Here's a quick question for you.
How many eigenvectors are there?
Not on the planet, in general.
Okay?
Okay, well, normally for
a real symmetric, and by the way,
these things are ty, typically symmetric
because if you take a look here, right?
I've got the sum of the squares,
but on the, off diagonal terms,
I've got the same, values.
The sum of the cross terms, right?
So if I have a real symmetric matrix
in general if it's sized end by end,
I normally would have
N real eigenvectors.
And distinct eigenvectors.
There are some degeneracies if you
have identical eigenvalues and
then you have, ambiguity We're not
going to worry about that too much but
just remember that normally,
there is i say normally is later
it's going to be a little different.
Normally with the N by N matrix,
im going to have N eigenvectors.
So graphically,
we can look at that like this.
So here i have a beautiful set of
points, aren't those delightful?
Distributed in two dimensions and
which way do you think is
the principal eigenvector?
Well, probably,
something along like that.
Right, that's going to be
the axis of least inertia, right,
or of greatest variance.
Get used to this idea that it's
greatest variance, and in fact,
that's what this nice
little picture shows us.
And this is showing us lambda 1,
meaning the first eigenvalue,
large eigenvalue and
what direction is the next eigenvector?
Well as you know,
it's perpendicular to that, right?
Cause eigenvectors are perpendicular
to each other, and go, and
the next one is the next most variance,
et cetera.
With two dimensions, it's hard to show.
There's only two here.
Why am I telling you all this?
