In the last video, we started to
talk about what eigenvalues and
eigenvectors were, are.
We basically gave the definition and
we're gonna continue exploring what
they are from the point of view of
diagonalization. See, we will say
that a matrix, or if you prefer a linear
operator is diagonalizable if you can
find a basis consisting of eigenvectors.
If you have a basis of Rn, consisting
of eigenvectors of the matrix A,
or a basis of the abstract vector 
space V consisting of eigenvectors
of linear operator L, we'll say that 
L or A is diagonalizable.
A fair question is alright, so what?
What's so great about a basis of
eigenvectors? And I'm about to show you.
I'm gonna do it in the case of a linear
operator and then we're gonna switch
back to matrices.
Let's suppose we have a linear operator
and let's suppose we have a basis and
let's suppose that every single basis
vector is an eigenvector.
So L applied to any basis vector gives you
a multiple of that basis vector.
Then let's try to figure out what the 
matrix of the linear transformation is.
You know how to figure out the matrix
of a linear transformation.
You have to apply L to all of the basis
vectors and you know that L(b_1)
is an eigenvector, so that gives you
lambda_1 b_1.
And L(b_2) gives you lambda_2 b_2.
And L(b_n) gives you lambda_n b_n.
That's the first step towards figuring
out what the matrix of a linear
transformation is.
The next step is to figure out what the
coordinates of all of these are.
Let's look at L(b_1).
Since L(b_1) is lambda_1 b_1 + 0b_2 
+ 0b_3 and so on, so the coordinates
of L(b_1) are lambda_1 0 0.
L(b_2) can be thought of as 0b_1 + 
lambda_2 b_2 + 0b_3, so on.
So the coordinates of L(b_2) 
are 0 lambda_2 0 and so on,
and you keep going.
And the coordinates of L(b_n) in the
B basis are just gonna be 0 0 0 and then
lambda_n.
The way we make the matrix a linear
transformation is we take -
the next page,
that's what we take.
Basis up to L(b_n) in the B basis
and concatenate them.
That's how we find matrices
of linear transformations.
In this case, L(b_1) is lambda_1 0 0 0.
L(b_2) is 0 lambda_2 0 0 0.
L(b_n) is 0 0 0 lambda_n and so
you wind up with this big diagonal matrix.
And what's so weird about 
diagonal matrices?
It means that if you have any problem
involving this matrix, the different
variables don't talk to each other.
If you work in a basis of eigenvectors,
then any problem having to do with
linear transformation is going to be
completely decoupled.
This is why you should care.
Now, there's another way of looking
at this which in terms of factorization.
So I said we'd get back to matrices 
and here we are.
Let's suppose that we've got a matrix A
or if you prefer, you can think of as
linear transformation that 
is multiplied by that matrix.
We've got a basis of Rn, consisting
of eigenvectors of A.
Then if you write the matrix of L
in the standard basis, that's just A.
We just said that if you write the
matrix of L in the B basis with
B as the basis of eigenvectors, 
then you get a diagonal matrix.
I'm gonna call this diagonal matrix
capital D for diagonal.
How are these related?
A matrix in one basis and a matrix in
another basis, they're related by
change of basis matrices.
You want to understand what A is.
You have to do a change of basis
from what it is in the B basis.
In other words, A, that's L in the
E basis, is some matrix, I'm gonna call -
use P as shorthand for P_EB.
PDP^-1.
What's P?
Well P is P_EB and how do you get P_EB?
You write down b_1 through b_n
and concatenate them.
There's this lovely factorization that if
you know the eigenvalues and the
eigenvectors of a matrix, 
you know the matrix.
You know what D is because D
is just all the eigenvalues.
You know what P is.
P is all the eigenvectors written 
side by side.
You know what P^-1 is. You just have
to take P and you take its inverse.
If you know the eigenvalues and 
eigenvectors of a matrix,
you know the matrix.
And there's this cute factorization that
turns out to be very useful.
That's it for talking about what is
diagonalization, what are eigenvalues
and eigenvectors.
The next step is gonna be getting ways
to figure out what are the eigenvalues
and eigenvectors of a matrix.
If somebody hands you A, how do you
figure out D? How do you figure out P?
That's in our next video.
