Eigenvectors and eigenvalues | Chapter 14, Essence of linear algebra

Eigenvectors and eigenvalues is one of those topics

that a lot of students find particularly unintuitive.

Questions like, why are we doing this and what does this actually mean,

are too often left just floating away in an unanswered sea of computations.

And as I've put out the videos of this series,

a lot of you have commented about looking forward to visualizing this topic in particular.

I suspect that the reason for this is not so much that

eigenthings are particularly complicated or poorly explained.

In fact, it's comparatively straightforward, and

I think most books do a fine job explaining it.

The issue is that it only really makes sense if you have a

solid visual understanding for many of the topics that precede it.

Most important here is that you know how to think about matrices as

linear transformations, but you also need to be comfortable with things

like determinants, linear systems of equations, and change of basis.

Confusion about eigenstuffs usually has more to do with a shaky foundation in

one of these topics than it does with eigenvectors and eigenvalues themselves.

To start, consider some linear transformation in two dimensions, like the one shown here.

It moves the basis vector i-hat to the coordinates 3, 0, and j-hat to 1, 2.

So it's represented with a matrix whose columns are 3, 0, and 1, 2.

Focus in on what it does to one particular vector,

and think about the span of that vector, the line passing through its origin and its tip.

Most vectors are going to get knocked off their span during the transformation.

I mean, it would seem pretty coincidental if the place where

the vector landed also happened to be somewhere on that line.

But some special vectors do remain on their own span,

meaning the effect that the matrix has on such a vector is just to stretch it or

squish it, like a scalar.

For this specific example, the basis vector i-hat is one such special vector.

The span of i-hat is the x-axis, and from the first column of the matrix,

we can see that i-hat moves over to 3 times itself, still on that x-axis.

What's more, because of the way linear transformations work,

any other vector on the x-axis is also just stretched by a factor of 3,

and hence remains on its own span.

A slightly sneakier vector that remains on its own

span during this transformation is negative 1, 1.

It ends up getting stretched by a factor of 2.

And again, linearity is going to imply that any other vector on the diagonal

line spanned by this guy is just going to get stretched out by a factor of 2.

And for this transformation, those are all the vectors

with this special property of staying on their span.

Those on the x-axis getting stretched out by a factor of 3,

and those on this diagonal line getting stretched by a factor of 2.

Any other vector is going to get rotated somewhat during the transformation,

knocked off the line that it spans.

As you might have guessed by now, these special vectors are called the eigenvectors of