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.