WEBVTT 00:00:00.046 --> 00:00:05.040 Multiply a vector by a matrix and, in general, two things happen to it. 00:00:05.352 --> 00:00:09.010 Its length changes, and its direction changes. 00:00:09.462 --> 00:00:13.665 Today I want to ask about the second of those, because for almost every matrix 00:00:13.665 --> 00:00:17.137 there are a few directions that simply refuse to be turned. 00:00:18.952 --> 00:00:22.343 So here is a piece of the plane, ruled into squares. 00:00:22.655 --> 00:00:25.396 And here is the matrix we will apply to it: 00:00:25.593 --> 00:00:28.275 two, one, one, two. 00:00:29.072 --> 00:00:31.534 And let me say what applying it means. 00:00:31.986 --> 00:00:33.902 It means multiplication. 00:00:34.354 --> 00:00:38.428 Take a point on the grid, write it as a column, put A in front, 00:00:38.511 --> 00:00:40.334 and out comes a different point: 00:00:40.473 --> 00:00:41.202 two 00:00:41.285 --> 00:00:44.572 x plus y, and x plus two y. 00:00:44.710 --> 00:00:47.892 A matrix is a map that carries one point to another. 00:00:48.642 --> 00:00:50.559 Take one point and try it. 00:00:50.802 --> 00:00:53.052 This one, at one, zero. 00:00:53.135 --> 00:00:54.785 Multiply it by A: 00:00:54.923 --> 00:00:57.394 two times one plus zero is two, 00:00:57.477 --> 00:01:00.137 and one plus two times zero is one. 00:01:00.333 --> 00:01:03.771 So A sends that point over to two, one. 00:01:04.567 --> 00:01:07.622 Now watch every point do that same sum at once. 00:01:07.818 --> 00:01:10.533 The whole thing stretches out along one diagonal, 00:01:10.616 --> 00:01:14.123 and our point lands exactly where the arithmetic said it would. 00:01:15.175 --> 00:01:19.739 Let me put it back, and follow one single arrow through the same motion. 00:01:19.843 --> 00:01:23.361 This one points straight up, at zero, one and a half. 00:01:23.813 --> 00:01:28.017 Put it through the same sum and the answer is one and a half, three. 00:01:29.069 --> 00:01:30.904 Now apply the matrix. 00:01:31.217 --> 00:01:33.502 The arrow gets longer, certainly. 00:01:33.585 --> 00:01:38.486 But look where it ends up. It has swung over to the right. The matrix turned it. 00:01:39.282 --> 00:01:41.187 Not every arrow does that. 00:01:41.384 --> 00:01:43.692 Let me put the plane back and try two others, 00:01:43.775 --> 00:01:46.841 one along this diagonal and one along the other. 00:01:47.455 --> 00:01:52.344 And this time let us do the multiplication first, before anything moves. 00:01:53.396 --> 00:01:57.992 Read those two answers. The first one is three times the vector we put in. 00:01:58.075 --> 00:02:02.789 The second one is the vector we put in, unchanged, which is one times it. 00:02:03.702 --> 00:02:07.253 Now apply the matrix and watch the arrows do exactly that. 00:02:07.336 --> 00:02:10.736 The red one triples and stays on the line it was already on. 00:02:10.819 --> 00:02:12.933 The green one does not move at all. 00:02:13.636 --> 00:02:17.074 Those two directions are the eigenvectors of this matrix. 00:02:17.270 --> 00:02:21.416 Everything else in the plane swings round toward the stretching direction. 00:02:21.520 --> 00:02:23.759 These two just sit there and get scaled, 00:02:23.842 --> 00:02:27.695 by three and by one, and those two numbers are their eigenvalues. 00:02:27.778 --> 00:02:30.669 So let us write down exactly what that means. 00:02:31.742 --> 00:02:34.472 Look again at what just happened to that red arrow. 00:02:34.784 --> 00:02:36.500 Here is the vector one, one. 00:02:36.584 --> 00:02:40.822 We put it through the matrix, and out came another vector: three, three. 00:02:41.619 --> 00:02:44.669 And here is the observation the whole subject is built on. 00:02:44.753 --> 00:02:48.295 That answer is nothing but three times the vector we started with. 00:02:48.608 --> 00:02:51.871 One vector, one plain number, no matrix. 00:02:52.668 --> 00:02:55.141 Which tells us something about the picture. 00:02:55.338 --> 00:02:58.775 Both arrows lie on this one line through the origin. 00:02:59.088 --> 00:03:03.500 So on this line, applying A is nothing but multiplying by three. 00:03:03.697 --> 00:03:07.575 The arrow got longer, but it was not turned by so much as a degree. 00:03:08.372 --> 00:03:10.819 Now try an arrow that is not on that line. 00:03:10.903 --> 00:03:13.261 Here is w, pointing straight up. 00:03:13.457 --> 00:03:15.687 A sends it to one, two. 00:03:16.391 --> 00:03:20.142 And one, two is not a multiple of zero, one. 00:03:20.338 --> 00:03:22.913 No number times zero is ever one. 00:03:22.997 --> 00:03:26.013 So w came back pointing somewhere else entirely, 00:03:26.097 --> 00:03:28.373 like almost every arrow in the plane. 00:03:29.170 --> 00:03:30.699 The arrows that survive, 00:03:30.783 --> 00:03:35.042 the ones A puts back on their own line, are the eigenvectors of A. 00:03:35.126 --> 00:03:40.072 And the number that does the stretching, three in this case, is its eigenvalue. 00:03:41.469 --> 00:03:44.523 Now say that for any matrix and any vector. 00:03:44.836 --> 00:03:48.145 Here is a general v, drawn on a general line. 00:03:48.342 --> 00:03:53.276 A vector v is an eigenvector of A when A times v comes out 00:03:53.276 --> 00:03:55.970 as a plain multiple of v itself. 00:03:56.167 --> 00:04:01.752 Written down, that is A v equals lambda v, and that is the whole definition. 00:04:02.061 --> 00:04:04.732 That lambda is the eigenvalue. 00:04:04.836 --> 00:04:09.477 And there is one piece of fine print: v is not allowed to be the zero vector. 00:04:09.561 --> 00:04:10.977 A times zero is zero 00:04:10.977 --> 00:04:15.564 for every matrix there is, so the zero vector would qualify for everything. 00:04:16.361 --> 00:04:18.726 Read the equation geometrically now. 00:04:18.810 --> 00:04:20.921 It says that when A gets hold of v, 00:04:21.005 --> 00:04:23.931 it puts it back down somewhere on that same line. 00:04:24.128 --> 00:04:28.216 If lambda is bigger than one, A stretches the arrow out along it. 00:04:28.641 --> 00:04:33.506 If lambda is between zero and one, it pulls the arrow in toward the origin. 00:04:33.819 --> 00:04:36.443 Shorter, yes. Turned, no. 00:04:36.868 --> 00:04:40.352 And now let lambda keep going, down past zero. 00:04:40.456 --> 00:04:44.540 Watch the arrow shrink as it goes, die away to nothing at the origin, 00:04:44.624 --> 00:04:47.968 and come back out the other side, pointing backwards. 00:04:48.881 --> 00:04:50.379 That still counts. 00:04:50.576 --> 00:04:53.142 Backwards along a line is the same line. 00:04:53.339 --> 00:04:55.859 Notice what the equation does not say: 00:04:56.056 --> 00:05:01.281 it does not say A leaves v alone, it says A leaves the line alone. 00:05:01.764 --> 00:05:05.221 So an eigenvector is a direction the matrix cannot turn, 00:05:05.305 --> 00:05:08.858 and the eigenvalue says what happens to lengths along it. 00:05:08.962 --> 00:05:13.212 Which leaves the real question: given a matrix, how do you find them? 00:05:14.327 --> 00:05:19.413 Right. We have a matrix, and we want the directions it cannot turn. 00:05:19.609 --> 00:05:22.977 So start from the definition and push the algebra around. 00:05:23.173 --> 00:05:25.206 A v equals lambda 00:05:25.310 --> 00:05:26.286 v. 00:05:27.187 --> 00:05:29.289 Move everything to one side. 00:05:29.393 --> 00:05:33.968 A v minus lambda v equals the zero vector. 00:05:34.164 --> 00:05:38.644 I would love to factor the v out, but I cannot subtract a number from a matrix, 00:05:38.728 --> 00:05:40.609 so I slip in the identity: 00:05:40.806 --> 00:05:45.323 lambda v is the same thing as lambda times I times v. 00:05:45.748 --> 00:05:47.432 And now it factors. 00:05:48.047 --> 00:05:53.609 The matrix A minus lambda I, acting on v, gives the zero vector. 00:05:54.405 --> 00:05:56.670 Look at what that line actually says. 00:05:56.774 --> 00:05:59.654 Something times something comes out zero. 00:05:59.850 --> 00:06:02.661 And one of the two somethings we already know about: 00:06:02.765 --> 00:06:05.343 v is not the zero vector. 00:06:05.539 --> 00:06:08.083 That was the fine print in the definition. 00:06:08.879 --> 00:06:12.395 So the zero on the right cannot have come from the vector. 00:06:12.479 --> 00:06:15.115 It has to have come from the matrix over here. 00:06:15.311 --> 00:06:20.397 A minus lambda I has to take a nonzero vector and crush it flat to nothing. 00:06:20.822 --> 00:06:25.351 And there is exactly one number that measures whether a matrix does that. 00:06:25.548 --> 00:06:29.191 If you want to say how much of a matrix there is, in a single number, 00:06:29.275 --> 00:06:31.400 that number is its determinant. 00:06:31.597 --> 00:06:37.135 A matrix that flattens a nonzero vector onto the origin has determinant zero. 00:06:37.932 --> 00:06:39.268 Formally, then: 00:06:39.372 --> 00:06:45.130 lambda is an eigenvalue of A exactly when the determinant of A minus lambda I comes 00:06:45.130 --> 00:06:50.971 out zero. That is one equation in one unknown, and we can actually solve it. 00:06:52.367 --> 00:06:54.481 Let us do it with our matrix. 00:06:54.585 --> 00:06:56.651 Subtract lambda from each entry 00:06:56.651 --> 00:07:02.190 on the diagonal, take the determinant of what is left, and set that equal to zero. 00:07:03.242 --> 00:07:07.574 The determinant of a two by two is read straight off the two diagonals. 00:07:07.677 --> 00:07:09.219 Multiply down this one. 00:07:09.303 --> 00:07:11.231 Then multiply down the other. 00:07:11.335 --> 00:07:14.168 And take the second product away from the first. 00:07:14.965 --> 00:07:19.981 So that is two minus lambda, all squared, take away one times one. 00:07:20.778 --> 00:07:23.553 Multiply the square out and keep every term. 00:07:23.866 --> 00:07:29.985 Lambda squared, minus four lambda, plus four, minus one, equals zero. 00:07:30.782 --> 00:07:35.798 And four minus one is three, so that line tidies itself up where it stands. 00:07:35.995 --> 00:07:39.351 Lambda squared minus four lambda plus three. 00:07:39.803 --> 00:07:41.220 This has a name: 00:07:41.324 --> 00:07:44.166 it is the characteristic polynomial of A, 00:07:44.250 --> 00:07:47.362 and its degree is always the size of the matrix. 00:07:48.158 --> 00:07:50.237 This one factors on sight: 00:07:50.434 --> 00:07:55.946 lambda minus three, times lambda minus one. So the eigenvalues are three and one. 00:07:56.030 --> 00:07:57.469 And there is our confirmation, 00:07:57.469 --> 00:08:01.092 because those are exactly the two numbers we watched on the grid. 00:08:01.405 --> 00:08:02.645 One arrow tripled. 00:08:02.729 --> 00:08:04.552 The other one stayed put. 00:08:05.653 --> 00:08:07.639 We have the eigenvalues. 00:08:07.952 --> 00:08:10.576 Now we want the directions that go with them. 00:08:11.191 --> 00:08:14.742 Go back to the factored equation, A minus lambda 00:08:14.825 --> 00:08:19.644 I times v equals zero, and put the first eigenvalue into it. 00:08:20.440 --> 00:08:25.701 With lambda equal to three, we subtract three from each entry on the diagonal. 00:08:25.898 --> 00:08:28.810 That gives us minus one, one, one, 00:08:28.893 --> 00:08:35.163 minus one, multiplying the column x, y, and the answer has to be the zero vector. 00:08:35.751 --> 00:08:37.911 Both rows say the same thing. 00:08:38.084 --> 00:08:43.448 Minus x plus y equals zero, or in other words y equals x. 00:08:43.622 --> 00:08:46.546 So any vector whose two entries are equal will do, 00:08:46.629 --> 00:08:49.381 and the tidiest one to write down is one, 00:08:49.578 --> 00:08:50.414 one. 00:08:51.211 --> 00:08:54.312 There is the line of them, and there is the arrow. 00:08:54.415 --> 00:08:57.493 And notice that an eigenvector is never alone. 00:08:57.689 --> 00:09:00.962 Every multiple of one, one is an eigenvector too, 00:09:01.045 --> 00:09:03.623 with the very same eigenvalue three. 00:09:03.936 --> 00:09:08.256 What we have really found is a whole line the matrix keeps to itself. 00:09:09.652 --> 00:09:11.696 Now the second eigenvalue. 00:09:11.800 --> 00:09:16.349 Lambda equals one, so we subtract one from the diagonal instead, 00:09:16.432 --> 00:09:19.533 and this time the matrix comes out as four ones. 00:09:20.329 --> 00:09:23.729 Every row now says x plus y equals zero, 00:09:23.812 --> 00:09:29.407 so y is minus x, and the tidiest eigenvector to pick is one, minus one. 00:09:29.490 --> 00:09:32.880 There is the second line, and there is the second arrow. 00:09:33.677 --> 00:09:35.533 Now hold on to both of those arrows, 00:09:35.616 --> 00:09:38.659 and let the matrix loose on the whole picture again. 00:09:39.363 --> 00:09:42.312 Every grid line has been dragged off somewhere new. 00:09:42.416 --> 00:09:46.073 The red arrow ran out along its own line and stopped three times further 00:09:46.073 --> 00:09:48.683 from the origin, which is its eigenvalue. 00:09:48.766 --> 00:09:52.770 The green one did not move at all, because its eigenvalue is one. 00:09:52.853 --> 00:09:55.245 Those two directions were untouchable. 00:09:56.342 --> 00:09:58.781 One more thing, and then a warning. 00:09:59.093 --> 00:10:02.368 Here is a matrix that rotates the plane through a quarter turn: 00:10:02.472 --> 00:10:06.127 zero, minus one, one, zero. 00:10:06.210 --> 00:10:09.021 And here are three arrows sitting in it. 00:10:09.817 --> 00:10:13.429 Every arrow in the plane gets swung round by ninety degrees. 00:10:13.626 --> 00:10:17.571 Watch these three go. Not one of them finishes on the line it started on. 00:10:17.654 --> 00:10:20.337 A rotation has no favourite direction. 00:10:21.134 --> 00:10:23.094 And the algebra says the same thing. 00:10:23.177 --> 00:10:25.242 Subtract lambda from the diagonal, 00:10:25.325 --> 00:10:29.842 take the determinant, and cross the two diagonals exactly as before. 00:10:30.383 --> 00:10:32.750 That comes out as lambda squared plus one, 00:10:32.833 --> 00:10:35.644 and over the real numbers that has no roots at all. 00:10:35.956 --> 00:10:38.218 This matrix has no eigenvectors. 00:10:38.301 --> 00:10:42.284 It is not a defect in the matrix, it is honest geometry. 00:10:43.081 --> 00:10:48.318 So keep that in your pocket. Eigenvectors are common, but they are not guaranteed. 00:10:48.514 --> 00:10:51.903 Now let me lift exactly this rotation up into three dimensions, 00:10:51.986 --> 00:10:54.471 where something rather lovely happens. 00:10:55.867 --> 00:10:58.838 Here is a rotation of space about the vertical axis, 00:10:58.921 --> 00:11:02.474 and here are two arrows lying flat in the horizontal plane. 00:11:03.387 --> 00:11:04.804 Watch them turn. 00:11:05.000 --> 00:11:06.938 Every one of them changes direction, 00:11:07.021 --> 00:11:09.703 exactly the way they did down in the plane. 00:11:10.616 --> 00:11:14.436 But now there is somewhere to stand that the rotation cannot reach. 00:11:14.633 --> 00:11:17.652 An arrow pointing straight up along the axis. 00:11:17.849 --> 00:11:20.879 The rotation carries the whole of space around it and leaves 00:11:20.879 --> 00:11:23.504 that one arrow precisely where it was. 00:11:24.045 --> 00:11:28.214 So that arrow is an eigenvector, and its eigenvalue is one. 00:11:28.317 --> 00:11:31.743 And this is a general fact about three dimensional space. 00:11:31.847 --> 00:11:37.594 Every rotation has an axis, and the axis is the eigenvector with eigenvalue one. 00:11:38.046 --> 00:11:42.215 Euler proved that, and the proof is essentially this picture. 00:11:43.298 --> 00:11:46.016 So there is the whole idea, in one line. 00:11:46.120 --> 00:11:49.836 A times v equals lambda times v. 00:11:50.632 --> 00:11:54.255 The eigenvectors are the directions the matrix cannot turn. 00:11:54.452 --> 00:11:55.625 The eigenvalue attached 00:11:55.625 --> 00:11:59.457 to each one says what happens to lengths along that direction. 00:12:00.253 --> 00:12:01.703 Find those directions, 00:12:01.786 --> 00:12:05.966 and you have found the frame in which your matrix is nothing but a list of stretches. 00:12:06.162 --> 00:12:10.468 Which is why, from vibration to stability to the ranking of the whole internet, 00:12:10.551 --> 00:12:14.371 this tends to be the first question anybody asks about a matrix.