:42,860 --> 00:27:44,760 are in California. 470 00:27:44,760 --> 00:27:49,620 Now at the following step, I'll multiply again by this matrix 471 00:27:49,620 --> 00:27:52,740 -- more people will move to California. 472 00:27:52,740 --> 00:27:54,340 Some people will move back. 473 00:27:54,340 --> 00:28:00,390 Twenty people will come back and, the -- 474 00:28:00,390 --> 00:28:03,010 the net result will be that the California population will be 475 00:28:03,010 --> 00:28:08,310 above 200 and the Massachusetts below 800 and they'll still add 476 00:28:08,310 --> 00:28:10,191 up to a thousand. 477 00:28:10,191 --> 00:28:10,690 Okay. 478 00:28:10,690 --> 00:28:14,100 I do that a few times. 479 00:28:14,100 --> 00:28:15,500 I do that 100 times. 480 00:28:15,500 --> 00:28:18,220 What's the population? 481 00:28:18,220 --> 00:28:20,770 Well, okay, to answer any question like that, 482 00:28:20,770 --> 00:28:22,700 I need the eigenvalues and eigenvectors, 483 00:28:22,700 --> 00:28:23,200 right? 484 00:28:23,200 --> 00:28:24,090 As soon as I've -- 485 00:28:24,090 --> 00:28:26,340 I've created an example, but as soon 486 00:28:26,340 --> 00:28:28,570 as I want to solve anything, I have 487 00:28:28,570 --> 00:28:31,610 to find eigenvalues and eigenvectors of that matrix. 488 00:28:31,610 --> 00:28:32,230 Okay. 489 00:28:32,230 --> 00:28:33,890 So let's do it. 490 00:28:33,890 --> 00:28:40,010 So there's the matrix .9, .2, .1, .8 and tell 491 00:28:40,010 --> 00:28:43,150 me its eigenvalues. 492 00:28:43,150 --> 00:28:46,390 Lambda equals -- so tell me one eigenvalue? 493 00:28:46,390 --> 00:28:48,680 One, thanks. 494 00:28:48,680 --> 00:28:49,960 And tell me the other one. 495 00:28:53,120 --> 00:28:55,300 What's the other eigenvalue -- 496 00:28:55,300 --> 00:28:59,370 from the trace or the determinant -- from the -- 497 00:28:59,370 --> 00:29:02,510 I -- the trace is what -- is, like, easier. 498 00:29:02,510 --> 00:29:06,110 So the trace of that matrix is one point seven. 499 00:29:06,110 --> 00:29:10,040 So the other eigenvalue is point seven. 500 00:29:10,040 --> 00:29:13,540 And it -- notice that it's less than one. 501 00:29:13,540 --> 00:29:17,780 And notice that that determinant is point 72-.02, 502 00:29:17,780 --> 00:29:18,960 which is point seven. 503 00:29:18,960 --> 00:29:19,490 Right. 504 00:29:19,490 --> 00:29:20,190 Okay. 505 00:29:20,190 --> 00:29:21,900 Now to find the eigenvectors. 506 00:29:21,900 --> 00:29:26,120 This is -- so that's lambda one and the eigenvector -- 507 00:29:26,120 --> 00:29:29,950 I'll subtract one from the diagonal, right? 508 00:29:29,950 --> 00:29:33,440 So can I do that in light let -- in light here? 509 00:29:33,440 --> 00:29:35,300 Subtract one from the diagonal, I 510 00:29:35,300 --> 00:29:38,410 have minus point one and minus point two, 511 00:29:38,410 --> 00:29:39,730 and of course these are still 512 00:29:39,730 --> 00:29:40,940 there. 513 00:29:40,940 --> 00:29:44,220 And I'm looking for its -- 514 00:29:44,220 --> 00:29:47,360 here's -- here's -- this is going to be x1. 515 00:29:47,360 --> 00:29:52,170 It's the null space of A minus I. 516 00:29:52,170 --> 00:29:56,820 Okay, everybody sees that it's two and one. 517 00:29:56,820 --> 00:29:57,940 Okay? 518 00:29:57,940 --> 00:29:59,820 And now how about -- so that -- and it -- 519 00:29:59,820 --> 00:30:03,160 notice that that eigenvector is positive. 520 00:30:03,160 --> 00:30:08,770 And actually, we can jump to infinity right now. 521 00:30:08,770 --> 00:30:14,100 What's the population at infinity? 522 00:30:14,100 --> 00:30:17,170 It's a multiple -- this is -- this eigenvector is giving 523 00:30:17,170 --> 00:30:19,910 the steady state. 524 00:30:19,910 --> 00:30:23,280 It's some multiple of this, and how is that multiple decided? 525 00:30:23,280 --> 00:30:26,940 By adding up to a thousand people. 526 00:30:26,940 --> 00:30:31,200 So the steady state, the c1x1 -- this is the x1, 527 00:30:31,200 --> 00:30:36,740 but that adds up to three, so I really want two -- 528 00:30:36,740 --> 00:30:39,820 it's going to be two thirds of a thousand and one third 529 00:30:39,820 --> 00:30:42,980 of a thousand, making a total of the thousand 530 00:30:42,980 --> 00:30:43,550 people. 531 00:30:43,550 --> 00:30:45,480 That'll be the steady state. 532 00:30:45,480 --> 00:30:48,390 That's really all I need to know at infinity. 533 00:30:48,390 --> 00:30:50,160 But if I want to know what's happened 534 00:30:50,160 --> 00:30:53,280 after just a finite number like 100 steps, 535 00:30:53,280 --> 00:30:55,380 I'd better find this eigenvector. 536 00:30:55,380 --> 00:30:57,840 So can I -- can I look at -- 537 00:30:57,840 --> 00:31:00,130 I'll subtract point seven time -- ti- 538 00:31:00,130 --> 00:31:06,250 from the diagonal and I'll get that and I'll look at the null 539 00:31:06,250 --> 00:31:11,280 space of that one and I -- and this is going to give me x2, 540 00:31:11,280 --> 00:31:13,540 now, and what is it? 541 00:31:13,540 --> 00:31:16,380 So what's in the null space of -- that's certainly singular, 542 00:31:16,380 --> 00:31:20,920 so I know my calculation is right, and -- 543 00:31:20,920 --> 00:31:24,290 one and minus one. 544 00:31:24,290 --> 00:31:26,240 One and minus one. 545 00:31:26,240 --> 00:31:30,580 So I'm prepared now to write down 546 00:31:30,580 --> 00:31:32,470 the solution after 100 time 547 00:31:32,470 --> 00:31:33,180 steps. 548 00:31:33,180 --> 00:31:35,670 The -- the populations after 100 time steps, 549 00:31:35,670 --> 00:31:36,420 right? 550 00:31:36,420 --> 00:31:39,290 Can -- can we remember the point one -- the -- 551 00:31:39,290 --> 00:31:43,100 the one with this two one eigenvector and the point seven 552 00:31:43,100 --> 00:31:44,930 with the minus one one eigenvector. 553 00:31:44,930 --> 00:31:46,160 So I'll -- let me -- 554 00:31:46,160 --> 00:31:48,960 I'll just write it above here. 555 00:31:48,960 --> 00:31:53,810 u after k steps is some multiple of one 556 00:31:53,810 --> 00:31:57,610 to the k times the two one eigenvector 557 00:31:57,610 --> 00:32:04,230 and some multiple of point seven to the k times the minus one 558 00:32:04,230 --> 00:32:07,430 one eigenvector. 559 00:32:07,430 --> 00:32:09,910 Right? 560 00:32:09,910 --> 00:32:11,290 That's -- I -- 561 00:32:11,290 --> 00:32:15,010 this is how I take -- how powers of a matrix work. 562 00:32:15,010 --> 00:32:21,360 When I apply those powers to a u0, what I -- so it's u0, 563 00:32:21,360 --> 00:32:26,390 which was zero a thousand -- 564 00:32:26,390 --> 00:32:29,220 that has to be corrected k=0. 565 00:32:29,220 --> 00:32:35,720 So I'm plugging in k=0 and I get c1 times two one and c2 times 566 00:32:35,720 --> 00:32:38,640 minus one one. 567 00:32:38,640 --> 00:32:44,990 Two equations, two constants, certainly 568 00:32:44,990 --> 00:32:49,710 independent eigenvectors, so there's a solution 569 00:32:49,710 --> 00:32:51,770 and you see what it is? 570 00:32:51,770 --> 00:32:56,450 Let's see, I guess we already figured that c1 was a thousand 571 00:32:56,450 --> 00:32:59,390 over three, I think -- did we think that had to be a thousand 572 00:32:59,390 --> 00:33:02,200 over three? 573 00:33:02,200 --> 00:33:05,840 And maybe c2 would be -- 574 00:33:05,840 --> 00:33:07,920 excuse me, let -- get an eraser -- 575 00:33:07,920 --> 00:33:08,930 we can -- 576 00:33:08,930 --> 00:33:11,330 I just -- I think we've -- get it here. 577 00:33:11,330 --> 00:33:15,590 c2, we want to get a zero here, so maybe we 578 00:33:15,590 --> 00:33:21,400 need plus two thousand over three. 579 00:33:21,400 --> 00:33:22,840 I think that has to work. 580 00:33:22,840 --> 00:33:26,010 Two times a thousand over three minus two thousand 581 00:33:26,010 --> 00:33:29,160 over three, that'll give us the zero and a thousand 582 00:33:29,160 --> 00:33:32,020 over three and the two thousand over three will give us 583 00:33:32,020 --> 00:33:33,290 three thousand over three, 584 00:33:33,290 --> 00:33:34,160 the thousand. 585 00:33:34,160 --> 00:33:37,820 So this is what we approach -- 586 00:33:37,820 --> 00:33:42,000 this part, with the point seven to the k-th power 587 00:33:42,000 --> 00:33:45,050 is the part that's disappearing. 588 00:33:45,050 --> 00:33:48,270 That's -- that's Markov matrices. 589 00:33:48,270 --> 00:33:48,770 Okay. 590 00:33:48,770 --> 00:33:52,610 That's an example of where they come from, 591 00:33:52,610 --> 00:34:00,220 from modeling movement of people with no gain or loss, 592 00:34:00,220 --> 00:34:03,690 with total -- total count conserved. 593 00:34:03,690 --> 00:34:04,440 Okay. 594 00:34:04,440 --> 00:34:07,260 I -- just if I can add one more comment, 595 00:34:07,260 --> 00:34:11,550 because you'll see Markov matrices in electrical 596 00:34:11,550 --> 00:34:16,949 engineering courses and often you'll see them -- sorry, 597 00:34:16,949 --> 00:34:19,679 here's my little comment. 598 00:34:19,679 --> 00:34:22,280 Sometimes -- in a lot of applications they prefer 599 00:34:22,280 --> 00:34:25,590 to work with row vectors. 600 00:34:25,590 --> 00:34:29,480 So they -- instead of -- this was natural for us, right? 601 00:34:29,480 --> 00:34:32,429 For all the eigenvectors to be column vectors. 602 00:34:32,429 --> 00:34:37,139 So our columns added to one in the Markov matrix. 603 00:34:37,139 --> 00:34:39,889 Just so you don't think, well, what -- 604 00:34:39,889 --> 00:34:41,820 what's going on? 605 00:34:41,820 --> 00:34:48,170 If we work with row vectors and we multiply vector times matrix 606 00:34:48,170 --> 00:34:51,179 -- so we're multiplying from the left -- 607 00:34:51,179 --> 00:34:55,679 then it'll be the then we'll be using the transpose of -- 608 00:34:55,679 --> 00:35:00,100 of this matrix and it'll be the rows that add to one. 609 00:35:00,100 --> 00:35:05,390 So in other textbooks, you'll see -- instead of col- 610 00:35:05,390 --> 00:35:08,200 columns adding to one, you'll see rows add to one. 611 00:35:08,200 --> 00:35:09,320 Okay. 612 00:35:09,320 --> 00:35:10,410 Fine. 613 00:35:10,410 --> 00:35:13,030 Okay, that's what I wanted to say about Markov, 614 00:35:13,030 --> 00:35:17,560 now I want to say something about projections and even 615 00:35:17,560 --> 00:35:22,190 leading in -- a little into Fourier series. 616 00:35:22,190 --> 00:35:24,960 Because -- but before any Fourier stuff, 617 00:35:24,960 --> 00:35:28,150 let me make a comment about projections. 618 00:35:28,150 --> 00:35:33,310 This -- so this is a comment about projections onto -- 619 00:35:33,310 --> 00:35:36,526 with an orthonormal basis. 620 00:35:42,420 --> 00:35:49,770 So, of course, the basis vectors are q1 up to qn. 621 00:35:49,770 --> 00:35:50,860 Okay. 622 00:35:50,860 --> 00:35:52,330 I have a vector b. 623 00:35:52,330 --> 00:35:58,700 Let -- let me imagine -- let me imagine this is a basis. 624 00:35:58,700 --> 00:36:00,640 Let -- let's say I'm in n by n. 625 00:36:00,640 --> 00:36:06,660 I'm -- I've got, eh, n orthonormal vectors, 626 00:36:06,660 --> 00:36:09,350 I'm in n dimensional space so they're a complete -- 627 00:36:09,350 --> 00:36:11,110 they're a basis -- 628 00:36:11,110 --> 00:36:16,170 any vector v could be expanded in this basis. 629 00:36:16,170 --> 00:36:22,050 So any vector v is some combination, some amount of q1 630 00:36:22,050 --> 00:36:28,060 plus some amount of q2 plus some amount of qn. 631 00:36:28,060 --> 00:36:35,220 So -- so any v. 632 00:36:35,220 --> 00:36:41,740 I just want you to tell me what those amounts are. 633 00:36:41,740 --> 00:36:46,510 What are x1 -- what's x1, for example? 634 00:36:46,510 --> 00:36:49,980 So I'm looking for the expansion. 635 00:36:49,980 --> 00:36:51,900 This is -- this is really our projection. 636 00:36:51,900 --> 00:36:56,330 I could -- I could really use the word expansion. 637 00:36:56,330 --> 00:37:01,450 I'm expanding the vector in the basis. 638 00:37:01,450 --> 00:37:05,770 And the special thing about the basis is that it's orthonormal. 639 00:37:05,770 --> 00:37:10,470 So that should give me a special formula for the answer, 640 00:37:10,470 --> 00:37:12,220 for the coefficients. 641 00:37:12,220 --> 00:37:14,010 So how do I get x1? 642 00:37:14,010 --> 00:37:15,510 What -- what's a formula for x1? 643 00:37:18,250 --> 00:37:23,040 I could -- I can go through the projection -- 644 00:37:23,040 --> 00:37:26,940 the Q transpose Q, all that -- 645 00:37:26,940 --> 00:37:31,450 normal equations, but -- 646 00:37:31,450 --> 00:37:32,140 and I'll get -- 647 00:37:32,140 --> 00:37:33,960 I'll come out with this nice answer 648 00:37:33,960 --> 00:37:36,060 that I think I can see right away. 649 00:37:36,060 --> 00:37:38,370 How can I pick -- 650 00:37:38,370 --> 00:37:42,920 get a hold of x1 and get these other x-s out of the equation? 651 00:37:42,920 --> 00:37:47,460 So how can I get a nice, simple formula for x1? 652 00:37:47,460 --> 00:37:50,751 And then we want to see, sure, we knew that all the time. 653 00:37:50,751 --> 00:37:51,250 Okay. 654 00:37:51,250 --> 00:37:52,610 So what's x1? 655 00:37:52,610 --> 00:37:59,020 The good way is take the inner product of everything with q1. 656 00:37:59,020 --> 00:38:02,430 Take the inner product of that whole equation, every term, 657 00:38:02,430 --> 00:38:03,850 with q1. 658 00:38:03,850 --> 00:38:08,370 What will happen to that last term? 659 00:38:08,370 --> 00:38:11,560 The inner product -- when -- if I take the dot product with q1 660 00:38:11,560 --> 00:38:13,970 I get zero, right? 661 00:38:13,970 --> 00:38:17,160 Because this basis was orthonormal. 662 00:38:17,160 --> 00:38:20,330 If I take the dot product with q2 I get zero. 663 00:38:20,330 --> 00:38:24,570 If I take the dot product with q1 I get one. 664 00:38:24,570 --> 00:38:28,740 So that tells me what x1 is. q1 transpose 665 00:38:28,740 --> 00:38:31,250 v, that's taking the dot product, 666 00:38:31,250 --> 00:38:41,060 is x1 times q1 transpose q1 plus a bunch of zeroes. 667 00:38:41,060 --> 00:38:45,330 And this is a one, so I can forget that. 668 00:38:45,330 --> 00:38:47,820 I get x1 immediately. 669 00:38:47,820 --> 00:38:49,840 So -- do you see what I'm saying -- 670 00:38:49,840 --> 00:38:53,070 is that I have an orthonormal basis, 671 00:38:53,070 --> 00:38:58,760 then the coefficient that I need for each basis vector is 672 00:38:58,760 --> 00:38:59,340 a cinch to 673 00:38:59,340 --> 00:39:00,070 find. 674 00:39:00,070 --> 00:39:02,310 Let me -- let me just -- 675 00:39:02,310 --> 00:39:05,580 I have to put this into matrix language, too, 676 00:39:05,580 --> 00:39:07,350 so you'll see it there also. 677 00:39:07,350 --> 00:39:10,690 If I write that first equation in matrix language, what -- 678 00:39:10,690 --> 00:39:12,010 what is it? 679 00:39:12,010 --> 00:39:13,870 I'm writing -- in matrix language, 680 00:39:13,870 --> 00:39:19,020 this equation says I'm taking these columns -- are -- 681 00:39:19,020 --> 00:39:20,620 are you guys good at this now? 682 00:39:20,620 --> 00:39:29,180 I'm taking those columns times the Xs and getting V, right? 683 00:39:29,180 --> 00:39:30,440 That's the matrix form. 684 00:39:30,440 --> 00:39:37,464 Okay, that's the matrix Q. Qx is v. 685 00:39:37,464 --> 00:39:39,005 What's the solution to that equation? 686 00:39:41,570 --> 00:39:44,920 It's -- of course, it's x equal Q inverse v. 687 00:39:44,920 --> 00:39:51,000 So x is Q inverse v, but what's the point? 688 00:39:51,000 --> 00:39:54,060 Q inverse in this case is going to -- is simple. 689 00:39:54,060 --> 00:39:58,220 I don't have to work to invert this matrix Q, 690 00:39:58,220 --> 00:40:03,110 because of the fact that the -- these columns are orthonormal, 691 00:40:03,110 --> 00:40:05,110 I know the inverse to that. 692 00:40:05,110 --> 00:40:10,070 And it is Q transpose. 693 00:40:10,070 --> 00:40:15,050 When you see a Q, a square matrix with that letter Q, 694 00:40:15,050 --> 00:40:17,090 the -- that just triggers -- 695 00:40:17,090 --> 00:40:19,670 Q inverse is the same as Q transpose. 696 00:40:19,670 --> 00:40:22,100 So the first component, then -- 697 00:40:22,100 --> 00:40:25,970 the first component of x is the first row times 698 00:40:25,970 --> 00:40:29,550 v, and what's that? 699 00:40:29,550 --> 00:40:32,650 The first component of this answer 700 00:40:32,650 --> 00:40:36,570 is the first row of Q transpose. 701 00:40:36,570 --> 00:40:42,110 That's just -- that's just q1 transpose times v. 702 00:40:42,110 --> 00:40:46,320 So that's what we concluded here, too. 703 00:40:46,320 --> 00:40:46,820 Okay. 704 00:40:49,590 --> 00:40:55,090 So -- so nothing Fourier here. 705 00:40:55,090 --> 00:40:59,650 The -- the key ingredient here was that the q-s are 706 00:40:59,650 --> 00:41:01,460 orthonormal. 707 00:41:01,460 --> 00:41:04,900 And now that's what Fourier series are built on. 708 00:41:04,900 --> 00:41:08,520 So now, in the remaining time, let 709 00:41:08,520 --> 00:41:11,760 me say something about Fourier series. 710 00:41:11,760 --> 00:41:12,400 Okay. 711 00:41:12,400 --> 00:41:20,120 So Fourier series is -- 712 00:41:20,120 --> 00:41:25,150 well, we've got a function f of x. 713 00:41:25,150 --> 00:41:28,160 And we want to write it as a combination of -- 714 00:41:28,160 --> 00:41:30,900 maybe it has a constant term. 715 00:41:30,900 --> 00:41:34,910 And then it has some cos(x) in it. 716 00:41:34,910 --> 00:41:38,120 And it has some sin(x) in it. 717 00:41:38,120 --> 00:41:42,330 And it has some cos(2x) in it. 718 00:41:42,330 --> 00:41:45,230 And a -- and some sin(2x), and forever. 719 00:41:50,780 --> 00:41:54,600 So what's -- what's the difference between this type 720 00:41:54,600 --> 00:41:56,900 problem and the one above it? 721 00:41:56,900 --> 00:42:02,090 This one's infinite, but the key property 722 00:42:02,090 --> 00:42:06,110 of things being orthogonal is still 723 00:42:06,110 --> 00:42:09,790 true for sines and cosines, so it's the property that 724 00:42:09,790 --> 00:42:11,170 makes Fourier series work. 725 00:42:11,170 --> 00:42:12,860 So that's called a Fourier series. 726 00:42:12,860 --> 00:42:14,620 Better write his name up. 727 00:42:14,620 --> 00:42:15,530 Fourier series. 728 00:42:22,170 --> 00:42:25,960 So it was Joseph Fourier who realized that, hey, I 729 00:42:25,960 --> 00:42:29,870 could work in function space. 730 00:42:29,870 --> 00:42:33,970 Instead of a vector v, I could have a function f of x. 731 00:42:33,970 --> 00:42:38,590 Instead of orthogonal vectors, q1, q2 , q3, 732 00:42:38,590 --> 00:42:42,180 I could have orthogonal functions, the constant, 733 00:42:42,180 --> 00:42:45,370 the cos(x), the sin(x), the s- cos(2x), 734 00:42:45,370 --> 00:42:47,450 but infinitely many of them. 735 00:42:47,450 --> 00:42:49,930 I need infinitely many, because my space 736 00:42:49,930 --> 00:42:52,440 is infinite dimensional. 737 00:42:52,440 --> 00:42:56,970 So this is, like, the moment in which we leave finite 738 00:42:56,970 --> 00:43:00,680 dimensional vector spaces and go to infinite dimensional vector 739 00:43:00,680 --> 00:43:03,090 spaces and our basis -- 740 00:43:03,090 --> 00:43:07,250 so the vectors are now functions -- 741 00:43:07,250 --> 00:43:09,650 and of course, there are so many functions that it's -- 742 00:43:09,650 --> 00:43:13,620 that we've got an infin- infinite dimensional space -- 743 00:43:13,620 --> 00:43:17,320 and the basis vectors are functions, too. 744 00:43:17,320 --> 00:43:25,400 a0, the constant function one -- so my basis is one cos(x), 745 00:43:25,400 --> 00:43:29,190 sin(x), cos(2x), sin(2x) and so on. 746 00:43:29,190 --> 00:43:32,710 And the reason Fourier series is a success 747 00:43:32,710 --> 00:43:35,170 is that those are orthogonal. 748 00:43:35,170 --> 00:43:35,760 Okay. 749 00:43:35,760 --> 00:43:37,190 Now what do I mean by orthogonal? 750 00:43:40,110 --> 00:43:44,490 I know what it means for two vectors to be orthogonal -- 751 00:43:44,490 --> 00:43:46,870 y transpose x equals zero, right? 752 00:43:46,870 --> 00:43:48,600 Dot product equals zero. 753 00:43:48,600 --> 00:43:52,070 But what's the dot product of functions? 754 00:43:52,070 --> 00:43:55,340 I'm claiming that whatever it is, the dot product -- 755 00:43:55,340 --> 00:43:59,840 or we would more likely use the word inner product of, say, 756 00:43:59,840 --> 00:44:02,430 cos(x) with sin(x) is zero. 757 00:44:02,430 --> 00:44:06,340 And cos(x) with cos(2x), also zero. 758 00:44:06,340 --> 00:44:08,780 So I -- let me tell you what I mean by that, 759 00:44:08,780 --> 00:44:10,410 by that dot product. 760 00:44:10,410 --> 00:44:12,850 Well, how do I compute a dot product? 761 00:44:12,850 --> 00:44:16,970 So, let's just remember for vectors v trans- 762 00:44:16,970 --> 00:44:23,050 v transpose w for vectors, so this was vectors, 763 00:44:23,050 --> 00:44:30,145 v transpose w was v1w1 +...+vnwn. 764 00:44:33,510 --> 00:44:34,140 Okay. 765 00:44:34,140 --> 00:44:34,750 Now functions. 766 00:44:40,100 --> 00:44:42,720 Now I have two functions, let's call them f and g. 767 00:44:45,260 --> 00:44:46,700 What's with them now? 768 00:44:46,700 --> 00:44:49,730 The vectors had n components, but the functions 769 00:44:49,730 --> 00:44:53,060 have a whole, like, continuum. 770 00:44:53,060 --> 00:44:55,640 To graph the function, I just don't have n points, 771 00:44:55,640 --> 00:44:57,780 I've got this whole graph. 772 00:44:57,780 --> 00:44:59,780 So I have functions -- 773 00:44:59,780 --> 00:45:01,590 I'm really trying to ask you what's 774 00:45:01,590 --> 00:45:03,840 the inner product of this function 775 00:45:03,840 --> 00:45:05,170 f with another function 776 00:45:05,170 --> 00:45:06,130 g? 777 00:45:06,130 --> 00:45:11,870 And I want to make it parallel to this the best I can. 778 00:45:11,870 --> 00:45:20,080 So the best parallel is to multiply f (x) times g(x) 779 00:45:20,080 --> 00:45:23,060 at every x -- 780 00:45:23,060 --> 00:45:25,120 and here I just had n multiplications, 781 00:45:25,120 --> 00:45:28,330 but here I'm going to have a whole range of x-s, 782 00:45:28,330 --> 00:45:31,980 and here I added the results. 783 00:45:31,980 --> 00:45:34,600 What do I do here? 784 00:45:34,600 --> 00:45:38,400 So what's the analog of addition when you have -- 785 00:45:38,400 --> 00:45:40,350 when you're in a continuum? 786 00:45:40,350 --> 00:45:41,830 It's integration. 787 00:45:41,830 --> 00:45:47,630 So that the -- the dot product of two functions will be 788 00:45:47,630 --> 00:45:49,525 the integral of those functions, dx. 789 00:45:52,240 --> 00:45:55,010 Now I have to say -- say, well, what are the limits 790 00:45:55,010 --> 00:45:56,800 of integration? 791 00:45:56,800 --> 00:46:03,980 And for this Fourier series, this function f(x) -- 792 00:46:03,980 --> 00:46:08,000 if I'm going to have -- if that right hand side is going to be 793 00:46:08,000 --> 00:46:11,920 f(x), that function that I'm seeing on the right, 794 00:46:11,920 --> 00:46:16,130 all those sines and cosines, they're all periodic, with -- 795 00:46:16,130 --> 00:46:18,210 with period two pi. 796 00:46:18,210 --> 00:46:21,650 So -- so that's what f(x) had better be. 797 00:46:21,650 --> 00:46:24,870 So I'll integrate from zero to two pi. 798 00:46:24,870 --> 00:46:29,160 My -- all -- everything -- is on the interval zero two pi now, 799 00:46:29,160 --> 00:46:33,690 because if I'm going to use these sines and cosines, 800 00:46:33,690 --> 00:46:39,950 then f(x) is equal to f(x+2pi). 801 00:46:39,950 --> 00:46:42,428 This is periodic -- 802 00:46:45,180 --> 00:46:48,250 periodic functions. 803 00:46:48,250 --> 00:46:49,800 Okay. 804 00:46:49,800 --> 00:46:52,770 So now I know what -- 805 00:46:52,770 --> 00:46:56,120 I've got all the right words now. 806 00:46:56,120 --> 00:47:00,490 I've got a vector space, but the vectors are functions. 807 00:47:00,490 --> 00:47:05,450 I've got inner products and -- and the inner product gives 808 00:47:05,450 --> 00:47:07,760 a number, all right. 809 00:47:07,760 --> 00:47:12,260 It just happens to be an integral instead of a sum. 810 00:47:12,260 --> 00:47:15,230 I've got -- and that -- then I have the idea of orthogonality 811 00:47:15,230 --> 00:47:15,860 -- 812 00:47:15,860 --> 00:47:18,470 because, actually, just -- let's just check. 813 00:47:18,470 --> 00:47:21,910 Orthogonality -- if I take the integral -- s- I -- 814 00:47:21,910 --> 00:47:25,890 let me do sin(x) times cos(x) -- 815 00:47:25,890 --> 00:47:31,106 sin(x) times cos(x) dx from zero to two pi -- 816 00:47:34,680 --> 00:47:36,700 I think we get zero. 817 00:47:36,700 --> 00:47:40,670 That's the differential of that, so it would be one half 818 00:47:40,670 --> 00:47:42,890 sine x squared, was that right? 819 00:47:47,220 --> 00:47:50,100 Between zero and two pi -- 820 00:47:50,100 --> 00:47:52,910 and, of course, we get zero. 821 00:47:52,910 --> 00:47:58,150 And the same would be true with a little more -- 822 00:47:58,150 --> 00:48:02,430 some trig identities to help us out -- of every other pair. 823 00:48:02,430 --> 00:48:05,650 So we have now an orthonormal infinite 824 00:48:05,650 --> 00:48:10,160 basis for function space, and all we want to do 825 00:48:10,160 --> 00:48:12,980 is express a function in that 826 00:48:12,980 --> 00:48:14,090 basis. 827 00:48:14,090 --> 00:48:20,120 And so I -- the end of my lecture is, okay, what is a1? 828 00:48:20,120 --> 00:48:24,210 What's the coefficient -- how much cos(x) is there 829 00:48:24,210 --> 00:48:30,230 in a function compared to the other harmonics? 830 00:48:30,230 --> 00:48:32,940 How much constant is in that function? 831 00:48:32,940 --> 00:48:35,310 That'll -- that would be an easy question. 832 00:48:35,310 --> 00:48:39,110 The answer a0 will come out to be the average value of f. 833 00:48:39,110 --> 00:48:40,620 That's the amount of the constant 834 00:48:40,620 --> 00:48:42,630 that's in there, its average value. 835 00:48:42,630 --> 00:48:45,350 But let's take a1 as more typical. 836 00:48:45,350 --> 00:48:48,170 How will I get -- here's the end of the lecture, then -- 837 00:48:48,170 --> 00:48:49,400 how do I get a1? 838 00:48:52,070 --> 00:48:54,720 The first Fourier coefficient. 839 00:48:54,720 --> 00:48:56,350 Okay. 840 00:48:56,350 --> 00:48:59,790 I do just as I did in the vector case. 841 00:48:59,790 --> 00:49:03,850 I take the inner product of everything with cos(x) 842 00:49:03,850 --> 00:49:07,010 Take the inner product of everything with cos(x). 843 00:49:07,010 --> 00:49:08,800 Then on the left -- 844 00:49:08,800 --> 00:49:13,990 on the left I have -- the inner product is the integral of f(x) 845 00:49:13,990 --> 00:49:15,365 times cos(x) cx. 846 00:49:18,740 --> 00:49:22,080 And on the right, what do I have? 847 00:49:22,080 --> 00:49:24,970 When I -- so what I -- when I say take the inner product with 848 00:49:24,970 --> 00:49:28,850 cos(x), let me put it in ordinary calculus words. 849 00:49:28,850 --> 00:49:32,610 Multiply by cos(x) and integrate. 850 00:49:32,610 --> 00:49:34,540 That's what inner products are. 851 00:49:34,540 --> 00:49:36,940 So if I multiply that whole thing by cos(x) 852 00:49:36,940 --> 00:49:40,990 and I integrate, I get a whole lot of zeroes. 853 00:49:40,990 --> 00:49:45,830 The only thing that survives is that term. 854 00:49:45,830 --> 00:49:47,250 All the others disappear. 855 00:49:47,250 --> 00:49:53,950 So -- and that term is a1 times the integral of cos(x) squared 856 00:49:53,950 --> 00:50:01,580 dx zero to 2pi equals -- so this was the left side and this is 857 00:50:01,580 --> 00:50:04,840 all that's left on the right-hand side. 858 00:50:04,840 --> 00:50:09,700 And this is not zero of course, because it's the length 859 00:50:09,700 --> 00:50:13,570 of the function squared, it's the inner product with itself, 860 00:50:13,570 --> 00:50:18,050 and -- and a simple calculation gives that answer to be pi. 861 00:50:18,050 --> 00:50:23,360 So that's an easy integral and it turns out to be pi, 862 00:50:23,360 --> 00:50:31,900 so that a1 is one over pi times there -- times this integral. 863 00:50:31,900 --> 00:50:35,490 So there is, actually -- that's Euler's famous formula 864 00:50:35,490 --> 00:50:39,110 for the -- or maybe Fourier found it -- 865 00:50:39,110 --> 00:50:41,405 for the coefficients in a Fourier series. 866 00:50:43,980 --> 00:50:47,940 And you see that it's exactly an expansion 867 00:50:47,940 --> 00:50:50,790 in an orthonormal basis. 868 00:50:50,790 --> 00:50:51,540 Okay, thanks. 869 00:50:51,540 --> 00:50:56,660 So I'll do a quiz review on Monday and then the quiz itself 870 00:50:56,660 --> 00:50:59,200 in Walker on Wednesday. 871 00:50:59,200 --> 00:51:00,590 Okay, see you Monday. 872 00:51:00,590 --> 00:51:02,140 Thanks.