MIT18_06S10_L21

MIT18_06S10_L21 - 18.06 Linear Algebra Spring 2010...

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18.06 Linear Algebra, Spring 2010 Transcript – Lecture 21 OK. So this is the first lecture on eigenvalues and eigenvectors, and that's a big subject that will take up most of the rest of the course. It's, again, matrices are square and we're looking now for some special numbers, the eigenvalues, and some special vectors, the eigenvectors. And so this lecture is mostly about what are these numbers, and then the other lectures are about how do we use them, why do we want them. OK, so what's an eigenvector? Maybe I'll start with eigenvector. What's an eigenvector? So I have a matrix A. OK. What does a matrix do? It acts on vectors. It multiplies vectors x. So the way that matrix acts is in goes a vector x and out comes a vector Ax. It's like a function. With a function in calculus, in goes a number x, out comes f(x). Here in linear algebra we're up in more dimensions. In goes a vector x, out comes a vector Ax. And the vectors I'm specially interested in are the ones the come out in the same direction that they went in. That won't be typical. Most vectors, Ax is in -- points in some different direction. But there are certain vectors where Ax comes out parallel to x. And those are the eigenvectors. So Ax parallel to x. Those are the eigenvectors. And what do I mean by parallel? Oh, much easier to just state it in an equation. Ax is some multiple -- and everybody calls that multiple lambda -- of x. That's our big equation. We look for special vectors -- and remember most vectors won't be eigenvectors -- that -- for which Ax is in the same direction as x, and by same direction I allow it to be the very opposite direction, I allow lambda to be negative or zero. Well, I guess we've met the eigenvectors that have eigenvalue zero. Those are in the same direction, but they're -- in a kind of very special way. So this -- the eigenvector x. Lambda, whatever this multiplying factor is, whether it's six or minus six or zero or even some imaginary number, that's the eigenvalue. So there's the eigenvalue, there's the eigenvector. Let's just take a second on eigenvalue zero.
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From the point of view of eigenvalues, that's no special deal. That's, we have an eigenvector. If the eigenvalue happened to be zero, that would mean that Ax was zero x, in other words zero. So what would x, where would we look for -- what are the x-s? What are the eigenvectors with eigenvalue zero? They're the guys in the null space, Ax equals zero. So if our matrix is singular, let me write this down. If, if A is singular, then that -- what does singular mean? It means that it takes some vector x into zero. Some non-zero vector, that's why -- will be the eigenvector into zero. Then lambda equals zero is an eigenvalue. But we're interested in all eigenvalues now, lambda equals zero is not, like, so special anymore. OK. So the question is, how do we find these x-s and lambdas? And notice -- we don't have an equation Ax equal B anymore.
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