1 00:00:00,000 --> 00:00:11,590 -- and lift-off on differential equations. 2 00:00:11,590 --> 00:00:16,660 So, this section is about how to solve 3 00:00:16,660 --> 00:00:22,760 a system of first order, first derivative, constant 4 00:00:22,760 --> 00:00:26,500 coefficient linear equations. 5 00:00:26,500 --> 00:00:32,740 And if we do it right, it turns directly into linear algebra. 6 00:00:32,740 --> 00:00:37,690 The key idea is the solutions to constant coefficient 7 00:00:37,690 --> 00:00:41,730 linear equations are exponentials. 8 00:00:41,730 --> 00:00:44,520 So if you look for an exponential, 9 00:00:44,520 --> 00:00:48,230 then all you have to find is what's up there in the exponent 10 00:00:48,230 --> 00:00:51,060 and what multiplies the exponential 11 00:00:51,060 --> 00:00:53,130 and that's the linear algebra. 12 00:00:53,130 --> 00:00:56,950 So -- and the result -- one thing we will fine -- 13 00:00:56,950 --> 00:01:01,380 it's completely parallel to powers of a matrix. 14 00:01:01,380 --> 00:01:06,090 So the last lecture was about how would you compute 15 00:01:06,090 --> 00:01:08,680 A to the K or A to the 100? 16 00:01:08,680 --> 00:01:12,120 How do you compute high powers of a matrix? 17 00:01:12,120 --> 00:01:18,590 Now it's not powers anymore, but it's exponentials. 18 00:01:18,590 --> 00:01:22,310 That's the natural thing for differential equation. 19 00:01:22,310 --> 00:01:22,810 Okay. 20 00:01:22,810 --> 00:01:26,730 But can I begin with an example? 21 00:01:26,730 --> 00:01:28,630 And I'll just go through the mechanics. 22 00:01:28,630 --> 00:01:31,700 How would I solve the differential -- 23 00:01:31,700 --> 00:01:33,707 two differential equations? 24 00:01:33,707 --> 00:01:34,790 So I'm going to make it -- 25 00:01:34,790 --> 00:01:38,160 I'll have a two by two matrix and the coefficients 26 00:01:38,160 --> 00:01:42,530 are minus one two, one minus two and I'd better give you 27 00:01:42,530 --> 00:01:44,200 some initial condition. 28 00:01:44,200 --> 00:01:50,570 So suppose it starts u at times zero -- this is u1, u2 -- 29 00:01:50,570 --> 00:01:52,810 let it -- let it -- 30 00:01:52,810 --> 00:01:57,150 suppose everything is in u1 at times zero. 31 00:01:57,150 --> 00:02:00,890 So -- at -- at the start, it's all in u1. 32 00:02:00,890 --> 00:02:07,640 But what happens as time goes on, du2/dt will -- 33 00:02:07,640 --> 00:02:10,690 will be positive, because of that u1 term, 34 00:02:10,690 --> 00:02:16,210 so flow will move into the u2 component and it will go out 35 00:02:16,210 --> 00:02:19,050 of the u1 component. 36 00:02:19,050 --> 00:02:22,330 So we'll just follow that movement as time 37 00:02:22,330 --> 00:02:28,260 goes forward by looking at the eigenvalues and eigenvectors 38 00:02:28,260 --> 00:02:30,220 of that matrix. 39 00:02:30,220 --> 00:02:31,430 That's a first job. 40 00:02:31,430 --> 00:02:34,210 Before you do anything else, find the -- 41 00:02:34,210 --> 00:02:39,510 find the matrix and its eigenvalues and eigenvectors. 42 00:02:39,510 --> 00:02:42,400 So let me do that. 43 00:02:42,400 --> 00:02:43,060 Okay. 44 00:02:43,060 --> 00:02:45,440 So here's our matrix. 45 00:02:45,440 --> 00:02:47,500 Maybe you can tell me right away what -- 46 00:02:47,500 --> 00:02:53,190 what are the eigenvalues and -- eigenvalues anyway. 47 00:02:53,190 --> 00:02:54,640 And then we can check. 48 00:02:54,640 --> 00:02:58,070 But can you spot any of the eigenvalues of that matrix? 49 00:02:58,070 --> 00:03:00,900 We're looking for two eigenvalues. 50 00:03:00,900 --> 00:03:02,090 Do you see -- 51 00:03:02,090 --> 00:03:04,530 I mean, if I just wrote that matrix down, what -- 52 00:03:04,530 --> 00:03:07,670 what do you notice about it? 53 00:03:07,670 --> 00:03:09,490 It's singular, right. 54 00:03:09,490 --> 00:03:11,380 That -- that's a singular matrix. 55 00:03:11,380 --> 00:03:14,190 That tells me right away that one of the eigenvalues 56 00:03:14,190 --> 00:03:18,860 is lambda equals zero. 57 00:03:18,860 --> 00:03:20,900 I can -- that's a singular matrix, 58 00:03:20,900 --> 00:03:25,440 the second column is minus two times the first column, 59 00:03:25,440 --> 00:03:28,600 the determinant is zero, it's -- it's singular, 60 00:03:28,600 --> 00:03:33,670 so zero is an eigenvalue and the other eigenvalue will be -- 61 00:03:33,670 --> 00:03:35,640 from the trace. 62 00:03:35,640 --> 00:03:37,510 I look at the trace, the sum down 63 00:03:37,510 --> 00:03:39,620 the diagonal is minus three. 64 00:03:39,620 --> 00:03:42,770 That has to agree with the sum of the eigenvalue, 65 00:03:42,770 --> 00:03:46,970 so that second eigenvalue better be minus three. 66 00:03:46,970 --> 00:03:48,610 I could, of course -- 67 00:03:48,610 --> 00:03:51,100 I could compute -- why don't I over here -- 68 00:03:51,100 --> 00:03:54,530 compute the determinant of A minus lambda I, 69 00:03:54,530 --> 00:04:01,220 the determinant of this minus one minus lambda two one minus 70 00:04:01,220 --> 00:04:03,680 two minus lambda matrix. 71 00:04:03,680 --> 00:04:05,930 But we know what's coming. 72 00:04:05,930 --> 00:04:09,700 When I do that multiplication, I get a lambda squared. 73 00:04:09,700 --> 00:04:14,200 I get a two lambda and a one lambda, that's a three lambda. 74 00:04:14,200 --> 00:04:16,750 And then -- now I'm going to get the determinant, 75 00:04:16,750 --> 00:04:19,910 which is two minus two which is zero. 76 00:04:19,910 --> 00:04:26,490 So there's my characteristic polynomial, this determinant. 77 00:04:26,490 --> 00:04:31,960 And of course I factor that into lambda times lambda plus three 78 00:04:31,960 --> 00:04:37,590 and I get the two eigenvalues that we saw coming. 79 00:04:37,590 --> 00:04:38,830 What else do I need? 80 00:04:38,830 --> 00:04:40,320 The eigenvectors. 81 00:04:40,320 --> 00:04:44,020 So before I even think about the differential equation or what 82 00:04:44,020 --> 00:04:47,780 -- how to solve it, let me find the eigenvectors for this 83 00:04:47,780 --> 00:04:48,910 matrix. 84 00:04:48,910 --> 00:04:49,430 Okay. 85 00:04:49,430 --> 00:04:52,970 So take lambda equals zero -- 86 00:04:52,970 --> 00:04:55,370 so that -- that's the first eigenvalue. 87 00:04:55,370 --> 00:04:58,690 Lambda one equals zero and the second eigenvalue 88 00:04:58,690 --> 00:05:03,130 will be lambda two equals minus three. 89 00:05:03,130 --> 00:05:05,210 By the way, I -- 90 00:05:05,210 --> 00:05:09,010 I already know something important about this. 91 00:05:09,010 --> 00:05:11,920 The eigenvalues are telling me something. 92 00:05:11,920 --> 00:05:15,820 You'll see how it comes out, but let me point to -- 93 00:05:15,820 --> 00:05:19,640 these numbers are -- this eigenvalue, 94 00:05:19,640 --> 00:05:23,700 a negative eigenvalue, is going to disappear. 95 00:05:23,700 --> 00:05:28,620 There's going to be an e to the minus three t in the answer. 96 00:05:28,620 --> 00:05:31,690 That e to the minus three t as times goes on 97 00:05:31,690 --> 00:05:34,070 is going to be very, very small. 98 00:05:34,070 --> 00:05:38,360 The other part of the answer will involve an e 99 00:05:38,360 --> 00:05:40,390 to the zero t. 100 00:05:40,390 --> 00:05:44,330 But e to the zero t is one and that's a constant. 101 00:05:44,330 --> 00:05:49,190 So I'm expecting that this solution'll have two parts, 102 00:05:49,190 --> 00:05:53,570 an e to the zero t part and an e to the minus three t part, 103 00:05:53,570 --> 00:05:57,900 and that -- and as time goes on, the second part'll disappear 104 00:05:57,900 --> 00:06:00,360 and the first part will be a steady 105 00:06:00,360 --> 00:06:02,030 It won't move. state. 106 00:06:02,030 --> 00:06:06,740 It will be -- at the end of -- as t approaches infinity, 107 00:06:06,740 --> 00:06:09,800 this part disappears and this is the -- 108 00:06:09,800 --> 00:06:13,330 the e to the zero t part is what I get. 109 00:06:13,330 --> 00:06:17,120 And I'm very interested in these steady states, so that's -- 110 00:06:17,120 --> 00:06:19,330 I get a steady state when I have a zero 111 00:06:19,330 --> 00:06:20,150 eigenvalue. 112 00:06:20,150 --> 00:06:20,650 Okay. 113 00:06:20,650 --> 00:06:22,950 What about those eigenvectors? 114 00:06:22,950 --> 00:06:26,170 So what's the eigenvector that goes with eigenvalue zero? 115 00:06:26,170 --> 00:06:26,670 Okay. 116 00:06:26,670 --> 00:06:31,280 The matrix is singular as it is, the eigenvector is -- 117 00:06:31,280 --> 00:06:34,940 is the guy in the null space, so what vector is in the null 118 00:06:34,940 --> 00:06:37,890 space of that matrix? 119 00:06:37,890 --> 00:06:38,860 Let's see. 120 00:06:38,860 --> 00:06:42,900 I guess I probably give the free variable the value one 121 00:06:42,900 --> 00:06:48,630 and I realize that if I want to get zero I need a two up here. 122 00:06:48,630 --> 00:06:49,260 Okay? 123 00:06:49,260 --> 00:06:52,560 So Ax1 is zero x1. 124 00:06:52,560 --> 00:06:55,550 A x1 is zero x1. 125 00:06:55,550 --> 00:06:56,370 Fine. 126 00:06:56,370 --> 00:06:57,520 Okay. 127 00:06:57,520 --> 00:06:59,520 What about the other eigenvalue? 128 00:06:59,520 --> 00:07:01,801 Lambda two is minus three. 129 00:07:01,801 --> 00:07:02,300 Okay. 130 00:07:02,300 --> 00:07:05,010 How do I get the other eigenvalue, then? 131 00:07:05,010 --> 00:07:08,080 For the moment -- can I mentally do it? 132 00:07:08,080 --> 00:07:12,850 I subtract minus three along the diagonal, 133 00:07:12,850 --> 00:07:15,070 which means I add three -- 134 00:07:15,070 --> 00:07:18,370 can I -- I'll just do it with an erase -- erase for the moment. 135 00:07:18,370 --> 00:07:20,950 So I'm going to add three to the diagonal. 136 00:07:20,950 --> 00:07:26,040 So this minus one will become a two and -- 137 00:07:26,040 --> 00:07:28,450 I'll make it in big loopy letters -- 138 00:07:28,450 --> 00:07:32,350 and when I add three to this guy, the minus two becomes -- 139 00:07:32,350 --> 00:07:36,200 well, I can't make one very loopy, but how's that? 140 00:07:36,200 --> 00:07:37,410 Okay. 141 00:07:37,410 --> 00:07:40,200 Now that's A minus three I -- 142 00:07:40,200 --> 00:07:41,540 A plus three I, sorry. 143 00:07:41,540 --> 00:07:43,400 That's A plus three I. 144 00:07:43,400 --> 00:07:45,510 It's supposed to be singular, right? 145 00:07:45,510 --> 00:07:48,260 I-- if things -- if I did it right, 146 00:07:48,260 --> 00:07:51,930 this matrix should be singular and the x2, 147 00:07:51,930 --> 00:07:55,521 the eigenvector should be in its null space. 148 00:07:55,521 --> 00:07:56,020 Okay. 149 00:07:56,020 --> 00:07:58,510 What do I get for the null space of this? 150 00:07:58,510 --> 00:08:02,140 Maybe minus one one, or one minus one. 151 00:08:02,140 --> 00:08:03,240 Doesn't matter. 152 00:08:03,240 --> 00:08:05,300 Those are both perfectly good. 153 00:08:05,300 --> 00:08:05,800 Right? 154 00:08:05,800 --> 00:08:07,550 Because that's in the null space of this. 155 00:08:07,550 --> 00:08:14,370 Now I'll -- because A times that vector is three times that 156 00:08:14,370 --> 00:08:15,600 vector. 157 00:08:15,600 --> 00:08:20,370 Ax2 is minus three x2. 158 00:08:20,370 --> 00:08:22,540 Good. 159 00:08:22,540 --> 00:08:23,040 Okay. 160 00:08:23,040 --> 00:08:26,380 Can I get A again so we see that correctly? 161 00:08:26,380 --> 00:08:29,680 That was a minus one and that was a minus two. 162 00:08:29,680 --> 00:08:31,730 Good. 163 00:08:31,730 --> 00:08:32,809 Okay. 164 00:08:32,809 --> 00:08:35,470 That -- that's the first job. 165 00:08:35,470 --> 00:08:37,970 eigenvalues and eigenvectors. 166 00:08:37,970 --> 00:08:41,179 And already the eigenvalues are telling me 167 00:08:41,179 --> 00:08:44,240 the most important information about the answer. 168 00:08:44,240 --> 00:08:46,540 But now, what is the answer? 169 00:08:46,540 --> 00:08:54,420 The answer is -- the solution will be U of T -- 170 00:08:54,420 --> 00:08:54,920 okay. 171 00:08:54,920 --> 00:08:59,750 Now, wh- now I use those eigenvalues and eigenvectors. 172 00:08:59,750 --> 00:09:04,210 The solution is some -- there are two eigenvalues. 173 00:09:04,210 --> 00:09:08,100 So I -- it -- so there're going to be two special solutions 174 00:09:08,100 --> 00:09:08,910 here. 175 00:09:08,910 --> 00:09:12,040 Two pure exponential solutions. 176 00:09:12,040 --> 00:09:17,300 The first one is going to be either the lambda one tx1 177 00:09:17,300 --> 00:09:23,440 and the -- so that solves the equation, and so does this one. 178 00:09:23,440 --> 00:09:29,290 They both are solutions to the differential equation. 179 00:09:29,290 --> 00:09:31,640 That's the general solution. 180 00:09:31,640 --> 00:09:33,900 The general solution is a combination 181 00:09:33,900 --> 00:09:37,010 of that pure exponential solution 182 00:09:37,010 --> 00:09:39,630 and that pure exponential solution. 183 00:09:39,630 --> 00:09:43,150 Can I just see that those guys do solve the equation? 184 00:09:43,150 --> 00:09:47,540 So let me just check -- check on this one, for example. 185 00:09:47,540 --> 00:09:53,360 I -- I want to check that the -- my equation -- 186 00:09:53,360 --> 00:09:53,860 let's 187 00:09:53,860 --> 00:09:57,190 Check. remember, the equation -- du/dt is Au. 188 00:09:57,190 --> 00:10:04,340 I plug in e to the lambda one t x1 189 00:10:04,340 --> 00:10:08,470 and let's just see that the equation's okay. 190 00:10:08,470 --> 00:10:12,370 I believe this is a solution to that equation. 191 00:10:12,370 --> 00:10:13,980 So just plug it in. 192 00:10:13,980 --> 00:10:17,870 On the left-hand side, I take the time derivative -- 193 00:10:17,870 --> 00:10:22,700 so the left-hand side will be lambda one, e to the lambda one 194 00:10:22,700 --> 00:10:25,540 t x1, right? 195 00:10:25,540 --> 00:10:28,990 The time derivative -- this is the term that depends on time, 196 00:10:28,990 --> 00:10:33,250 it's just ordinary exponential, its derivative brings down 197 00:10:33,250 --> 00:10:34,470 a lambda one. 198 00:10:34,470 --> 00:10:37,460 On the other side of the equation it's A times this 199 00:10:37,460 --> 00:10:38,380 thing. 200 00:10:38,380 --> 00:10:44,790 A times either the lambda one t x one, and does that check out? 201 00:10:44,790 --> 00:10:47,550 Do we have equality there? 202 00:10:47,550 --> 00:10:52,190 Yes, because either the lambda one t appears on both sides 203 00:10:52,190 --> 00:10:58,030 and the other one is Ax1 equal lambda one x1 -- check. 204 00:10:58,030 --> 00:11:02,590 Do you -- so, the -- we've come to the first point to remember. 205 00:11:02,590 --> 00:11:05,030 These pure solutions. 206 00:11:05,030 --> 00:11:10,760 Those pure solutions are the -- those pure exponentials are 207 00:11:10,760 --> 00:11:14,280 the differential equations analogue of -- 208 00:11:14,280 --> 00:11:17,040 last time we had pure powers. 209 00:11:17,040 --> 00:11:19,540 Last time -- so -- 210 00:11:19,540 --> 00:11:24,200 so last time, the analog was lambda -- 211 00:11:24,200 --> 00:11:29,020 lambda one to the K-th power x1, some amount of that, 212 00:11:29,020 --> 00:11:35,130 plus some amount of lambda two to the K-th power x2. 213 00:11:35,130 --> 00:11:37,590 That was our formula from last time. 214 00:11:37,590 --> 00:11:41,690 I put it up just to -- so your eye compares those two 215 00:11:41,690 --> 00:11:42,880 formulas. 216 00:11:42,880 --> 00:11:45,550 Powers of lambda in the -- 217 00:11:45,550 --> 00:11:48,610 in the difference equation -- that -- this was in the -- 218 00:11:48,610 --> 00:11:55,480 this was for the equation uk plus one equals A uk. 219 00:11:55,480 --> 00:11:59,890 That was for the finite step -- stepping by one. 220 00:11:59,890 --> 00:12:02,330 And we got powers, now this is the one 221 00:12:02,330 --> 00:12:04,920 we're interested in, the exponentials. 222 00:12:04,920 --> 00:12:08,750 So -- so that's -- that's the solution -- 223 00:12:08,750 --> 00:12:10,660 what are c1 and c2? 224 00:12:10,660 --> 00:12:12,300 Then we're through. 225 00:12:12,300 --> 00:12:14,060 What are c1 and c2? 226 00:12:14,060 --> 00:12:17,340 Well, of course we know these actual things. 227 00:12:17,340 --> 00:12:22,260 Let me just -- let me come back to this. 228 00:12:22,260 --> 00:12:26,790 c1 is -- we haven't figured out yet, but e to the lambda one t, 229 00:12:26,790 --> 00:12:32,890 the lambda one is zero so that's just a one times x1 which is 230 00:12:32,890 --> 00:12:34,260 two one. 231 00:12:34,260 --> 00:12:39,590 So it's c1 times this one that's not moving times the vector, 232 00:12:39,590 --> 00:12:44,660 the eigenvector two one and c2 times -- 233 00:12:44,660 --> 00:12:47,730 what's e to the lambda two t? 234 00:12:47,730 --> 00:12:51,830 Lambda two is minus three. 235 00:12:51,830 --> 00:12:54,010 So this is the term that has the minus 236 00:12:54,010 --> 00:12:58,360 three t and its eigenvector is this one minus one. 237 00:13:01,520 --> 00:13:06,640 So this vector solves the equation 238 00:13:06,640 --> 00:13:08,580 and any multiple of it. 239 00:13:08,580 --> 00:13:12,190 This vector solves the equation if it's got that factor 240 00:13:12,190 --> 00:13:14,620 e to the minus three t. 241 00:13:14,620 --> 00:13:17,980 We've got the answer except for c1 and c2. 242 00:13:17,980 --> 00:13:22,970 So -- so everything I've done is immediate as soon as you know 243 00:13:22,970 --> 00:13:25,570 the eigenvalues and eigenvectors. 244 00:13:25,570 --> 00:13:27,640 So how do we get c1 and c2? 245 00:13:27,640 --> 00:13:30,930 That has to come from the initial condition. 246 00:13:30,930 --> 00:13:38,740 So now I -- now I use -- u of zero is given as one zero. 247 00:13:41,780 --> 00:13:46,346 So this is the initial condition that will find c1 and c2. 248 00:13:46,346 --> 00:13:48,095 So let me do that on the board underneath. 249 00:13:51,080 --> 00:13:52,935 At t equals zero, then -- 250 00:13:56,500 --> 00:14:05,280 I get c1 times this guy plus c2 and now I'm at times zero. 251 00:14:05,280 --> 00:14:08,320 So that's a one and this is a one minus one 252 00:14:08,320 --> 00:14:12,920 and that's supposed to agree with u of zero one zero. 253 00:14:19,550 --> 00:14:20,850 Okay. 254 00:14:20,850 --> 00:14:23,090 That should be two equations. 255 00:14:23,090 --> 00:14:26,500 That should give me c1 and c2 and then I'm through. 256 00:14:26,500 --> 00:14:28,540 So what are c1 and c2? 257 00:14:28,540 --> 00:14:30,430 Let's see. 258 00:14:30,430 --> 00:14:33,000 I guess we could actually spot them by eye 259 00:14:33,000 --> 00:14:36,800 or we could solve two equations in two unknowns. 260 00:14:36,800 --> 00:14:38,090 Let's see. 261 00:14:38,090 --> 00:14:40,940 If these were both ones -- so I'm just adding -- 262 00:14:40,940 --> 00:14:43,630 then I would get three zero. 263 00:14:43,630 --> 00:14:46,740 So what's the -- what's the solution, then? 264 00:14:49,970 --> 00:14:53,910 If -- if c1 and c2 are both ones, I get three zero, 265 00:14:53,910 --> 00:14:55,990 so I want, like, one third of that, 266 00:14:55,990 --> 00:14:57,750 because I want to get one zero. 267 00:14:57,750 --> 00:15:02,220 So I think it's c1 equals a third, c2 equals a third. 268 00:15:05,460 --> 00:15:08,030 So finally I have the answer. 269 00:15:08,030 --> 00:15:11,000 Let me keep it in the -- in this board here. 270 00:15:11,000 --> 00:15:20,530 Finally the answer is one third of this plus one third of this. 271 00:15:24,450 --> 00:15:27,990 Do you see what -- what's actually happening with this 272 00:15:27,990 --> 00:15:28,880 flow? 273 00:15:28,880 --> 00:15:32,630 This flow started out at -- the solution started out at one 274 00:15:32,630 --> 00:15:34,140 zero. 275 00:15:34,140 --> 00:15:36,790 Started at one zero. 276 00:15:36,790 --> 00:15:41,290 Then as time went on, people moved, essentially. 277 00:15:41,290 --> 00:15:46,190 Some fraction of this one moved here. 278 00:15:46,190 --> 00:15:52,030 And -- and in the limit, there's -- there's the limit, as -- 279 00:15:52,030 --> 00:15:52,530 right? 280 00:15:52,530 --> 00:15:55,540 As t goes to infinity, as t gets very large, 281 00:15:55,540 --> 00:15:59,110 this disappears and this is the steady state. 282 00:15:59,110 --> 00:16:02,550 So the steady state is -- 283 00:16:02,550 --> 00:16:04,272 so the steady state -- 284 00:16:08,700 --> 00:16:14,190 u -- we could call it u at infinity is one third of two 285 00:16:14,190 --> 00:16:15,040 and one. 286 00:16:15,040 --> 00:16:17,110 It's -- it's two thirds of one third. 287 00:16:19,970 --> 00:16:22,790 So that's the -- we really -- 288 00:16:22,790 --> 00:16:25,280 I mean, you're getting, like, total, 289 00:16:25,280 --> 00:16:29,790 insight into the behavior of the solution, 290 00:16:29,790 --> 00:16:32,050 what the differential equation does. 291 00:16:32,050 --> 00:16:37,480 Of course, we don't -- wouldn't always have a steady state. 292 00:16:37,480 --> 00:16:40,580 Sometimes we would approach zero. 293 00:16:40,580 --> 00:16:42,320 Sometimes we would blow up. 294 00:16:42,320 --> 00:16:45,250 Can we straighten out those cases? 295 00:16:45,250 --> 00:16:47,510 The eigenvalue should tell us. 296 00:16:47,510 --> 00:16:50,170 So when do we get -- 297 00:16:50,170 --> 00:16:54,440 so -- so let me ask first, when do we get stability? 298 00:16:57,220 --> 00:17:00,250 That's u of t going to zero. 299 00:17:03,070 --> 00:17:05,660 When would the solution go to zero no matter 300 00:17:05,660 --> 00:17:09,230 what the initial condition is? 301 00:17:09,230 --> 00:17:11,140 Negative eigenvalues, right. 302 00:17:11,140 --> 00:17:12,609 Negative eigenvalues. 303 00:17:12,609 --> 00:17:13,630 But now I have to -- 304 00:17:13,630 --> 00:17:16,950 I have to ask you for one more step. 305 00:17:16,950 --> 00:17:20,420 Suppose the eigenvalues are complex numbers? 306 00:17:20,420 --> 00:17:22,680 Because we know they could be. 307 00:17:22,680 --> 00:17:27,760 Then we want stability -- this -- this -- we want -- 308 00:17:27,760 --> 00:17:35,260 we need all these e to the lambda t-s all going to zero 309 00:17:35,260 --> 00:17:40,920 and somehow that asks us to have lambda negative. 310 00:17:40,920 --> 00:17:43,470 But suppose lambda is a complex number? 311 00:17:43,470 --> 00:17:45,690 Then what's the test? 312 00:17:45,690 --> 00:17:50,340 What -- if lambda's a complex number like, oh, 313 00:17:50,340 --> 00:17:54,730 suppose lambda is negative plus an imaginary part? 314 00:17:54,730 --> 00:17:59,810 Say lambda is minus three plus six i? 315 00:17:59,810 --> 00:18:01,120 What -- what happens then? 316 00:18:01,120 --> 00:18:03,530 Can we just, like, do a -- a case here? 317 00:18:03,530 --> 00:18:11,550 If -- if this lambda is minus three plus six it, 318 00:18:11,550 --> 00:18:14,170 how big is that number? 319 00:18:14,170 --> 00:18:18,450 Does this -- does this imaginary part play a -- play a -- 320 00:18:18,450 --> 00:18:20,840 play a role here or not? 321 00:18:20,840 --> 00:18:22,850 Or how big is -- 322 00:18:22,850 --> 00:18:25,700 what's the absolute value of that -- of that quantity? 323 00:18:28,530 --> 00:18:32,670 It's just e to the minus three t, right? 324 00:18:32,670 --> 00:18:36,880 Because this other part, this -- the -- the magnitude -- the -- 325 00:18:36,880 --> 00:18:41,795 this -- e to the six it -- what -- that has absolute value one. 326 00:18:44,680 --> 00:18:45,180 Right? 327 00:18:45,180 --> 00:18:50,540 That's just this cosine of six t plus i, sine of six t. 328 00:18:50,540 --> 00:18:53,060 And the absolute value squared will 329 00:18:53,060 --> 00:18:56,230 be cos squared plus sine squared will be one. 330 00:18:56,230 --> 00:18:59,680 This is -- this complex number is running around the unit 331 00:18:59,680 --> 00:19:00,660 circle. 332 00:19:00,660 --> 00:19:04,770 This com- this -- the -- it's the real part that matters. 333 00:19:04,770 --> 00:19:07,020 This is what I'm trying to do. 334 00:19:07,020 --> 00:19:10,980 Real part of lambda has to be negative. 335 00:19:10,980 --> 00:19:14,880 If lambda's a complex number, it's the real part, 336 00:19:14,880 --> 00:19:19,200 the minus three, that either makes us go to zero 337 00:19:19,200 --> 00:19:24,940 or doesn't, or let -- or blows up. 338 00:19:24,940 --> 00:19:27,380 The imaginary part won't -- will just, like, 339 00:19:27,380 --> 00:19:30,690 oscillate between the two components. 340 00:19:30,690 --> 00:19:31,360 Okay. 341 00:19:31,360 --> 00:19:33,230 So that's stability. 342 00:19:33,230 --> 00:19:36,040 And what about -- 343 00:19:36,040 --> 00:19:37,305 what about a steady state? 344 00:19:42,130 --> 00:19:45,490 When would we have, a steady state, 345 00:19:45,490 --> 00:19:47,390 always in the same direction? 346 00:19:47,390 --> 00:19:48,160 So let me -- 347 00:19:48,160 --> 00:19:51,280 I'll take this part away -- 348 00:19:51,280 --> 00:19:54,280 when -- so that was, like, checking that it's -- 349 00:19:54,280 --> 00:19:57,830 that it's the real part that we care about. 350 00:19:57,830 --> 00:20:01,530 Now, we have a steady state when -- 351 00:20:01,530 --> 00:20:12,500 when lambda one is zero and the other eigenvalues have what? 352 00:20:12,500 --> 00:20:14,990 So I'm looking -- like, that example was, like, 353 00:20:14,990 --> 00:20:18,910 perfect for a steady state. 354 00:20:18,910 --> 00:20:22,760 We have a zero eigenvalue and the other eigenvalues, 355 00:20:22,760 --> 00:20:25,070 we want those to disappear. 356 00:20:25,070 --> 00:20:28,975 So the other eigenvalues have real part negative. 357 00:20:31,880 --> 00:20:35,700 And we blow up, for sure -- 358 00:20:35,700 --> 00:20:45,380 we blow up if any real part of lambda is positive. 359 00:20:49,090 --> 00:20:54,240 So if I -- if I reverse the sign of A -- of the matrix A -- 360 00:20:54,240 --> 00:20:57,420 suppose instead of the matrix I had, the A that I had, 361 00:20:57,420 --> 00:20:58,390 I changed it -- 362 00:20:58,390 --> 00:21:00,770 I changed all its sines. 363 00:21:00,770 --> 00:21:04,950 What would that do to the eigenvalues and eigenvectors? 364 00:21:04,950 --> 00:21:08,090 If I -- if -- if I know the eigenvalues and eigenvectors 365 00:21:08,090 --> 00:21:11,520 of A, tell me about minus A. 366 00:21:11,520 --> 00:21:14,780 What happens to the eigenvalues? 367 00:21:14,780 --> 00:21:18,410 For minus A, they'll all change sine. 368 00:21:18,410 --> 00:21:20,660 So I'll have blow up. 369 00:21:20,660 --> 00:21:23,020 This -- instead of the e to the minus three t, 370 00:21:23,020 --> 00:21:26,460 if I change that to minus -- if I change the sines in that 371 00:21:26,460 --> 00:21:30,810 matrix, I would change the trace to plus three, 372 00:21:30,810 --> 00:21:34,020 I would have an e to the plus three t and I would have blow 373 00:21:34,020 --> 00:21:36,150 up. 374 00:21:36,150 --> 00:21:39,430 Of course the zero eigenvalue would stay at zero, 375 00:21:39,430 --> 00:21:42,490 but the other guy is taking off in -- 376 00:21:42,490 --> 00:21:45,091 if I reversed all the sines. 377 00:21:45,091 --> 00:21:45,590 Okay. 378 00:21:45,590 --> 00:21:51,090 So this is -- this is the stability picture. 379 00:21:51,090 --> 00:21:56,680 And for a two by two matrix, we can actually 380 00:21:56,680 --> 00:22:01,220 pin down even more closely what that means. 381 00:22:01,220 --> 00:22:02,710 Can I -- let -- can I do that? 382 00:22:02,710 --> 00:22:04,410 Let me do that -- 383 00:22:04,410 --> 00:22:05,810 I want to -- 384 00:22:05,810 --> 00:22:11,230 for a two by two matrix, I can tell whether the real part 385 00:22:11,230 --> 00:22:14,740 of the eigenvalues is negative, I -- well, let me -- 386 00:22:14,740 --> 00:22:18,480 let me tell you what I have in mind for that. 387 00:22:18,480 --> 00:22:21,040 So two by two stability -- 388 00:22:21,040 --> 00:22:25,750 let me -- this is just a little comment. 389 00:22:25,750 --> 00:22:27,506 Two by two stability. 390 00:22:31,240 --> 00:22:35,930 So my matrix, now, is just a b c d 391 00:22:35,930 --> 00:22:41,770 and I'm looking for the real parts of both eigenvalues 392 00:22:41,770 --> 00:22:42,910 to be negative. 393 00:22:47,480 --> 00:22:47,980 Okay. 394 00:22:52,330 --> 00:22:55,300 What -- how can I tell from looking at the matrix, 395 00:22:55,300 --> 00:22:58,230 without computing its eigenvalues, 396 00:22:58,230 --> 00:23:02,150 whether the two eigenvalues are negative, 397 00:23:02,150 --> 00:23:04,930 or at least their real parts are negative? 398 00:23:04,930 --> 00:23:07,260 What would that tell me about the trace? 399 00:23:07,260 --> 00:23:10,830 So -- so the trace -- 400 00:23:10,830 --> 00:23:14,930 that's this a plus d -- 401 00:23:14,930 --> 00:23:19,470 what can you tell me about the trace in the case of a two 402 00:23:19,470 --> 00:23:21,760 by two stable matrix? 403 00:23:21,760 --> 00:23:25,320 That means the eigenvalues have -- are negative, 404 00:23:25,320 --> 00:23:28,660 or at least the real parts of those eigenvalues are negative 405 00:23:28,660 --> 00:23:33,140 -- then, when I take the -- when I look at the matrix and find 406 00:23:33,140 --> 00:23:36,930 its trace, what -- what do I know about that? 407 00:23:36,930 --> 00:23:38,360 It's negative, right. 408 00:23:38,360 --> 00:23:40,940 This is the sum of the -- this equals -- 409 00:23:40,940 --> 00:23:47,010 this equals lambda one plus lambda two, so it's negative. 410 00:23:47,010 --> 00:23:49,590 The two eigenvalues, by the way, will have -- 411 00:23:49,590 --> 00:23:54,990 if they're complex -- will have plus six i and minus six i. 412 00:23:54,990 --> 00:23:59,860 The complex parts will -- will be conjugates of each other 413 00:23:59,860 --> 00:24:04,720 and disappear when we add and the trace will be negative. 414 00:24:04,720 --> 00:24:06,710 Okay, the trace has to be negative. 415 00:24:06,710 --> 00:24:09,030 Is that enough -- 416 00:24:09,030 --> 00:24:14,670 is a negative trace enough to make the matrix stable? 417 00:24:14,670 --> 00:24:16,180 Shouldn't be enough, right? 418 00:24:16,180 --> 00:24:19,270 Can I -- can you make -- what's a matrix that has a negative 419 00:24:19,270 --> 00:24:24,040 trace but still it's not stable? 420 00:24:24,040 --> 00:24:27,500 So it -- it has a blow -- it still has a blow-up factor 421 00:24:27,500 --> 00:24:30,900 and a -- and a -- and a decaying one. 422 00:24:30,900 --> 00:24:33,820 So what would be a -- so just -- just to see -- 423 00:24:33,820 --> 00:24:35,920 maybe I just put that here. 424 00:24:35,920 --> 00:24:40,790 This -- now I'm looking for an example where the trace could 425 00:24:40,790 --> 00:24:48,080 be negative but still blow up. 426 00:24:48,080 --> 00:24:52,830 Of course -- yeah, let's just take one. 427 00:24:52,830 --> 00:24:57,990 Oh, look, let me -- let me make it minus two zero zero one. 428 00:24:57,990 --> 00:25:00,140 Okay. 429 00:25:00,140 --> 00:25:04,810 There's a case where that matrix has negative trace -- 430 00:25:04,810 --> 00:25:06,390 I know its eigenvalues of course. 431 00:25:06,390 --> 00:25:09,750 They're minus two and one and it blows up. 432 00:25:09,750 --> 00:25:12,780 It's got -- it's got a plus one eigenvalue here, 433 00:25:12,780 --> 00:25:17,280 so there would be an e to the plus t in the solution 434 00:25:17,280 --> 00:25:21,170 and it'll blow up if it has any second component at all. 435 00:25:21,170 --> 00:25:23,900 I need another condition. 436 00:25:23,900 --> 00:25:25,615 And it's a condition on the determinant. 437 00:25:28,240 --> 00:25:29,560 And what's that condition? 438 00:25:29,560 --> 00:25:32,730 If I know that the two eigenvalues -- 439 00:25:32,730 --> 00:25:36,090 suppose I know they're negative, both negative. 440 00:25:36,090 --> 00:25:39,790 What does that tell me about the determinant? 441 00:25:39,790 --> 00:25:41,260 Let me ask again. 442 00:25:41,260 --> 00:25:44,990 If I know both the eigenvalues are negative, 443 00:25:44,990 --> 00:25:47,760 then I know the trace is negative 444 00:25:47,760 --> 00:25:53,170 but the determinant is positive, because it's 445 00:25:53,170 --> 00:25:56,450 the product of the two eigenvalues. 446 00:25:56,450 --> 00:26:00,580 So this determinant is lambda one times lambda two. 447 00:26:00,580 --> 00:26:04,540 This is -- this is lambda one times lambda two 448 00:26:04,540 --> 00:26:08,060 and if they're both negative, the product is positive. 449 00:26:08,060 --> 00:26:11,860 So positive determinant, negative trace. 450 00:26:11,860 --> 00:26:17,200 I can easily track down the -- this condition also for the -- 451 00:26:17,200 --> 00:26:20,380 if -- if there's an imaginary part hanging around. 452 00:26:20,380 --> 00:26:20,880 Okay. 453 00:26:20,880 --> 00:26:25,550 So that's a -- like a small but quite useful, 454 00:26:25,550 --> 00:26:29,820 because two by two is always important -- 455 00:26:29,820 --> 00:26:33,100 comment on stability. 456 00:26:33,100 --> 00:26:33,750 Okay. 457 00:26:33,750 --> 00:26:40,170 So let's just look at the picture again. 458 00:26:40,170 --> 00:26:41,290 Okay. 459 00:26:41,290 --> 00:26:43,980 The main part of my lecture, the one thing 460 00:26:43,980 --> 00:26:46,700 you want to be able to, like, just do automatically 461 00:26:46,700 --> 00:26:51,750 is this step of solving the system. 462 00:26:51,750 --> 00:26:54,190 Find the eigenvalues, find the eigenvectors, 463 00:26:54,190 --> 00:26:56,000 find the coefficients. 464 00:26:56,000 --> 00:27:01,090 And notice -- what's the matrix -- in this linear system here, 465 00:27:01,090 --> 00:27:05,590 I can't help -- if I -- if I have equations like that -- 466 00:27:05,590 --> 00:27:08,900 that's my equations column at a time -- 467 00:27:08,900 --> 00:27:11,710 what's the matrix form of that equation? 468 00:27:11,710 --> 00:27:18,580 So -- so this -- this system of equations is -- 469 00:27:18,580 --> 00:27:26,710 is some matrix multiplying c1, c2 to give u of zero. 470 00:27:26,710 --> 00:27:29,370 One zero. 471 00:27:29,370 --> 00:27:30,510 What's the matrix? 472 00:27:30,510 --> 00:27:35,400 Well, it's obviously two one, one minus one, 473 00:27:35,400 --> 00:27:37,760 but we have a name, or at least a letter -- 474 00:27:37,760 --> 00:27:40,090 actually a name for that matrix. 475 00:27:40,090 --> 00:27:42,670 Wh- what matrix are we s- are we -- 476 00:27:42,670 --> 00:27:47,390 are we dealing with here in this step of finding the c-s? 477 00:27:47,390 --> 00:27:50,690 Its letter is S -- 478 00:27:50,690 --> 00:27:52,340 it's the eigenvector matrix. 479 00:27:52,340 --> 00:27:52,970 Of course. 480 00:27:52,970 --> 00:27:55,230 These are the eigenvectors, there 481 00:27:55,230 --> 00:27:57,150 in the columns of our matrix. 482 00:27:57,150 --> 00:28:02,520 So this is S c equals u of zero -- 483 00:28:02,520 --> 00:28:09,640 is the -- that step where you find the actual coefficients, 484 00:28:09,640 --> 00:28:14,690 you find out how much of each pure exponential is 485 00:28:14,690 --> 00:28:16,910 in the solution. 486 00:28:16,910 --> 00:28:20,660 By getting it right at the start, then it stays right 487 00:28:20,660 --> 00:28:21,320 forever. 488 00:28:21,320 --> 00:28:24,820 I -- you're seeing this picture that each -- 489 00:28:24,820 --> 00:28:29,090 each pure exponential goes on its own way once you start it 490 00:28:29,090 --> 00:28:29,620 from u of 491 00:28:29,620 --> 00:28:30,490 zero. 492 00:28:30,490 --> 00:28:33,810 So you start it by figuring out how much 493 00:28:33,810 --> 00:28:37,880 each one is present in u of zero and then off they go. 494 00:28:37,880 --> 00:28:38,740 Okay. 495 00:28:38,740 --> 00:28:45,110 So -- so that's a system of two equations in two unknowns 496 00:28:45,110 --> 00:28:48,070 coupled -- 497 00:28:48,070 --> 00:28:53,630 the matrix sort of couples u1 and u2 and the eigenvalues 498 00:28:53,630 --> 00:28:57,770 and eigenvectors uncouple it, diagonalize it. 499 00:28:57,770 --> 00:29:00,830 Actually -- okay, now can I -- 500 00:29:00,830 --> 00:29:04,600 can I think in terms of S and lambda? 501 00:29:04,600 --> 00:29:07,330 So I want to write the solution down, 502 00:29:07,330 --> 00:29:10,840 again in terms of S and lambda. 503 00:29:10,840 --> 00:29:11,340 Okay. 504 00:29:11,340 --> 00:29:14,890 I'll do that on this far board. 505 00:29:14,890 --> 00:29:15,890 Okay. 506 00:29:15,890 --> 00:29:20,540 So I'm coming back to -- 507 00:29:20,540 --> 00:29:26,470 I'm coming back to our equation du/dt equals Au. 508 00:29:26,470 --> 00:29:35,890 Now this matrix A couples them. 509 00:29:35,890 --> 00:29:39,010 The whole point of eigenvectors is to uncouple. 510 00:29:39,010 --> 00:29:46,510 So one way to see that is introduce set u equal A -- 511 00:29:46,510 --> 00:29:47,010 not 512 00:29:47,010 --> 00:29:54,580 A. It's S, the eigenvector matrix that uncouples it. 513 00:29:54,580 --> 00:29:58,890 So I'm -- I'm taking this equation as I'm given, 514 00:29:58,890 --> 00:30:04,150 coupled with -- with A has -- is probably full of non-zeroes, 515 00:30:04,150 --> 00:30:08,130 but I'm -- by uncoupling it, I mean I'm diagonalizing it. 516 00:30:08,130 --> 00:30:11,630 If I can get a diagonal matrix, I'm -- 517 00:30:11,630 --> 00:30:12,470 I'm in. 518 00:30:12,470 --> 00:30:13,120 Okay. 519 00:30:13,120 --> 00:30:14,950 So I plug that in. 520 00:30:14,950 --> 00:30:18,720 This is A S v. 521 00:30:18,720 --> 00:30:20,813 And this is S dv/dt. 522 00:30:25,810 --> 00:30:26,890 S is a constant. 523 00:30:26,890 --> 00:30:30,050 It's -- this it the eigenvector matrix. 524 00:30:30,050 --> 00:30:31,635 This is the eigenvector matrix. 525 00:30:34,820 --> 00:30:35,330 Okay. 526 00:30:35,330 --> 00:30:37,550 Now I'm going to bring S inverse over here. 527 00:30:40,390 --> 00:30:41,285 And what have I got? 528 00:30:45,160 --> 00:30:53,470 That combination S inverse A S is lambda, the diagonal matrix. 529 00:30:53,470 --> 00:30:56,940 That's -- that's the point, that in -- 530 00:30:56,940 --> 00:31:01,400 if I'm using the eigenvectors as my basis, 531 00:31:01,400 --> 00:31:06,700 then my system of equations is just diagonal. 532 00:31:06,700 --> 00:31:10,420 I -- each -- there's no coupling anymore -- 533 00:31:10,420 --> 00:31:13,130 dv1/dt is lambda one v1. 534 00:31:13,130 --> 00:31:20,890 So let's just write that down. dv1/ dt is lambda one v1 535 00:31:20,890 --> 00:31:26,470 and so on for all n of the equations. 536 00:31:26,470 --> 00:31:29,610 It's a system of equations but they're not connected, 537 00:31:29,610 --> 00:31:32,865 so they're easy to solve and why don't I just 538 00:31:32,865 --> 00:31:33,865 write down the solution? 539 00:31:36,880 --> 00:31:47,160 v -- well, v is now some e to the lambda one t -- 540 00:31:47,160 --> 00:31:49,600 well, okay -- 541 00:31:49,600 --> 00:31:56,330 I guess my idea here now is to use, the natural notation -- 542 00:31:56,330 --> 00:32:04,740 v of T is e to the lambda tv of zero. 543 00:32:04,740 --> 00:32:16,150 And u of t will be Se to the lambda t S inverse, u of zero. 544 00:32:16,150 --> 00:32:20,990 This is the -- this is the, formula I'm headed for. 545 00:32:25,220 --> 00:32:29,530 This -- this matrix, S e to the lambda t S inverse, 546 00:32:29,530 --> 00:32:32,040 that's my exponential. 547 00:32:32,040 --> 00:32:40,483 That's my e to the A t, is this S e to the lambda t S inverse. 548 00:32:43,600 --> 00:32:47,310 So my -- my job really now is to explain what's going on with 549 00:32:47,310 --> 00:32:49,620 this matrix up in the exponential. 550 00:32:49,620 --> 00:32:51,610 What does that mean? 551 00:32:51,610 --> 00:32:54,110 What does it mean to say e to a matrix? 552 00:32:58,100 --> 00:33:01,680 This ought to be easier because this is e to a diagonal matrix, 553 00:33:01,680 --> 00:33:03,970 but still it's a matrix. 554 00:33:03,970 --> 00:33:07,350 What do we mean by e to the A t? 555 00:33:07,350 --> 00:33:13,120 Because really e to the A t is my answer here. 556 00:33:13,120 --> 00:33:18,620 My -- my answer to this equation is -- 557 00:33:18,620 --> 00:33:26,170 this u of t, this is my -- this is my e to the A t u of zero. 558 00:33:26,170 --> 00:33:30,907 So it's -- my job is really now to say what's -- 559 00:33:30,907 --> 00:33:31,740 what does that mean? 560 00:33:31,740 --> 00:33:33,780 What's the exponential of a matrix 561 00:33:33,780 --> 00:33:38,390 and why is that formula correct? 562 00:33:38,390 --> 00:33:38,950 Okay. 563 00:33:38,950 --> 00:33:42,190 So I'll put that on the board underneath. 564 00:33:42,190 --> 00:33:45,210 What's the exponential of a matrix? 565 00:33:45,210 --> 00:33:47,430 Let me come back here. 566 00:33:47,430 --> 00:33:49,460 So I'm talking about matrix exponentials. 567 00:33:55,040 --> 00:33:57,540 e to the At. 568 00:33:57,540 --> 00:33:58,350 Okay. 569 00:33:58,350 --> 00:34:01,090 How are we going to define the exponential of a -- 570 00:34:01,090 --> 00:34:01,735 of something? 571 00:34:04,490 --> 00:34:09,940 The trick -- the idea is -- the thing to go back to is 572 00:34:09,940 --> 00:34:15,860 exponential -- there's a power series for exponentials. 573 00:34:15,860 --> 00:34:19,690 That's how you -- you -- the good way to define e to the x 574 00:34:19,690 --> 00:34:25,469 is the power series one plus x plus one half x squared, 575 00:34:25,469 --> 00:34:29,489 one six x cubed and we'll do it now when the -- 576 00:34:29,489 --> 00:34:30,770 when we have a matrix. 577 00:34:30,770 --> 00:34:34,739 So the one becomes the identity, the x is At, 578 00:34:34,739 --> 00:34:42,620 the x squared is At squared and I divide by two. 579 00:34:42,620 --> 00:34:47,600 The cube, the x cube is At cubed over six, 580 00:34:47,600 --> 00:34:50,880 and what's the general term in here? 581 00:34:50,880 --> 00:34:55,929 At to the n-th power divided by -- 582 00:34:55,929 --> 00:34:57,710 and it goes on. 583 00:34:57,710 --> 00:35:01,430 But what do I divide by? 584 00:35:01,430 --> 00:35:05,350 So, you see the pattern -- here I divided by one, 585 00:35:05,350 --> 00:35:10,080 here I divided by one by two by six, those are the factorials. 586 00:35:10,080 --> 00:35:10,965 It's n factorial. 587 00:35:14,110 --> 00:35:17,445 That was, like, the one beautiful Taylor series. 588 00:35:20,300 --> 00:35:23,180 The one beautiful Taylor series -- well, there are two -- 589 00:35:23,180 --> 00:35:25,960 there are two beautiful Taylor series in this world. 590 00:35:25,960 --> 00:35:29,550 The Taylor series for e to the x is 591 00:35:29,550 --> 00:35:35,090 the s with x to the n-th over n factorial. 592 00:35:35,090 --> 00:35:38,680 And all I'm doing is doing the same thing for matrixes. 593 00:35:38,680 --> 00:35:40,850 The other beautiful Taylor series 594 00:35:40,850 --> 00:35:47,920 is just the sum of x to the n-th not divided by n factorial. 595 00:35:47,920 --> 00:35:51,300 Can you -- do you know what function that one is? 596 00:35:51,300 --> 00:35:53,990 So if I take -- this is the series, 597 00:35:53,990 --> 00:35:58,070 all these sums are going from zero to infinity. 598 00:35:58,070 --> 00:35:59,960 What's -- what function have I got -- 599 00:35:59,960 --> 00:36:02,770 this is like a side comment -- 600 00:36:02,770 --> 00:36:06,750 this is one plus x plus x squared plus x cubed plus x 601 00:36:06,750 --> 00:36:09,430 to the fourth not divided by anything, what's -- 602 00:36:09,430 --> 00:36:11,800 what's that function? 603 00:36:11,800 --> 00:36:15,380 One plus x plus x squared plus x cubed plus x fourth forever 604 00:36:15,380 --> 00:36:18,700 is one over one minus x. 605 00:36:18,700 --> 00:36:24,120 It's the geometric series, the nicest power series of all. 606 00:36:24,120 --> 00:36:28,850 So, actually, of course, there would be a similar thing here. 607 00:36:28,850 --> 00:36:36,810 If -- if I wanted, I minus A t inverse would be -- 608 00:36:36,810 --> 00:36:39,060 now I've got matrixes. 609 00:36:39,060 --> 00:36:43,150 I've got matrixes everywhere, but it's just like that series 610 00:36:43,150 --> 00:36:46,690 with -- and just like this one without the divisions. 611 00:36:46,690 --> 00:36:56,310 It's I plus At plus At squared plus At cubed and forever. 612 00:36:59,200 --> 00:37:02,460 So that's actually a -- a reasonable way to find 613 00:37:02,460 --> 00:37:04,660 the inverse of a matrix. 614 00:37:04,660 --> 00:37:07,500 If I chop it off -- 615 00:37:07,500 --> 00:37:10,120 well, it's reasonable if t is small. 616 00:37:10,120 --> 00:37:13,330 If t is a small number, then -- 617 00:37:13,330 --> 00:37:15,940 then t squared is extremely small, 618 00:37:15,940 --> 00:37:19,350 t cubed is even smaller, so approximately 619 00:37:19,350 --> 00:37:22,480 that inverse is I plus At. 620 00:37:22,480 --> 00:37:24,660 I can keep more terms if I like. 621 00:37:24,660 --> 00:37:25,830 Do you see what I'm doing? 622 00:37:25,830 --> 00:37:31,440 I'm just saying we can do the same thing for matrixes that we 623 00:37:31,440 --> 00:37:35,600 do for ordinary functions and the good thing about 624 00:37:35,600 --> 00:37:38,470 the exponential series -- so in a way, 625 00:37:38,470 --> 00:37:41,990 this series is better than this one. 626 00:37:41,990 --> 00:37:43,290 Why? 627 00:37:43,290 --> 00:37:45,410 Because this one always converges. 628 00:37:45,410 --> 00:37:48,440 I'm dividing by these bigger and bigger numbers, 629 00:37:48,440 --> 00:37:54,770 so whatever matrix A and however large t is, that series -- 630 00:37:54,770 --> 00:37:57,430 these terms go to zero. 631 00:37:57,430 --> 00:38:01,960 The series adds up to a finite sum, e to the At is a -- is -- 632 00:38:01,960 --> 00:38:04,390 is completely defined. 633 00:38:04,390 --> 00:38:08,710 Whereas this second guy could fail, right? 634 00:38:08,710 --> 00:38:11,730 If At is big -- 635 00:38:11,730 --> 00:38:15,190 somehow if At has an eigenvalue larger than one, 636 00:38:15,190 --> 00:38:18,690 then when I square it it'll have that eigenvalue squared, 637 00:38:18,690 --> 00:38:21,820 when I cube it the eigenvalue will be cubed -- 638 00:38:21,820 --> 00:38:26,560 that series will blow up unless the eigenvalues of At 639 00:38:26,560 --> 00:38:28,500 are smaller than one. 640 00:38:28,500 --> 00:38:32,150 So when the eigenvalues of At are smaller than one -- 641 00:38:32,150 --> 00:38:33,610 so I'd better put that in. 642 00:38:33,610 --> 00:38:38,260 The -- all eigenvalues of At below one -- 643 00:38:38,260 --> 00:38:42,230 then that series converges and gives me the inverse. 644 00:38:42,230 --> 00:38:42,970 Okay. 645 00:38:42,970 --> 00:38:47,590 So this is the guy I'm chiefly interested in, and I wanted 646 00:38:47,590 --> 00:38:51,820 to connect it to -- 647 00:38:51,820 --> 00:38:52,400 oh, okay. 648 00:38:52,400 --> 00:38:55,810 What's -- how do I -- how do I get -- this is my, like, 649 00:38:55,810 --> 00:38:58,160 main thing now to do -- 650 00:38:58,160 --> 00:39:02,270 how do I get e to the At -- 651 00:39:02,270 --> 00:39:05,760 how do I see that e to the At is the same as this? 652 00:39:10,450 --> 00:39:16,410 In other words, I can find e to the At by finding S and lambda, 653 00:39:16,410 --> 00:39:18,710 because now e to the lambda t 654 00:39:18,710 --> 00:39:21,350 is easy. 655 00:39:21,350 --> 00:39:24,820 Lambda's a diagonal matrix and we can write down either 656 00:39:24,820 --> 00:39:27,250 the lambda t -- and will right -- in a minute. 657 00:39:27,250 --> 00:39:29,810 But how -- do you see what -- 658 00:39:29,810 --> 00:39:33,290 do you see that we're hoping for a -- 659 00:39:33,290 --> 00:39:38,540 we're hoping that we can compute either the A T from S 660 00:39:38,540 --> 00:39:41,450 and lambda -- 661 00:39:41,450 --> 00:39:44,980 and I have to look at that definition and say, okay, 662 00:39:44,980 --> 00:39:48,660 how do -- how do I get an S and the lambda to come out of that? 663 00:39:48,660 --> 00:39:50,670 Okay, can -- do you see how I -- 664 00:39:50,670 --> 00:39:54,830 I want to connect that to that, from that definition. 665 00:39:54,830 --> 00:39:59,090 So let me erase this -- the geometric series, 666 00:39:59,090 --> 00:40:08,250 which isn't part of the differential equations case 667 00:40:08,250 --> 00:40:14,200 and get the S and the lambda into this picture. 668 00:40:14,200 --> 00:40:15,900 Oh, okay. 669 00:40:15,900 --> 00:40:16,420 Here we go. 670 00:40:19,210 --> 00:40:22,930 So identity is fine. 671 00:40:22,930 --> 00:40:26,880 Now -- all right, you -- you -- you'll see how I'm -- 672 00:40:26,880 --> 00:40:31,900 how I'm -- how I going to get A replaced by S and S is 673 00:40:31,900 --> 00:40:32,550 in lambda's? 674 00:40:32,550 --> 00:40:36,520 Well I use the fundamental formula of this whole chapter. 675 00:40:36,520 --> 00:40:43,540 A is S lambda S inverse and then times t. 676 00:40:43,540 --> 00:40:45,451 That's At. 677 00:40:45,451 --> 00:40:45,950 Okay. 678 00:40:45,950 --> 00:40:48,910 What's A squared t? 679 00:40:48,910 --> 00:40:51,450 I can -- I've got the divide by two, 680 00:40:51,450 --> 00:40:56,580 I've got the t squared and I've got an A squared. 681 00:40:56,580 --> 00:41:02,580 All right, I -- so I've got a -- there's A -- there's A. 682 00:41:02,580 --> 00:41:04,390 Now square it. 683 00:41:04,390 --> 00:41:05,880 So what happens when I square it? 684 00:41:05,880 --> 00:41:08,050 We've seen that before. 685 00:41:08,050 --> 00:41:16,120 When I square it, I get S lambda squared S inverse, right? 686 00:41:16,120 --> 00:41:20,070 When I square that thing, the -- there's an S and a -- 687 00:41:20,070 --> 00:41:23,840 an S cancels out an S inverse. 688 00:41:23,840 --> 00:41:25,900 I'm left with the S on the left, the S 689 00:41:25,900 --> 00:41:29,140 inverse on the right and lambda squared in the middle. 690 00:41:29,140 --> 00:41:31,660 And so on. 691 00:41:31,660 --> 00:41:36,680 The next one'll be S lambda cubed, S inverse -- 692 00:41:36,680 --> 00:41:39,590 times t cubed over three factorial. 693 00:41:39,590 --> 00:41:45,190 And now -- what do I do now? 694 00:41:45,190 --> 00:41:48,440 I want to pull an S out from everything. 695 00:41:48,440 --> 00:41:53,010 I want an S out of the whole thing. 696 00:41:53,010 --> 00:41:57,020 Well, look, I'd better write the identity how? 697 00:41:57,020 --> 00:42:01,250 I -- I want to be able to pull an S out and an S inverse out 698 00:42:01,250 --> 00:42:04,840 from the other side, so I just write the identity as S times S 699 00:42:04,840 --> 00:42:05,820 inverse. 700 00:42:05,820 --> 00:42:11,120 So I have an S there and an S inverse from this side 701 00:42:11,120 --> 00:42:13,240 and what have I got in the middle? 702 00:42:16,170 --> 00:42:18,241 If I pull out an S and an S inverse, 703 00:42:18,241 --> 00:42:19,490 what have I got in the middle? 704 00:42:19,490 --> 00:42:23,260 I've got the identity, a lambda t, 705 00:42:23,260 --> 00:42:26,650 a lambda squared t squared over two -- 706 00:42:26,650 --> 00:42:30,780 I've got e to the lambda t. 707 00:42:30,780 --> 00:42:32,600 That's what's in the middle. 708 00:42:32,600 --> 00:42:36,630 That's my formula for e to the At. 709 00:42:36,630 --> 00:42:39,120 Oh, now I have to ask you. 710 00:42:39,120 --> 00:42:42,290 Does this formula always work? 711 00:42:42,290 --> 00:42:45,420 This formula always works -- 712 00:42:45,420 --> 00:42:48,540 well, except it's an infinite series. 713 00:42:48,540 --> 00:42:51,800 But what do I mean by always work? 714 00:42:51,800 --> 00:42:55,300 And this one doesn't always work and I just 715 00:42:55,300 --> 00:42:58,460 have to remind you of what assumption 716 00:42:58,460 --> 00:43:00,660 is built into this formula that's 717 00:43:00,660 --> 00:43:03,580 not built into the original. 718 00:43:03,580 --> 00:43:07,900 The assumption that A can be diagonalized. 719 00:43:07,900 --> 00:43:11,660 You'll remember that there are some small -- 720 00:43:11,660 --> 00:43:14,770 sm- some subset of matrixes that don't 721 00:43:14,770 --> 00:43:18,000 have n independent eigenvectors, so we 722 00:43:18,000 --> 00:43:20,590 don't have an S inverse for those matrixes 723 00:43:20,590 --> 00:43:24,780 and the whole diagonalization breaks down. 724 00:43:24,780 --> 00:43:26,860 We could still make it triangular. 725 00:43:26,860 --> 00:43:28,010 I'll tell you about that. 726 00:43:28,010 --> 00:43:32,930 But diagonal we can't do for those particular degenerate 727 00:43:32,930 --> 00:43:37,310 matrixes that don't have enough independent eigenvectors. 728 00:43:37,310 --> 00:43:40,240 But otherwise, this is golden. 729 00:43:40,240 --> 00:43:40,970 Okay. 730 00:43:40,970 --> 00:43:44,780 So that's the formula -- that's the matrix exponential. 731 00:43:44,780 --> 00:43:48,680 Now it just remains for me to say what is e to the lambda t? 732 00:43:48,680 --> 00:43:50,460 Can I just do that? 733 00:43:50,460 --> 00:43:55,280 Let me just put that in the corner here. 734 00:43:55,280 --> 00:44:02,180 What is the exponential of a diagonal matrix? 735 00:44:02,180 --> 00:44:10,140 Remember lambda is diagonal, lambda one down to lambda n. 736 00:44:10,140 --> 00:44:14,520 What's the exponential of that diagonal matrix? 737 00:44:14,520 --> 00:44:19,550 Because our whole point is that this ought to be simple. 738 00:44:19,550 --> 00:44:23,240 Our whole point is that to take the exponential of a diagonal 739 00:44:23,240 --> 00:44:27,560 matrix ought to be completely decoupled -- 740 00:44:27,560 --> 00:44:30,300 it ought to be diagonal, in other words, and it is. 741 00:44:30,300 --> 00:44:37,010 It's just e to the lambda one t, e to the lambda two t, 742 00:44:37,010 --> 00:44:41,070 e to the lambda n t, all zeroes. 743 00:44:41,070 --> 00:44:47,620 So -- so if we have a diagonal matrix and I plug it into this 744 00:44:47,620 --> 00:44:52,140 exponential formula, everything's diagonal 745 00:44:52,140 --> 00:44:55,710 and the diagonal terms are just the ordinary scaler 746 00:44:55,710 --> 00:44:58,930 exponentials e to the lambda one t. 747 00:44:58,930 --> 00:45:01,840 Okay, so that's -- that's -- 748 00:45:01,840 --> 00:45:06,050 in a sense, I'm doing here, on this board, with -- with, like, 749 00:45:06,050 --> 00:45:11,100 formulas what I did on the -- 750 00:45:11,100 --> 00:45:15,940 in the first half of the lecture with specific matrix A 751 00:45:15,940 --> 00:45:19,270 and specific eigenvalues and eigenvectors. 752 00:45:19,270 --> 00:45:21,990 The -- so let me show you the formulas again. 753 00:45:21,990 --> 00:45:24,770 But the -- so you -- 754 00:45:24,770 --> 00:45:27,300 I guess -- what should you know from this 755 00:45:27,300 --> 00:45:28,300 lecture? 756 00:45:28,300 --> 00:45:34,180 You should know what this matrix exponential is and, like, 757 00:45:34,180 --> 00:45:36,670 when does it go to zero? 758 00:45:36,670 --> 00:45:38,390 Tell me again, now, the answer to that. 759 00:45:38,390 --> 00:45:41,350 When does e to the At approach -- 760 00:45:41,350 --> 00:45:45,510 get smaller and smaller as t increases? 761 00:45:45,510 --> 00:45:49,070 Well, the S and the S inverse aren't moving. 762 00:45:49,070 --> 00:45:51,780 It's this one that has to get smaller and smaller 763 00:45:51,780 --> 00:45:57,160 and that one has this simple diagonal form. 764 00:45:57,160 --> 00:46:01,740 And it goes to zero provided every one of these lambdas -- 765 00:46:01,740 --> 00:46:04,840 I -- I need to have each one of these guys go to zero, 766 00:46:04,840 --> 00:46:09,930 so I need every real part of every eigenvalue negative. 767 00:46:12,650 --> 00:46:13,230 Right? 768 00:46:13,230 --> 00:46:15,690 If the real part is negative, that's -- 769 00:46:15,690 --> 00:46:19,160 that takes the exponential -- that forces -- 770 00:46:19,160 --> 00:46:21,420 the exponential goes to zero. 771 00:46:21,420 --> 00:46:24,480 Okay, so that -- that's really the difference. 772 00:46:24,480 --> 00:46:33,250 If I can just draw the -- here's a picture of the -- of the -- 773 00:46:33,250 --> 00:46:36,780 this is the complex plain. 774 00:46:36,780 --> 00:46:42,080 Here's the real axis and here's the imaginary axis. 775 00:46:42,080 --> 00:46:43,960 And where do the eigenvalues have 776 00:46:43,960 --> 00:46:47,570 to be for stability in differential equations? 777 00:46:47,570 --> 00:46:52,470 They have to be over here, in the left half plain. 778 00:46:52,470 --> 00:46:55,910 So the left half plain is this plain, real part of lambda, 779 00:46:55,910 --> 00:46:58,820 less than zero. 780 00:46:58,820 --> 00:47:01,190 Where do the ma- where do the eigenvalues have 781 00:47:01,190 --> 00:47:06,500 to be for powers of the matrix to go to zero? 782 00:47:06,500 --> 00:47:11,960 Powers of the matrix go to zero if the eigenvalues are in here. 783 00:47:11,960 --> 00:47:17,800 So this is the stability region for powers -- 784 00:47:17,800 --> 00:47:22,490 this is the region -- absolute value of lambda, less than one. 785 00:47:22,490 --> 00:47:26,740 That's the stability for -- that tells us that the powers of A 786 00:47:26,740 --> 00:47:30,260 go to zero, this tells us that the exponential of A goes 787 00:47:30,260 --> 00:47:31,220 to zero. 788 00:47:31,220 --> 00:47:31,790 Okay. 789 00:47:31,790 --> 00:47:33,700 One final example. 790 00:47:33,700 --> 00:47:38,580 Let me just write down how to deal with a final example. 791 00:47:38,580 --> 00:47:39,980 Let's see. 792 00:47:44,480 --> 00:47:48,930 So my final example will be a single equation, u''+bu'+Ku=0. 793 00:47:57,040 --> 00:48:01,000 One equation, second order. 794 00:48:01,000 --> 00:48:03,490 How do I -- 795 00:48:03,490 --> 00:48:05,290 and maybe I should have used -- 796 00:48:05,290 --> 00:48:08,170 I'll use -- I prefer to use y here, 797 00:48:08,170 --> 00:48:12,170 because that's what you see in differential equations. 798 00:48:12,170 --> 00:48:14,790 And I want u to be a vector. 799 00:48:14,790 --> 00:48:23,730 So how do I change one second order equation into a two 800 00:48:23,730 --> 00:48:28,250 by two first order system? 801 00:48:28,250 --> 00:48:30,540 Just the way I did for Fibonacci. 802 00:48:30,540 --> 00:48:38,180 I'll let u be y prime and y. 803 00:48:38,180 --> 00:48:43,210 What I'm going to do is I'm going to add an extra equation, 804 00:48:43,210 --> 00:48:46,620 y prime equals y prime. 805 00:48:46,620 --> 00:48:50,800 So I take this -- so by -- 806 00:48:50,800 --> 00:48:55,110 so using this as the vector unknown, 807 00:48:55,110 --> 00:48:58,830 now my equation is u prime. 808 00:48:58,830 --> 00:49:00,800 My first order system is u prime, 809 00:49:00,800 --> 00:49:06,160 that'll be y double prime y prime, the derivative of u, 810 00:49:06,160 --> 00:49:10,940 okay, now the differential equation is telling me that y 811 00:49:10,940 --> 00:49:14,430 double prime is m- so I'm just looking for -- 812 00:49:14,430 --> 00:49:17,160 what's this matrix? 813 00:49:17,160 --> 00:49:19,410 y prime y. 814 00:49:19,410 --> 00:49:23,220 I'm looking for the matrix A. 815 00:49:23,220 --> 00:49:28,460 What's the matrix in case I have a single first order equation 816 00:49:28,460 --> 00:49:31,140 and I want to make it into a two by two system? 817 00:49:31,140 --> 00:49:32,270 Okay, simple. 818 00:49:32,270 --> 00:49:35,920 The first row of the matrix is given by the equation. 819 00:49:35,920 --> 00:49:43,800 So y''-by'-ky -- no problem. 820 00:49:43,800 --> 00:49:47,240 And what's the second row on the matrix? 821 00:49:47,240 --> 00:49:48,660 Then we're done. 822 00:49:48,660 --> 00:49:50,710 The second row of the matrix I want just 823 00:49:50,710 --> 00:49:54,490 to be the trivial equation y prime equals y prime, 824 00:49:54,490 --> 00:49:56,340 so I need a one and a zero there. 825 00:49:59,240 --> 00:50:03,950 So matrixes like these, the gen- the general case, 826 00:50:03,950 --> 00:50:09,050 if I had a five by five -- if I had a fifth order equation 827 00:50:09,050 --> 00:50:11,590 and I wanted a five by five matrix, 828 00:50:11,590 --> 00:50:15,850 I would see the coefficients of the equation up there and then 829 00:50:15,850 --> 00:50:21,260 my four trivial equations would put ones here. 830 00:50:21,260 --> 00:50:27,110 This is the kind of matrix that goes from a fifth order 831 00:50:27,110 --> 00:50:32,060 to a five by five first order. 832 00:50:35,140 --> 00:50:40,010 So the -- and the eigenvalues will come out in a natural way 833 00:50:40,010 --> 00:50:41,350 connected to the differential 834 00:50:41,350 --> 00:50:41,970 equation. 835 00:50:41,970 --> 00:50:45,840 Okay, that's differential equations. 836 00:50:45,840 --> 00:50:49,890 The -- a parallel lecture compared to powers of a matrix 837 00:50:49,890 --> 00:50:52,060 we can now do exponentials. 838 00:50:52,060 --> 00:50:53,610 Thanks.