In this video we are gonna talk about
the geometry of eigenvalues,
both real and complex. So for examples,
suppose I give you a matrix A. I tell you
its eigenvalues are 3, 5, 2+i and 2-i.
Just knowing the eigenvalues, what can
you tell me about the matrix?
Well, from the real eigenvalues, you can 
immediately tell there are certain
special directions that all the matrix
does in that direction is stretched by
a factor of 3 or stretched by factor of 5.
I've drawn if Ax is λx, then for any
vector in the x direction, it just 
multiplies it by a factor of λ,
in the direction of the, 
this eigenvector stretches by 3.
In the direction of this eigenvector
it stretches by 5.
And that's all it's doing, its stretching
in a particular direction.
Now things are more interesting
if we look at the complex eigenvalues.
If you have eigenvalue say 2+i, 2-i,
or in general, we talk about eigenvalues
a +- bi. What can you figure out about
what the matrix is doing?
We saw an example of that with
the matrix that was doing stretching
and rotating and in fact, it turns out
that it always involves some stretching
and rotation.
So let's suppose we've got a complex
eigenvalue and a corresponding complex
eigenvector I denote by the Greek letter
ξ. And since it's a complex eigenvector
it got a real part. We shall denote it 
as x and an imaginary part that I will
denote as y. So let's write down
the eigenvalue equation.
A (x + iy) is (a + bi) (x + iy).
And of course this is Ax + iAy.
And we multiply this out.
a times x + x times iy + b times ix + 
bi times iy that gives you - by.
And we look at this and say, "oh but 
this is a real and imaginary part."
The real part of the left hand side
has to equal the real part of
the right hand side. The imaginary part
of the left hand side has to equal
the imaginary part of the right hand side.
And A was a real matrix so Ax is real
and Ay is real. So the real part of 
this equation is that Ax is ax -by.
The imaginary part is that Ay is bx+ay.
Okay? So now what we are gonna do
is we are gonna look at the plane,
spanned by x and y, except actually
I am gonna take a basis, with the first
basis vector is y. It's the imaginary
part of our eigenvector.
The second basis vector is x. It's 
the real part of our eigenvector.
And relative to that basis. Let's see.
Relative to that basis, we have that Ay
was ay + bx.
So this, our first eigenvector was y.
And the second eigenvector was x.
And Ab_2 is -bb_1 + ab_2 and
I apologize for using the same letter,
a and b for entries of the matrix
and b_1 and b_2 for basis vectors.
A little bit confusing.
Sorry about that.
So what that means is that if we want
the matrix of A in the b basis,
or let's see the coordinates of this 
in the b basis are just a, b.
And the coordinates of this in 
the b basis are -b, a.
So the matrix of A in the b basis
at least A restricted to this plane,
is given by this. That's exactly
a rotation and a stretch.
It takes the first basis vector to
a times that basis vector plus
b times the other basis vector.
And it takes second basis vector
to a times the second basis vector
minus b times the our first basis vector.
This looks exactly like
a rotation and a stretch.
So whenever you see an eigenvalue
of a + bi, you should always think
there is some rotation and stretching 
going on, stretching by a^2 + b^2,
rotating by tan^-1(b/a).
So now let's go back and look at
our original matrix, 2 -1 1 2
and see how this fits into the scheme.
We solved that the eigenvector was i 1.
You can write that as (0 1) + i (1 0).
So this is what we called x and 
this is what we called y.
So our first basis vector was 
the imaginary part of that.
That's 1 0. The second basis vector
was the real part of it, which is 0 1.
And so the plane in which things are
rotating is the plane spanned by
1 0 and 0 1. Or is that just r2?
And in fact, we already knew that 
this was rotation and stretching.
But now we see how by looking
at the imaginary and real part
of the eigenvector, you get this picture.
By the way, another thing you could do
is you could look at the eigenvector 
with eigenvalue a - bi in which case
you would want to take the real part
first and then the imaginary part.
I prefer to do it with a + bi and
doing imaginary than real.
