1 00:00:08,280 --> 00:00:09,650 Okay. 2 00:00:09,650 --> 00:00:15,450 This lecture is mostly about the idea of similar matrixes. 3 00:00:15,450 --> 00:00:18,520 I'm going to tell you what that word similar means 4 00:00:18,520 --> 00:00:22,990 and in what way two matrixes are called similar. 5 00:00:22,990 --> 00:00:25,870 But before I do that, I have a little more 6 00:00:25,870 --> 00:00:28,310 to say about positive definite matrixes. 7 00:00:28,310 --> 00:00:34,880 You can tell this is a subject I think is really important and I 8 00:00:34,880 --> 00:00:39,170 told you what positive definite meant -- 9 00:00:39,170 --> 00:00:41,430 it means that this -- 10 00:00:41,430 --> 00:00:45,490 this expression, this quadratic form, x transpose I 11 00:00:45,490 --> 00:00:48,010 x is always positive. 12 00:00:48,010 --> 00:00:51,710 But the direct way to test it was with eigenvalues 13 00:00:51,710 --> 00:00:52,940 or pivots or determinants. 14 00:00:55,850 --> 00:00:58,730 So I -- we know what it means, we know how to test it, 15 00:00:58,730 --> 00:01:03,820 but I didn't really say where positive definite matrixes come 16 00:01:03,820 --> 00:01:05,140 from. 17 00:01:05,140 --> 00:01:10,470 And so one thing I want to say is that they come from least 18 00:01:10,470 --> 00:01:15,780 squares in -- and all sorts of physical problems start with 19 00:01:15,780 --> 00:01:19,520 a rectangular matrix -- well, you remember in least squares 20 00:01:19,520 --> 00:01:23,810 the crucial combination was A transpose A. 21 00:01:23,810 --> 00:01:27,970 So I want to show that that's a positive definite matrix. 22 00:01:27,970 --> 00:01:28,727 Can -- so I -- 23 00:01:28,727 --> 00:01:31,060 I'm going to speak a little more about positive definite 24 00:01:31,060 --> 00:01:34,730 matrixes, just recapping -- 25 00:01:34,730 --> 00:01:38,400 so let me ask a question. 26 00:01:38,400 --> 00:01:40,910 It may be on the homework. 27 00:01:40,910 --> 00:01:45,330 Suppose a matrix A is positive definite. 28 00:01:45,330 --> 00:01:48,360 I mean by that it's all -- 29 00:01:48,360 --> 00:01:50,100 I'm assuming it's symmetric. 30 00:01:50,100 --> 00:01:52,760 That's always built into the definition. 31 00:01:52,760 --> 00:01:56,440 So we have a symmetric positive definite matrix. 32 00:01:56,440 --> 00:01:58,610 What about its inverse? 33 00:01:58,610 --> 00:02:03,180 Is the inverse of a symmetric positive definite matrix also 34 00:02:03,180 --> 00:02:06,310 symmetric positive definite? 35 00:02:06,310 --> 00:02:09,789 So you quickly think, okay, what do I 36 00:02:09,789 --> 00:02:15,240 know about the pivots of the inverse matrix? 37 00:02:15,240 --> 00:02:17,390 Not much. 38 00:02:17,390 --> 00:02:19,620 What do I know about the eigenvalues 39 00:02:19,620 --> 00:02:22,540 of the inverse matrix? 40 00:02:22,540 --> 00:02:24,480 Everything, right? 41 00:02:24,480 --> 00:02:27,310 The eigenvalues of the inverse are 42 00:02:27,310 --> 00:02:31,290 one over the eigenvalues of the matrix. 43 00:02:31,290 --> 00:02:34,680 So if my matrix starts out positive definite, 44 00:02:34,680 --> 00:02:39,490 then right away I know that its inverse is positive definite, 45 00:02:39,490 --> 00:02:42,440 because those positive eigenvalues -- 46 00:02:42,440 --> 00:02:47,560 then one over the eigenvalue is also positive. 47 00:02:47,560 --> 00:02:51,820 What if I know that A -- a matrix A and a matrix B are 48 00:02:51,820 --> 00:02:53,280 both positive definite? 49 00:02:53,280 --> 00:02:54,780 But let me ask you this. 50 00:02:54,780 --> 00:03:04,140 Suppose if A and B are positive definite, what about -- 51 00:03:04,140 --> 00:03:05,940 what about A plus B? 52 00:03:09,020 --> 00:03:11,600 In some way, you hope that that would be true. 53 00:03:11,600 --> 00:03:15,730 It's -- positive definite for a matrix is kind of like positive 54 00:03:15,730 --> 00:03:18,540 for a real number. 55 00:03:18,540 --> 00:03:22,300 But we don't know the eigenvalues of A plus B. 56 00:03:22,300 --> 00:03:24,860 We don't know the pivots of A plus B. 57 00:03:24,860 --> 00:03:27,870 So we just, like, have to go down this list of, all right, 58 00:03:27,870 --> 00:03:31,130 which approach to positive definite 59 00:03:31,130 --> 00:03:33,050 can we get a handle on? 60 00:03:33,050 --> 00:03:35,570 And this is a good one. 61 00:03:35,570 --> 00:03:36,670 This is a good one. 62 00:03:36,670 --> 00:03:39,190 Can we -- how would we decide that -- 63 00:03:39,190 --> 00:03:43,610 if A was like this and if B was like this, 64 00:03:43,610 --> 00:03:48,410 then we would look at x transpose A plus B x. 65 00:03:48,410 --> 00:03:51,280 I'm sure this is in the homework. 66 00:03:51,280 --> 00:03:55,320 Now -- so we have x transpose A x bigger than zero, 67 00:03:55,320 --> 00:03:59,940 x transpose B x positive for all -- for all x, 68 00:03:59,940 --> 00:04:02,180 so now I ask you about this 69 00:04:02,180 --> 00:04:03,090 guy. 70 00:04:03,090 --> 00:04:06,800 And of course, you just add that and that 71 00:04:06,800 --> 00:04:09,940 and we get what we want. 72 00:04:09,940 --> 00:04:14,320 If A and B are positive definites, so is A plus B. 73 00:04:14,320 --> 00:04:15,810 So that's what I've shown. 74 00:04:15,810 --> 00:04:20,910 So is A plus B. 75 00:04:20,910 --> 00:04:27,310 Just -- be sort of ready for all the approaches through 76 00:04:27,310 --> 00:04:31,260 eigenvalues and through this expression. 77 00:04:31,260 --> 00:04:35,500 And now, finally, one more thought about positive definite 78 00:04:35,500 --> 00:04:39,020 is this combination that came up in least squares. 79 00:04:39,020 --> 00:04:41,260 Can I do that? 80 00:04:41,260 --> 00:04:48,290 So now -- now suppose A is rectangular, m by n. 81 00:04:52,890 --> 00:04:58,380 I -- so I'm sorry that I've used the same letter A 82 00:04:58,380 --> 00:05:02,730 for the positive definite matrixes in the eigenvalue 83 00:05:02,730 --> 00:05:07,660 chapter that I used way back in earlier chapters when 84 00:05:07,660 --> 00:05:09,620 the matrix was rectangular. 85 00:05:09,620 --> 00:05:12,750 Now, that matrix -- a rectangular matrix, 86 00:05:12,750 --> 00:05:14,569 no way its positive definite. 87 00:05:14,569 --> 00:05:15,360 It's not symmetric. 88 00:05:18,240 --> 00:05:20,450 It's not even square in general. 89 00:05:20,450 --> 00:05:25,540 But you remember that the key for these rectangular ones 90 00:05:25,540 --> 00:05:28,147 was A transpose A. 91 00:05:28,147 --> 00:05:28,730 That's square. 92 00:05:34,922 --> 00:05:35,630 That's symmetric. 93 00:05:38,690 --> 00:05:41,310 Those are things we knew -- 94 00:05:41,310 --> 00:05:44,890 we knew back when we met this thing 95 00:05:44,890 --> 00:05:49,020 in the least square stuff, in the projection stuff. 96 00:05:49,020 --> 00:05:52,340 But now we know something more -- 97 00:05:52,340 --> 00:05:56,450 we can ask a more important question, a deeper question -- 98 00:05:56,450 --> 00:05:57,920 is it positive definite? 99 00:06:01,110 --> 00:06:02,770 And we sort of hope so. 100 00:06:02,770 --> 00:06:04,730 Like, we -- we might -- 101 00:06:04,730 --> 00:06:09,200 in analogy with numbers, this is like -- 102 00:06:09,200 --> 00:06:15,360 sort of like the square of a number, and that's positive. 103 00:06:15,360 --> 00:06:19,930 So now I want to ask the matrix question. 104 00:06:19,930 --> 00:06:24,350 Is A transpose A positive definite? 105 00:06:24,350 --> 00:06:27,740 Okay, now it's -- so again, it's a rectangular A that 106 00:06:27,740 --> 00:06:31,790 I'm starting with, but it's the combination A transpose A 107 00:06:31,790 --> 00:06:36,150 that's the square, symmetric and hopefully positive definite 108 00:06:36,150 --> 00:06:36,780 matrix. 109 00:06:36,780 --> 00:06:41,700 So how -- how do I see that it is positive definite, 110 00:06:41,700 --> 00:06:43,750 or at least positive semi-definite? 111 00:06:43,750 --> 00:06:44,440 You'll see that. 112 00:06:47,180 --> 00:06:52,280 Well, I don't know the eigenvalues of this product. 113 00:06:52,280 --> 00:06:54,250 I don't want to work with the pivots. 114 00:06:54,250 --> 00:07:00,060 The right thing -- the right quantity to look at is this, 115 00:07:00,060 --> 00:07:01,800 x transpose Ax -- 116 00:07:01,800 --> 00:07:07,450 A -- x transpose times my matrix times x. 117 00:07:07,450 --> 00:07:09,280 I'd like to see that this thing -- 118 00:07:09,280 --> 00:07:13,450 that that expression is always positive. 119 00:07:13,450 --> 00:07:16,310 I'm not doing it with numbers, I'm doing it with symbols. 120 00:07:16,310 --> 00:07:21,890 Do you see -- how do I see that that expression comes out 121 00:07:21,890 --> 00:07:23,620 positive? 122 00:07:23,620 --> 00:07:29,260 I'm taking a rectangular matrix A and an A transpose -- 123 00:07:29,260 --> 00:07:31,270 that gives me something square symmetric, 124 00:07:31,270 --> 00:07:34,030 but now I want to see that if I multiply -- 125 00:07:34,030 --> 00:07:34,970 that if I do this -- 126 00:07:34,970 --> 00:07:38,140 I form this quadratic expression that I 127 00:07:38,140 --> 00:07:42,160 get this positive thing that goes upwards when I graph it. 128 00:07:42,160 --> 00:07:44,210 How do I see that that's positive, 129 00:07:44,210 --> 00:07:47,860 or absolutely it isn't negative anyway? 130 00:07:47,860 --> 00:07:51,990 We'll have to, like, spend a minute on the question 131 00:07:51,990 --> 00:07:54,620 could it be zero, but it can't be negative. 132 00:07:54,620 --> 00:07:56,310 Why can this never be negative? 133 00:07:59,060 --> 00:08:02,590 The argument is -- 134 00:08:02,590 --> 00:08:08,930 like the one key idea in so many steps in linear algebra -- 135 00:08:08,930 --> 00:08:12,960 put those parentheses in a good way. 136 00:08:12,960 --> 00:08:19,560 Put the parentheses around Ax and what's the first part? 137 00:08:19,560 --> 00:08:22,280 What's this x transpose A transpose? 138 00:08:22,280 --> 00:08:25,540 That is Ax transpose. 139 00:08:29,000 --> 00:08:30,810 So what do we have? 140 00:08:30,810 --> 00:08:35,650 We have the length squared of Ax. 141 00:08:35,650 --> 00:08:39,909 We have -- that's the column vector Ax that's the row vector 142 00:08:39,909 --> 00:08:48,380 Ax, its length squared, certainly greater than 143 00:08:48,380 --> 00:08:51,310 or possibly equal to zero. 144 00:08:51,310 --> 00:08:53,860 So we have to deal with this little possibility. 145 00:08:53,860 --> 00:08:55,070 Could it be equal? 146 00:08:55,070 --> 00:08:58,970 Well, when could the length squared be zero? 147 00:08:58,970 --> 00:09:02,320 Only if the vector is zero, right? 148 00:09:02,320 --> 00:09:08,340 That's the only vector that has length squared zero. 149 00:09:08,340 --> 00:09:11,080 So we have -- we would like to -- 150 00:09:11,080 --> 00:09:15,230 I would like to get that possibility out of there. 151 00:09:15,230 --> 00:09:18,920 So I want to have Ax never -- never be zero, 152 00:09:18,920 --> 00:09:21,340 except of course for the zero vector. 153 00:09:21,340 --> 00:09:24,280 How do I assure that Ax is never zero? 154 00:09:24,280 --> 00:09:27,650 The -- in other words, how do I show that there's no null space 155 00:09:27,650 --> 00:09:28,370 of A? 156 00:09:28,370 --> 00:09:33,420 The rank should be -- 157 00:09:33,420 --> 00:09:39,830 so now remember -- what's the rank when there's no null 158 00:09:39,830 --> 00:09:41,600 space? 159 00:09:41,600 --> 00:09:43,680 By no null space, you know what I mean. 160 00:09:43,680 --> 00:09:46,370 Only the zero vector in the null space. 161 00:09:46,370 --> 00:09:52,850 So if I have a -- if I have an 11 by 5 matrix -- 162 00:09:52,850 --> 00:09:59,230 so it's got 11 rows, 5 columns, when is there no null space? 163 00:09:59,230 --> 00:10:04,380 So the columns should be independent -- what's the rank? 164 00:10:04,380 --> 00:10:06,810 n 5 -- rank n. 165 00:10:06,810 --> 00:10:12,170 Independent columns, when -- so if I -- 166 00:10:12,170 --> 00:10:15,830 then I conclude yes, positive definite. 167 00:10:15,830 --> 00:10:19,020 And this was the assumption -- then A transpose A is 168 00:10:19,020 --> 00:10:23,210 invertible -- 169 00:10:23,210 --> 00:10:27,330 the least squares equations all work fine. 170 00:10:27,330 --> 00:10:33,870 And more than that -- the matrix is even positive definite. 171 00:10:33,870 --> 00:10:38,150 And I just to say one comment about numerical things, 172 00:10:38,150 --> 00:10:41,770 with a positive definite matrix, you never 173 00:10:41,770 --> 00:10:45,010 have to do row exchanges. 174 00:10:45,010 --> 00:10:50,230 You never run into unsuitably small numbers or zeroes 175 00:10:50,230 --> 00:10:51,560 in the pivot position. 176 00:10:51,560 --> 00:10:54,960 They're the right -- they're the great matrixes to compute with, 177 00:10:54,960 --> 00:10:57,250 and they're the great matrixes to study. 178 00:10:57,250 --> 00:11:05,900 So that's -- I wanted to take this first ten minutes of grab 179 00:11:05,900 --> 00:11:12,150 the first ten minutes away from similar matrixes and continue 180 00:11:12,150 --> 00:11:14,710 a -- this much more with positive definite. 181 00:11:14,710 --> 00:11:19,270 I'm really at this point, now, coming close 182 00:11:19,270 --> 00:11:24,300 to the end of the heart of linear algebra. 183 00:11:24,300 --> 00:11:28,160 The positive definiteness brought everything together. 184 00:11:28,160 --> 00:11:32,920 Similar matrixes, which is coming the rest of this hour 185 00:11:32,920 --> 00:11:38,400 is a key topic, and please come on Monday. 186 00:11:38,400 --> 00:11:42,690 Monday is about what's called the SVD, singular values. 187 00:11:42,690 --> 00:11:49,340 It's the -- has become a central fact in -- 188 00:11:49,340 --> 00:11:53,530 a central part of linear algebra. 189 00:11:53,530 --> 00:11:57,940 I mean, you can come after Monday also, but -- 190 00:11:57,940 --> 00:12:04,800 Monday is, -- that singular value thing has made it 191 00:12:04,800 --> 00:12:05,990 into this course. 192 00:12:05,990 --> 00:12:08,870 Ten years ago, five years ago it wasn't in the course, 193 00:12:08,870 --> 00:12:10,640 now it has to be. 194 00:12:10,640 --> 00:12:11,490 Okay. 195 00:12:11,490 --> 00:12:16,560 So can I begin today's lecture proper with this idea 196 00:12:16,560 --> 00:12:18,610 of similar matrixes. 197 00:12:18,610 --> 00:12:22,220 This is what similar matrixes mean. 198 00:12:22,220 --> 00:12:23,680 So here -- let's start again. 199 00:12:23,680 --> 00:12:24,720 I'll write it again. 200 00:12:24,720 --> 00:12:27,690 So A and B are similar. 201 00:12:27,690 --> 00:12:33,550 A and B are -- now I'm -- these matrixes -- 202 00:12:33,550 --> 00:12:38,720 I'm no longer talking about symmetric matrixes, in -- 203 00:12:38,720 --> 00:12:42,750 at least no longer expecting symmetric matrixes. 204 00:12:42,750 --> 00:12:46,960 I'm talking about two square matrixes n by n. 205 00:12:46,960 --> 00:12:51,790 A and B, they're n by n matrixes. 206 00:12:54,760 --> 00:12:59,110 And I'm introducing this word similar. 207 00:12:59,110 --> 00:13:00,880 So I'm going to say what does it mean? 208 00:13:00,880 --> 00:13:08,520 It means that they're connected in the way -- 209 00:13:08,520 --> 00:13:12,920 well, in the way I've written here, so let me rewrite it. 210 00:13:12,920 --> 00:13:24,260 That means that for some matrix M, which has to be invertible, 211 00:13:24,260 --> 00:13:26,140 because you'll see that -- 212 00:13:26,140 --> 00:13:29,680 this one matrix is -- 213 00:13:29,680 --> 00:13:32,980 take the other matrix, multiply on the right 214 00:13:32,980 --> 00:13:39,700 by M and on the left by M inverse. 215 00:13:39,700 --> 00:13:43,810 So the question is, why that combination? 216 00:13:43,810 --> 00:13:46,890 But part of the answer you know already. 217 00:13:46,890 --> 00:13:52,070 You remember -- we've done this -- we've taken a matrix A -- 218 00:13:52,070 --> 00:13:58,620 so let's do an example of similar. 219 00:13:58,620 --> 00:14:06,340 Suppose A -- the matrix A -- suppose it has a full set 220 00:14:06,340 --> 00:14:09,090 of eigenvectors. 221 00:14:09,090 --> 00:14:12,910 They go in this eigenvector matrix S. 222 00:14:12,910 --> 00:14:15,220 Then what was the main point of the whole -- 223 00:14:15,220 --> 00:14:19,960 the main calculation of the whole chapter was -- is -- 224 00:14:19,960 --> 00:14:25,380 use that eigenvector matrix S and its inverse 225 00:14:25,380 --> 00:14:33,720 comes over there to produce the nicest possible matrix lambda. 226 00:14:33,720 --> 00:14:38,160 Nicest possible because it's diagonal. 227 00:14:38,160 --> 00:14:47,615 So in our new language, this is saying A is similar to lambda. 228 00:14:51,440 --> 00:14:56,440 A is similar to lambda, because there is a matrix, 229 00:14:56,440 --> 00:14:58,970 and this particular -- 230 00:14:58,970 --> 00:15:01,140 there is an M and this particular M 231 00:15:01,140 --> 00:15:05,220 is this important guy, this eigenvector matrix. 232 00:15:05,220 --> 00:15:13,040 But if I take a different matrix M and I look at M inverse A M, 233 00:15:13,040 --> 00:15:15,350 the result won't come out diagonal, 234 00:15:15,350 --> 00:15:21,140 but it will come out a matrix B that's similar to A. 235 00:15:21,140 --> 00:15:25,870 Do you see that I'm -- what I'm doing is, like -- 236 00:15:25,870 --> 00:15:28,700 I'm putting these matrixes into families. 237 00:15:28,700 --> 00:15:34,060 All the matrixes in one -- in the family are similar to each 238 00:15:34,060 --> 00:15:35,070 other. 239 00:15:35,070 --> 00:15:39,080 They're all -- each one in this family is connected to each 240 00:15:39,080 --> 00:15:42,550 other one by some matrix M and the -- 241 00:15:42,550 --> 00:15:47,660 like the outstanding member of the family is the diagonal guy. 242 00:15:47,660 --> 00:15:50,700 I mean, that's the simplest, neatest matrix 243 00:15:50,700 --> 00:15:55,680 in this family of all the matrixes that are similar to A, 244 00:15:55,680 --> 00:15:58,530 the best one is lambda. 245 00:15:58,530 --> 00:16:01,720 But there are lots of others, because I can take different -- 246 00:16:01,720 --> 00:16:04,870 instead of S, I can take any old matrix M, 247 00:16:04,870 --> 00:16:08,440 any old invertible matrix and -- and do it. 248 00:16:08,440 --> 00:16:10,020 I'd better do an example. 249 00:16:10,020 --> 00:16:10,520 Okay. 250 00:16:10,520 --> 00:16:16,901 Suppose I take A as the matrix two one one two. 251 00:16:16,901 --> 00:16:17,400 Okay. 252 00:16:21,510 --> 00:16:25,470 Do you know the eigenvalue matrix for that? 253 00:16:25,470 --> 00:16:28,630 The eigenvalues of that matrix are -- 254 00:16:28,630 --> 00:16:33,040 well, three and one. 255 00:16:33,040 --> 00:16:37,430 So that -- and the eigenvectors would be easy to find. 256 00:16:37,430 --> 00:16:40,780 So this matrix is similar to this one. 257 00:16:40,780 --> 00:16:43,080 But my point is -- 258 00:16:43,080 --> 00:16:49,410 but also, I can also take my matrix, two one one two, 259 00:16:49,410 --> 00:16:52,090 I could multiply it by -- let's see, what -- 260 00:16:52,090 --> 00:16:55,580 I'm just going to cook up a matrix M here. 261 00:16:55,580 --> 00:17:00,350 I'm -- I'll -- let me just invent -- one four one zero. 262 00:17:00,350 --> 00:17:02,710 And over here I'll put M inverse, 263 00:17:02,710 --> 00:17:05,839 and because I happened to make that triangular, 264 00:17:05,839 --> 00:17:09,940 I know that its inverse is that, right? 265 00:17:09,940 --> 00:17:13,819 So there's M inverse A M, that's going to produce some matrix -- 266 00:17:13,819 --> 00:17:17,200 oh, well, I've got to do the multiplication, 267 00:17:17,200 --> 00:17:19,380 so hang on a second, let -- 268 00:17:19,380 --> 00:17:22,880 I'll just copy that one minus four zero one 269 00:17:22,880 --> 00:17:33,930 and multiply these guys so I'm getting two nine one and six, 270 00:17:33,930 --> 00:17:36,010 I think. 271 00:17:36,010 --> 00:17:39,680 Can you check it as I go, because you -- see I'm just -- 272 00:17:39,680 --> 00:17:43,850 so that's two minus four, I'm getting a minus two nine 273 00:17:43,850 --> 00:17:48,550 minus 24 is a minus 15, my God, how did I get this? 274 00:17:48,550 --> 00:17:50,935 And that's probably one and six. 275 00:17:54,560 --> 00:17:57,810 So there's my matrix B. 276 00:17:57,810 --> 00:18:02,180 And there's my matrix lambda, there's my matrix A 277 00:18:02,180 --> 00:18:04,715 and my point is these are all similar matrixes. 278 00:18:07,250 --> 00:18:09,880 They all have something in common, 279 00:18:09,880 --> 00:18:13,130 besides being just two by two. 280 00:18:13,130 --> 00:18:17,190 They have something in common. 281 00:18:17,190 --> 00:18:21,310 And that's -- and what is it? 282 00:18:21,310 --> 00:18:26,540 What's the point about two matrixes that are built out 283 00:18:26,540 --> 00:18:27,840 of -- 284 00:18:27,840 --> 00:18:32,440 the B is built out of M inverse A M. 285 00:18:32,440 --> 00:18:34,840 What is it that A and B have in common? 286 00:18:34,840 --> 00:18:38,127 That's the main -- now I'm telling you the main fact about 287 00:18:38,127 --> 00:18:38,835 similar matrixes. 288 00:18:41,590 --> 00:18:44,910 They have the same eigenvalues. 289 00:18:44,910 --> 00:18:47,550 This is -- this chapter is about eigenvalues, 290 00:18:47,550 --> 00:18:51,350 and that's why we're interested in this family of matrixes that 291 00:18:51,350 --> 00:18:53,140 have the same eigenvalues. 292 00:18:53,140 --> 00:18:56,640 What are the eigenvalues in this example? 293 00:18:56,640 --> 00:18:57,570 Lambda. 294 00:18:57,570 --> 00:19:01,070 The eigenvalues of that I could compute. 295 00:19:01,070 --> 00:19:06,370 The eigenvalues of that I can compute really fast. 296 00:19:06,370 --> 00:19:10,330 So the eigenvalues are three and one -- 297 00:19:10,330 --> 00:19:11,940 for this for sure. 298 00:19:11,940 --> 00:19:15,520 Now did we -- do you see why the eigenvalues are three and one 299 00:19:15,520 --> 00:19:17,930 for that one? 300 00:19:17,930 --> 00:19:21,870 If I tell you the eigenvalues are three and one, you prick -- 301 00:19:21,870 --> 00:19:26,190 quickly process the trace, which is -- and four -- 302 00:19:26,190 --> 00:19:29,510 agrees with four and you process the determinant, 303 00:19:29,510 --> 00:19:31,990 three times one -- 304 00:19:31,990 --> 00:19:35,930 the determinant is three and you say yes, it's right. 305 00:19:35,930 --> 00:19:39,540 Now I'm hoping that the eigenvalues of this thing 306 00:19:39,540 --> 00:19:41,770 are three and one. 307 00:19:41,770 --> 00:19:45,450 May I process the trace and the determinant for that one? 308 00:19:45,450 --> 00:19:47,830 What's the trace here? 309 00:19:47,830 --> 00:19:52,510 The trace of this matrix is four minus two and six, 310 00:19:52,510 --> 00:19:54,320 and that's what it should be. 311 00:19:54,320 --> 00:19:59,210 What's the determinant minus twelve plus fifteen is three. 312 00:19:59,210 --> 00:20:00,390 The determinant is three. 313 00:20:00,390 --> 00:20:04,190 The eigenvalues of that matrix are also three and one. 314 00:20:04,190 --> 00:20:07,460 And you see I created this matrix just like -- 315 00:20:07,460 --> 00:20:11,600 I just took any M, like, one that popped into my head 316 00:20:11,600 --> 00:20:16,030 and computed M inverse A M, got that matrix, 317 00:20:16,030 --> 00:20:22,980 it didn't look anything special but it's -- 318 00:20:22,980 --> 00:20:26,290 like A itself, it has those eigenvalues three and one. 319 00:20:26,290 --> 00:20:31,370 So that's the main fact and let me write it down. 320 00:20:31,370 --> 00:20:45,290 Similar matrixes have the same eigenvalues. 321 00:20:45,290 --> 00:20:51,590 So I'll just put that as an important point. 322 00:20:51,590 --> 00:20:53,890 And think why. 323 00:20:56,690 --> 00:20:57,430 Why is that? 324 00:20:57,430 --> 00:21:01,120 So that's what that family of matrixes is. 325 00:21:01,120 --> 00:21:03,940 The matrixes that are similar to this A 326 00:21:03,940 --> 00:21:09,610 here are all the matrixes with eigenvalues three and one. 327 00:21:09,610 --> 00:21:12,450 Every matrix with eigenvalues three and one, 328 00:21:12,450 --> 00:21:16,870 there's some M that connects this guy 329 00:21:16,870 --> 00:21:19,810 to the one you think of. 330 00:21:19,810 --> 00:21:22,840 And then of course, the most special guy in the whole family 331 00:21:22,840 --> 00:21:26,690 is the diagonal one with eigenvalues three and one 332 00:21:26,690 --> 00:21:28,630 sitting there on the diagonal. 333 00:21:28,630 --> 00:21:30,590 But also, I could find -- 334 00:21:30,590 --> 00:21:34,130 I mean, tell me just a couple more members of the family. 335 00:21:34,130 --> 00:21:38,070 Another -- tell me another matrix that has eigenvalues 336 00:21:38,070 --> 00:21:38,680 three and one. 337 00:21:41,450 --> 00:21:44,740 Well, let's see, I -- oh, I'll just make it triangular. 338 00:21:47,610 --> 00:21:49,180 That's in the family. 339 00:21:49,180 --> 00:21:53,840 There is some M that -- that connects to this one. 340 00:21:53,840 --> 00:21:58,310 And -- and also this. 341 00:21:58,310 --> 00:22:02,630 There's some matrix M -- so that M inverse A M comes out to be 342 00:22:02,630 --> 00:22:03,180 that. 343 00:22:03,180 --> 00:22:06,290 There's a whole family here. 344 00:22:06,290 --> 00:22:11,710 And they all share the same eigenvalues. 345 00:22:11,710 --> 00:22:14,320 So why is that? 346 00:22:14,320 --> 00:22:14,930 Okay. 347 00:22:14,930 --> 00:22:21,880 I'm going to start -- the only possibility is to start with Ax 348 00:22:21,880 --> 00:22:24,180 equal lambda x. 349 00:22:24,180 --> 00:22:27,790 Okay, so suppose A has the eigenvalue lambda. 350 00:22:30,970 --> 00:22:33,880 Now I want to get B into the picture here somehow. 351 00:22:33,880 --> 00:22:37,930 You remember B is M inverse A M. 352 00:22:37,930 --> 00:22:39,910 Let's just remember that over here. 353 00:22:39,910 --> 00:22:44,390 B is M inverse A M. 354 00:22:44,390 --> 00:22:48,030 And I want to see its eigenvalues. 355 00:22:48,030 --> 00:22:51,520 How I going to get M inverse A M into this equation? 356 00:22:51,520 --> 00:22:54,830 Let me just sort of do it. 357 00:22:54,830 --> 00:22:59,970 I'll put an M times an M inverse in there, right? 358 00:22:59,970 --> 00:23:01,870 That was -- 359 00:23:01,870 --> 00:23:04,100 I haven't changed the left-hand side, 360 00:23:04,100 --> 00:23:06,420 so I better not change the right-hand side. 361 00:23:09,570 --> 00:23:12,850 So everybody's okay so far, I just put in there -- see, 362 00:23:12,850 --> 00:23:16,240 I want to get a -- so now I'll multiply on the left by M 363 00:23:16,240 --> 00:23:17,870 inverse -- 364 00:23:17,870 --> 00:23:20,200 I have to do the same to this side 365 00:23:20,200 --> 00:23:22,880 and that number lambda's just a number, 366 00:23:22,880 --> 00:23:25,380 so it factors out in the front. 367 00:23:25,380 --> 00:23:32,370 So what I have here is this was safe. 368 00:23:32,370 --> 00:23:34,760 I did the same thing to both sides. 369 00:23:34,760 --> 00:23:37,090 And now I've got B. 370 00:23:37,090 --> 00:23:38,270 There's B. 371 00:23:38,270 --> 00:23:42,210 That's B times this vector M inverse 372 00:23:42,210 --> 00:23:47,670 x is equal to lambda times this vector M inverse x. 373 00:23:47,670 --> 00:23:49,920 So what have I learned? 374 00:23:49,920 --> 00:23:53,980 I've learned that B times some vector 375 00:23:53,980 --> 00:23:55,600 is lambda times that vector. 376 00:23:55,600 --> 00:23:58,910 I've learned that lambda is an eigenvalue of B also. 377 00:23:58,910 --> 00:24:01,370 So this is -- if -- so this is -- 378 00:24:01,370 --> 00:24:05,600 if lambda's an eigenvalue of A, then I can write it this way 379 00:24:05,600 --> 00:24:10,110 and I discover that lambda's an eigenvalue of B. 380 00:24:10,110 --> 00:24:12,240 That's the end of the proof. 381 00:24:12,240 --> 00:24:16,670 The eigenvector didn't stay the same. 382 00:24:16,670 --> 00:24:19,410 Of course I don't expect the eigenvectors to stay the same. 383 00:24:19,410 --> 00:24:22,240 If all the eigenvalues are the same and all the eigenvectors 384 00:24:22,240 --> 00:24:25,120 are the same, then probably the matrix is the same. 385 00:24:27,960 --> 00:24:31,260 Here the eigenvector changes, so the eigenvector -- 386 00:24:31,260 --> 00:24:38,370 so the point is then the eigenvector of B -- 387 00:24:38,370 --> 00:24:44,880 of B is M inverse times the eigenvector of A. 388 00:24:44,880 --> 00:24:45,380 Okay. 389 00:24:51,989 --> 00:24:53,280 That's all that this says here. 390 00:24:53,280 --> 00:24:58,940 The eigenvector of A was X, and so the M inverse -- 391 00:24:58,940 --> 00:25:01,930 similar matrixes, then have the same eigenvalues 392 00:25:01,930 --> 00:25:05,070 and their eigenvectors are just moved around. 393 00:25:05,070 --> 00:25:08,790 Of course, that's what we -- that's what happened way back 394 00:25:08,790 --> 00:25:09,580 -- 395 00:25:09,580 --> 00:25:15,070 and the most important similar matrixes are to diagonalize. 396 00:25:15,070 --> 00:25:18,140 So what was the point when we diagonalized? 397 00:25:18,140 --> 00:25:20,830 The eigenvalues stayed the same, of course. 398 00:25:20,830 --> 00:25:22,380 Three and one. 399 00:25:22,380 --> 00:25:25,150 What about the eigenvectors? 400 00:25:25,150 --> 00:25:29,190 The eigenvectors were whatever they were for the matrix A, 401 00:25:29,190 --> 00:25:31,340 but then what were the eigenvectors 402 00:25:31,340 --> 00:25:34,630 for the diagonal matrix? 403 00:25:34,630 --> 00:25:37,370 They're just -- what are the eigenvectors of a diagonal 404 00:25:37,370 --> 00:25:37,900 matrix? 405 00:25:37,900 --> 00:25:41,130 They're just one zero and zero one. 406 00:25:41,130 --> 00:25:44,290 So this step made the eigenvectors nice, 407 00:25:44,290 --> 00:25:49,750 didn't change the eigenvalues, and every time we 408 00:25:49,750 --> 00:25:51,580 don't change the eigenvalues. 409 00:25:51,580 --> 00:25:53,021 Same eigenvalues. 410 00:25:53,021 --> 00:25:53,520 Okay. 411 00:25:56,150 --> 00:26:01,100 Now -- so I've got all these matrixes in -- 412 00:26:01,100 --> 00:26:07,380 I've got this family of matrixes with eigenvalues three and one. 413 00:26:07,380 --> 00:26:08,610 Fine. 414 00:26:08,610 --> 00:26:10,270 That's a nice family. 415 00:26:10,270 --> 00:26:15,520 It's nice because those two eigenvalues are different. 416 00:26:15,520 --> 00:26:17,280 I now have to -- 417 00:26:17,280 --> 00:26:19,460 to get into that -- 418 00:26:19,460 --> 00:26:25,320 the -- into the less happy possibility that the two 419 00:26:25,320 --> 00:26:26,960 eigenvalues could be the 420 00:26:26,960 --> 00:26:29,680 same. 421 00:26:29,680 --> 00:26:32,940 And then it's a little trickier, because you remember 422 00:26:32,940 --> 00:26:35,360 when two eigenvalues are the same, 423 00:26:35,360 --> 00:26:38,400 what's the bad possibility? 424 00:26:38,400 --> 00:26:42,030 That there might not be enough -- 425 00:26:42,030 --> 00:26:44,730 a full set of eigenvectors and we might not be able 426 00:26:44,730 --> 00:26:46,080 to diagonalize. 427 00:26:46,080 --> 00:26:50,550 So I need to discuss the bad case. 428 00:26:50,550 --> 00:26:53,070 So the bad -- can I just say bad? 429 00:26:57,500 --> 00:27:03,707 If lambda one equals lambda two, then the matrix 430 00:27:03,707 --> 00:27:04,873 might not be diagonalizable. 431 00:27:07,410 --> 00:27:11,010 Suppose lambda one equals lambda two equals four, 432 00:27:11,010 --> 00:27:11,510 say. 433 00:27:14,460 --> 00:27:20,990 Now if I look at the family of matrixes with eigenvalues four 434 00:27:20,990 --> 00:27:25,840 and four, well, one possibility occurs to me. 435 00:27:25,840 --> 00:27:38,120 One family with eigenvalues four and four has this matrix in it, 436 00:27:38,120 --> 00:27:41,420 four times the identity. 437 00:27:41,420 --> 00:27:46,400 Then another -- but now I want to ask also about the matrix 438 00:27:46,400 --> 00:27:48,930 four four one zero. 439 00:27:51,800 --> 00:27:55,290 And my point -- here's the whole point of this -- 440 00:27:55,290 --> 00:28:00,107 of this bad stuff, is that this guy is not in the same family 441 00:28:00,107 --> 00:28:00,690 with that one. 442 00:28:00,690 --> 00:28:05,310 The family of a -- of matrixes that have eigenvalues four 443 00:28:05,310 --> 00:28:08,760 and four is two families. 444 00:28:08,760 --> 00:28:16,081 There's this total loner here who's in a family off -- 445 00:28:16,081 --> 00:28:16,580 right? 446 00:28:16,580 --> 00:28:18,890 Just by himself. 447 00:28:18,890 --> 00:28:23,070 And all the others are in with this guy. 448 00:28:23,070 --> 00:28:35,090 So the big family includes this one. 449 00:28:35,090 --> 00:28:39,720 And it includes a whole lot of other matrixes, all -- 450 00:28:39,720 --> 00:28:44,310 in fact, in this two by two case, it -- you see where -- 451 00:28:44,310 --> 00:28:48,290 what do I mean -- so what I using, this word family -- 452 00:28:48,290 --> 00:28:51,980 in a family, I mean they're similar. 453 00:28:51,980 --> 00:28:56,570 So my point is that the only matrix that's similar to this 454 00:28:56,570 --> 00:28:58,840 is itself. 455 00:28:58,840 --> 00:29:02,240 The only matrix that's similar to four times the identity 456 00:29:02,240 --> 00:29:03,650 is four times the identity. 457 00:29:03,650 --> 00:29:05,320 It's off by itself. 458 00:29:05,320 --> 00:29:07,360 Why is that? 459 00:29:07,360 --> 00:29:11,750 The -- if this is my matrix, four times the identity, 460 00:29:11,750 --> 00:29:17,150 and I take it, I multiply on the right by any matrix M, 461 00:29:17,150 --> 00:29:22,430 I multiply on the left by M inverse, what do I get? 462 00:29:22,430 --> 00:29:28,720 This is any M, but what's the result? 463 00:29:28,720 --> 00:29:30,930 Well, factoring out a four, that's 464 00:29:30,930 --> 00:29:34,710 just the identity matrix in there. 465 00:29:34,710 --> 00:29:37,250 So then the M inverse cancels the M, 466 00:29:37,250 --> 00:29:39,420 so I've just got this matrix back again. 467 00:29:42,910 --> 00:29:45,310 So whatever the M is, I'm not getting 468 00:29:45,310 --> 00:29:47,610 any more members of the family. 469 00:29:47,610 --> 00:29:57,680 So this is one small family, because it only has one person. 470 00:29:57,680 --> 00:29:59,610 One matrix, excuse me. 471 00:29:59,610 --> 00:30:02,200 I think of these matrixes as people by this point, 472 00:30:02,200 --> 00:30:04,110 in eighteen oh six. 473 00:30:04,110 --> 00:30:08,990 Okay, the other family includes all the rest -- 474 00:30:08,990 --> 00:30:13,930 all other matrixes that have eigenvalues four and four. 475 00:30:13,930 --> 00:30:20,290 This is somehow the best one in that family. 476 00:30:20,290 --> 00:30:21,870 See, I can't make it diagonal. 477 00:30:21,870 --> 00:30:24,580 If I -- if it's diagonal, it's this one. 478 00:30:24,580 --> 00:30:27,020 It's in its own, by itself. 479 00:30:27,020 --> 00:30:29,250 So I have to think, okay, what's the nearest I 480 00:30:29,250 --> 00:30:32,480 can get to diagonal? 481 00:30:32,480 --> 00:30:36,380 But it will not be diagonalizable. 482 00:30:36,380 --> 00:30:39,670 That -- do you know that that matrix is not diagonalizable? 483 00:30:39,670 --> 00:30:42,050 Of course, because if it was diagonalizable, 484 00:30:42,050 --> 00:30:46,150 it would be similar to that, which it isn't. 485 00:30:46,150 --> 00:30:48,540 The eigenvalues of this are four and four, 486 00:30:48,540 --> 00:30:52,960 but what's the catch with that matrix? 487 00:30:52,960 --> 00:30:55,430 It's only got one eigenvector. 488 00:30:55,430 --> 00:30:57,560 That's a non-diagonalizable matrix. 489 00:30:57,560 --> 00:30:59,340 Only one eigenvector. 490 00:30:59,340 --> 00:31:06,700 And somehow, if I made that one into a ten or to a million, 491 00:31:06,700 --> 00:31:10,540 I could find an M, it's in the family, it's similar. 492 00:31:10,540 --> 00:31:14,570 But the best -- so the best guy in this family is this one. 493 00:31:14,570 --> 00:31:19,810 And this is called the Jordan -- 494 00:31:19,810 --> 00:31:24,800 so this guy Jordan picked out -- so he, like, studied, 495 00:31:24,800 --> 00:31:29,250 these families of matrixes, and each family, 496 00:31:29,250 --> 00:31:34,740 he picked out the nicest, most diagonal one. 497 00:31:34,740 --> 00:31:37,960 But not completely diagonal, because there's nobody -- 498 00:31:37,960 --> 00:31:40,430 there isn't a diagonal matrix in this family, 499 00:31:40,430 --> 00:31:44,780 so there's a one up there in the Jordan form. 500 00:31:44,780 --> 00:31:46,010 Okay. 501 00:31:46,010 --> 00:31:50,699 I think we've got to see some more matrixes in that family. 502 00:31:50,699 --> 00:31:52,990 So, all right, let me -- let's just think of some other 503 00:31:52,990 --> 00:31:58,570 matrixes whose eigenvalues are four and four but they're not 504 00:31:58,570 --> 00:32:01,220 four times the identity. 505 00:32:01,220 --> 00:32:03,130 So -- and I believe that -- 506 00:32:03,130 --> 00:32:06,660 that this -- that all the examples we pick up will be 507 00:32:06,660 --> 00:32:13,320 similar to each other and -- do you see why -- 508 00:32:13,320 --> 00:32:17,340 in this topic of similar matrixes, 509 00:32:17,340 --> 00:32:21,820 the climax is the Jordan form. 510 00:32:21,820 --> 00:32:23,910 So it says that every matrix -- 511 00:32:23,910 --> 00:32:29,310 I'll write down what the Jordan form -- what Jordan discovered. 512 00:32:29,310 --> 00:32:35,030 He found the best looking matrix in each family. 513 00:32:35,030 --> 00:32:41,080 And that's -- then we've got -- then we've covered all matrixes 514 00:32:41,080 --> 00:32:44,540 including the non-diagonalizable one. 515 00:32:44,540 --> 00:32:47,060 That -- that's the point, that in some way, 516 00:32:47,060 --> 00:32:50,430 Jordan completed the diagonalization by coming 517 00:32:50,430 --> 00:32:54,890 as near as he could, which is his Jordan form. 518 00:32:54,890 --> 00:32:57,340 And therefore, if you want to cover all matrixes, 519 00:32:57,340 --> 00:32:59,720 you've got to get him in the picture. 520 00:32:59,720 --> 00:33:03,030 It used to be -- when I took eighteen oh six, 521 00:33:03,030 --> 00:33:07,530 that was the climax of the course, this Jordan form stuff. 522 00:33:07,530 --> 00:33:11,490 I think it's not the climax of linear algebra anymore, 523 00:33:11,490 --> 00:33:15,930 because -- 524 00:33:15,930 --> 00:33:20,590 it's not easy to find this Jordan form 525 00:33:20,590 --> 00:33:25,340 for a general matrix, because it depends on these eigenvalues 526 00:33:25,340 --> 00:33:27,900 being exactly the same. 527 00:33:27,900 --> 00:33:30,620 You'd have to know exactly the eigenvalues and it -- 528 00:33:30,620 --> 00:33:35,270 and you'd have to know exactly the rank and the slightest 529 00:33:35,270 --> 00:33:39,180 change in numbers will change those eigenvalues, 530 00:33:39,180 --> 00:33:43,320 change the rank and therefore the whole thing is numerically 531 00:33:43,320 --> 00:33:46,970 not an -- a good thing. 532 00:33:46,970 --> 00:33:51,180 But for algebra, it's the right thing 533 00:33:51,180 --> 00:33:52,660 to understand this family. 534 00:33:52,660 --> 00:33:56,600 So just tell me another matrix -- a few more matrixes -- 535 00:33:56,600 --> 00:34:03,380 so more members of the family. 536 00:34:06,510 --> 00:34:11,380 Let me put down again what the best one is. 537 00:34:11,380 --> 00:34:12,400 Okay. 538 00:34:12,400 --> 00:34:13,409 All right. 539 00:34:13,409 --> 00:34:14,690 Some more matrixes. 540 00:34:14,690 --> 00:34:17,659 Let's see, what I looking for? 541 00:34:17,659 --> 00:34:22,750 I'm looking for matrixes whose trace is what? 542 00:34:22,750 --> 00:34:25,190 So if I'm looking for more matrixes in the family, 543 00:34:25,190 --> 00:34:28,290 they'll all have the same eigenvalues, four and four. 544 00:34:28,290 --> 00:34:30,449 So their trace will be eight. 545 00:34:30,449 --> 00:34:34,120 So why don't I just take, like, five and three -- 546 00:34:34,120 --> 00:34:40,540 I've got the trace right, now the determinant should be what? 547 00:34:40,540 --> 00:34:41,469 Sixteen. 548 00:34:41,469 --> 00:34:45,370 So I just fix this up -- shall I put maybe a one and a minus one 549 00:34:45,370 --> 00:34:46,771 there? 550 00:34:46,771 --> 00:34:47,270 Okay. 551 00:34:47,270 --> 00:34:52,139 There's a matrix with eigenvalues four and four, 552 00:34:52,139 --> 00:34:56,719 because the trace is eight and the determinant is sixteen. 553 00:34:56,719 --> 00:35:00,670 And I don't think it's diagonalizable. 554 00:35:00,670 --> 00:35:03,800 Do you know why it's not diagonalizable? 555 00:35:03,800 --> 00:35:06,470 Because if it was diagonalizable, 556 00:35:06,470 --> 00:35:09,505 the diagonal form would have to be this. 557 00:35:12,460 --> 00:35:14,980 But I can't get to that form, because whatever 558 00:35:14,980 --> 00:35:17,880 I do with any M inverse and M I stay with that form. 559 00:35:17,880 --> 00:35:20,320 I could never get -- connect those. 560 00:35:20,320 --> 00:35:22,470 So I can put down more members -- here -- 561 00:35:22,470 --> 00:35:23,810 here's another easy one. 562 00:35:23,810 --> 00:35:26,960 I could put the four and the four and a seventeen 563 00:35:26,960 --> 00:35:27,580 down there. 564 00:35:30,270 --> 00:35:32,080 All these matrixes are similar. 565 00:35:32,080 --> 00:35:35,350 If I'm -- I could find an M that would show that that one is 566 00:35:35,350 --> 00:35:37,350 similar to that one. 567 00:35:37,350 --> 00:35:40,420 And in -- you can see the general picture is I can take 568 00:35:40,420 --> 00:35:45,710 any a and any 8-a here and any -- oh, I don't know, 569 00:35:45,710 --> 00:35:48,200 whatever you put it'd be -- anyway, you can see. 570 00:35:48,200 --> 00:35:54,890 I can fill this in, fill this in to make the trace equal eight, 571 00:35:54,890 --> 00:36:01,160 the determinant equal 16, I get all that family of matrixes 572 00:36:01,160 --> 00:36:02,635 and they're all similar. 573 00:36:05,170 --> 00:36:08,250 So we see what eigenvalues do. 574 00:36:08,250 --> 00:36:13,020 They're all similar and they all have only one eigenvector. 575 00:36:13,020 --> 00:36:16,610 So I -- if I'm -- if you were going to -- 576 00:36:16,610 --> 00:36:20,780 allow me to add to this picture, they have the same lambdas 577 00:36:20,780 --> 00:36:25,799 and they also have the same number of independent 578 00:36:25,799 --> 00:36:26,340 eigenvectors. 579 00:36:29,880 --> 00:36:33,910 Because if I get an eigenvector for x I get one for -- for A, 580 00:36:33,910 --> 00:36:36,260 I get one for B also. 581 00:36:36,260 --> 00:36:43,170 So -- and same number of eigenvectors. 582 00:36:43,170 --> 00:36:45,740 But even more than that -- 583 00:36:45,740 --> 00:36:47,048 even more than that -- 584 00:36:47,048 --> 00:36:49,173 I mean, it's not enough just to count eigenvectors. 585 00:36:51,970 --> 00:36:54,450 Yes, let me give you an example why it's not even 586 00:36:54,450 --> 00:36:58,460 enough to count eigenvectors. 587 00:36:58,460 --> 00:37:00,170 So another example. 588 00:37:00,170 --> 00:37:04,210 So here are some matrixes -- 589 00:37:04,210 --> 00:37:07,600 oh, let me make them four by four -- 590 00:37:07,600 --> 00:37:09,260 okay, here -- here's a matrix. 591 00:37:09,260 --> 00:37:11,570 I mean, like if you want nightmares, 592 00:37:11,570 --> 00:37:14,480 think about matrixes like these. 593 00:37:14,480 --> 00:37:21,670 Uh, so a one off the diagonal -- say a one there, how many -- 594 00:37:21,670 --> 00:37:24,600 what are the eigenvalues of that matrix? 595 00:37:24,600 --> 00:37:29,040 Oh, I mean -- 596 00:37:29,040 --> 00:37:30,480 okay. 597 00:37:30,480 --> 00:37:32,550 What are the eigenvalues of that matrix? 598 00:37:35,740 --> 00:37:37,910 Please. 599 00:37:37,910 --> 00:37:40,160 Four 0s, right? 600 00:37:40,160 --> 00:37:44,930 So we're really getting bad matrixes now. 601 00:37:44,930 --> 00:37:47,080 So I mean, this is, like -- 602 00:37:47,080 --> 00:37:53,190 Jordan was a good guy, but he had to think about matrixes 603 00:37:53,190 --> 00:37:58,590 that all -- that had, like -- an eigenvalue repeated four times. 604 00:37:58,590 --> 00:38:01,140 How many eigenvectors does that matrix have? 605 00:38:04,750 --> 00:38:07,980 Well, I'm -- eigenvectors will be -- 606 00:38:07,980 --> 00:38:11,710 since the eigenvalue is zero, eigenvectors will be 607 00:38:11,710 --> 00:38:13,350 in the null space, right? 608 00:38:13,350 --> 00:38:17,870 I'm -- eigenvectors have got to be A x equal zero x. 609 00:38:17,870 --> 00:38:21,340 So what's the dimension of the null space? 610 00:38:21,340 --> 00:38:22,110 Two. 611 00:38:22,110 --> 00:38:23,920 Somebody said two. 612 00:38:23,920 --> 00:38:24,680 And that's right. 613 00:38:24,680 --> 00:38:26,690 How -- why? 614 00:38:26,690 --> 00:38:29,390 Because you ask what's the rank of that matrix, 615 00:38:29,390 --> 00:38:32,410 the rank is obviously two. 616 00:38:32,410 --> 00:38:35,050 The number of independent rows is two, 617 00:38:35,050 --> 00:38:36,880 the number of independent columns is two, 618 00:38:36,880 --> 00:38:41,430 the rank is two so the null -- the dimension of the null space 619 00:38:41,430 --> 00:38:45,950 is four minus two, so it's got two eigenvectors. 620 00:38:45,950 --> 00:38:47,500 Two eigenvectors. 621 00:38:47,500 --> 00:38:49,441 Two independent eigenvectors. 622 00:38:49,441 --> 00:38:49,940 All right. 623 00:38:49,940 --> 00:38:55,260 The dimension of the null space is two. 624 00:38:59,910 --> 00:39:03,905 Now, suppose I change this zero to a seven. 625 00:39:09,050 --> 00:39:11,840 The eigenvalues are all still zero, how -- what about -- 626 00:39:11,840 --> 00:39:12,756 how many eigenvectors? 627 00:39:15,449 --> 00:39:17,990 What's the dimension of the -- what's the rank of this matrix 628 00:39:17,990 --> 00:39:19,160 now? 629 00:39:19,160 --> 00:39:20,890 Still two, right? 630 00:39:20,890 --> 00:39:22,820 So it's okay. 631 00:39:22,820 --> 00:39:26,370 And actually, this would be similar to the one that 632 00:39:26,370 --> 00:39:27,800 had a zero in there. 633 00:39:27,800 --> 00:39:31,740 But it's not as beautiful, Jordan picked this one. 634 00:39:31,740 --> 00:39:34,850 He picked -- he put ones -- 635 00:39:34,850 --> 00:39:39,000 we have a one on the -- above the diagonal for every missing 636 00:39:39,000 --> 00:39:42,330 eigenvector, and here we're missing two because we've got 637 00:39:42,330 --> 00:39:45,550 two, so we've got two eigenvectors and two are 638 00:39:45,550 --> 00:39:53,860 missing, because it's a four by four matrix. 639 00:39:53,860 --> 00:39:58,910 Okay, now -- but I was going to give you this second example. 640 00:40:01,690 --> 00:40:05,145 0 1 0 0, let me just move the one. 641 00:40:09,030 --> 00:40:11,500 Oop, not there. 642 00:40:11,500 --> 00:40:15,710 Off the diagonal and zero zero zero zero zero. 643 00:40:15,710 --> 00:40:16,210 Okay. 644 00:40:19,420 --> 00:40:22,570 So now tell me about this matrix. 645 00:40:22,570 --> 00:40:27,460 Its eigenvalues are four zeroes again. 646 00:40:27,460 --> 00:40:31,910 Its rank is two again. 647 00:40:31,910 --> 00:40:36,670 So it has two eigenvectors and two missing. 648 00:40:36,670 --> 00:40:40,890 But the darn thing is not similar to that one. 649 00:40:40,890 --> 00:40:44,430 A -- a count of eigenvectors looks like these could be 650 00:40:44,430 --> 00:40:47,130 similar, but they're not. 651 00:40:47,130 --> 00:40:52,720 Jordan -- see, this is like -- a little three by three block 652 00:40:52,720 --> 00:40:55,880 and a little one by one block. 653 00:40:55,880 --> 00:40:58,770 And this one is like a two by two block and a two 654 00:40:58,770 --> 00:41:03,080 by two block, and those blocks are called Jordan blocks. 655 00:41:03,080 --> 00:41:06,770 So let me say what is a Jordan block? 656 00:41:11,150 --> 00:41:17,400 J block number I has -- 657 00:41:17,400 --> 00:41:22,660 so a Jordan block has a repeated eigenvalue, lambda I, lambda I 658 00:41:22,660 --> 00:41:24,530 on the diagonal. 659 00:41:24,530 --> 00:41:26,905 Zeroes below and ones above. 660 00:41:30,060 --> 00:41:34,160 So there's a block with this guy repeated, 661 00:41:34,160 --> 00:41:37,070 but it only has one eigenvector. 662 00:41:37,070 --> 00:41:40,900 So a Jordan block has one eigenvector only. 663 00:41:43,970 --> 00:41:47,720 This one has one eigenvector, this block has one eigenvector 664 00:41:47,720 --> 00:41:49,580 and we get two. 665 00:41:49,580 --> 00:41:52,170 This block has one eigenvector and that block has 666 00:41:52,170 --> 00:41:54,980 one eigenvector and we get two. 667 00:41:54,980 --> 00:42:02,020 So -- but the blocks are different sizes. 668 00:42:02,020 --> 00:42:05,540 And that -- it turns out Jordan worked out -- 669 00:42:05,540 --> 00:42:14,190 then this is not similar, not similar to this one. 670 00:42:18,020 --> 00:42:22,140 So the -- so I'm, like, giving you the whole story -- 671 00:42:22,140 --> 00:42:25,780 well, not the whole story, but the main themes of the story -- 672 00:42:25,780 --> 00:42:29,700 is here's Jordan's theorem. 673 00:42:29,700 --> 00:42:49,060 Every square matrix A is similar to A Jordan matrix J. 674 00:42:49,060 --> 00:42:51,890 And what's a Jordan matrix J? 675 00:42:51,890 --> 00:42:56,670 It's a matrix with these blocks, block -- 676 00:42:56,670 --> 00:43:03,850 Jordan block number one, Jordan block number two and so on. 677 00:43:03,850 --> 00:43:07,950 And let's say Jordan block number d. 678 00:43:11,360 --> 00:43:14,760 And those Jordan blocks look like that, 679 00:43:14,760 --> 00:43:17,760 so the eigenvalues are sitting on the diagonal, 680 00:43:17,760 --> 00:43:22,750 but we've got some of these ones above the diagonal. 681 00:43:22,750 --> 00:43:25,010 We've got the number of -- 682 00:43:25,010 --> 00:43:27,530 so the number of blocks -- 683 00:43:27,530 --> 00:43:37,510 the number of blocks is the number of eigenvectors, 684 00:43:37,510 --> 00:43:42,510 because we get one eigenvector per block. 685 00:43:42,510 --> 00:43:47,250 So what I'm -- so if I summarize Jordan's idea -- 686 00:43:47,250 --> 00:43:49,570 start with any A. 687 00:43:49,570 --> 00:43:54,080 If its eigenvalues are distinct, then what's it similar 688 00:43:54,080 --> 00:43:54,700 to? 689 00:43:54,700 --> 00:43:56,430 This is the good case. 690 00:43:56,430 --> 00:44:00,710 if I start with a matrix A and it has different eigenvalues -- 691 00:44:00,710 --> 00:44:03,450 it's n eigenvalues, none of them are repeated, 692 00:44:03,450 --> 00:44:09,350 then that's a diagonal -- diagonalizable matrix -- 693 00:44:09,350 --> 00:44:13,990 the Jordan blocks is -- has -- the Jordan matrix is diagonal. 694 00:44:13,990 --> 00:44:15,900 It's lambda. 695 00:44:15,900 --> 00:44:17,840 So the good case -- 696 00:44:17,840 --> 00:44:22,590 the good case, J is lambda. 697 00:44:27,020 --> 00:44:29,400 All -- there are -- 698 00:44:29,400 --> 00:44:29,900 d=n. 699 00:44:29,900 --> 00:44:33,830 There are n eigenvectors, n blocks, diagonal, everything 700 00:44:33,830 --> 00:44:35,640 great. 701 00:44:35,640 --> 00:44:40,990 But Jordan covered all cases by including 702 00:44:40,990 --> 00:44:44,460 these cases of repeated eigenvalues and missing 703 00:44:44,460 --> 00:44:47,100 eigenvectors. 704 00:44:47,100 --> 00:44:47,810 Okay. 705 00:44:47,810 --> 00:44:49,640 That's a description of Jordan. 706 00:44:49,640 --> 00:44:51,030 That -- that's -- 707 00:44:51,030 --> 00:44:54,300 I haven't told you how to compute this thing, 708 00:44:54,300 --> 00:44:56,500 and it isn't easy. 709 00:44:56,500 --> 00:45:00,920 Whereas the good case is the -- the good case is what 18.06 is 710 00:45:00,920 --> 00:45:01,660 about. 711 00:45:01,660 --> 00:45:07,040 The -- this case is what 18.06 was about 20 years ago. 712 00:45:07,040 --> 00:45:12,220 So you can see you probably won't have on the final exam 713 00:45:12,220 --> 00:45:18,540 the computation of a Jordan matrix for some horrible thing 714 00:45:18,540 --> 00:45:21,720 with four repeated eigenvalues. 715 00:45:21,720 --> 00:45:28,730 I'm not that crazy about the Jordan form. 716 00:45:28,730 --> 00:45:34,950 But I'm very positive about positive definite matrixes 717 00:45:34,950 --> 00:45:38,880 and about the idea that's coming Monday, 718 00:45:38,880 --> 00:45:40,964 the singular value decomposition. 719 00:45:40,964 --> 00:45:43,130 So I'll see you on Monday, and have a great weekend. 720 00:45:43,130 --> 00:45:44,680 Bye.