In this video, we're gonna talk about
complex eigenvalues and eigenvectors.
We'll start with an example.
Let's take a look at the matrix
2, 1, -1, 2.
So does that have any eigenvalues
and eigenvectors?
And if you think about
how this acts in the plane,
you might think the answer would be no.
Because what this matrix does,
is it sends 1,0 to 2,1
and it sends 0,1 to -1,2.
Now here's a picture of that.
It sends 1,0 to 2,1
and 0,1 to -1,2.
So what it seems to be doing
is it seems to be rotating by this angle,
the arctan(1/2), and stretching
by this length, which is √5.
So if it stretches and rotates,
no matter what vector you start out with,
you start out with a vector here,
it's gonna rotate it
and you're not gonna get
a multiple of that vector.
You start out with a vector here,
it's gonna rotate it.
You're gonna get something
that's not a multiple of that vector.
No matter what vector in R2 you pick,
Ax cannot possibly be parallel to x.
So it would seem that the matrix doesn't
have any eigenvalues and eigenvectors.
Except that's not quite right.
It doesn't have any eigenvectors in R2.
It doesn't have any real eigenvectors.
Let's suppose we look at
vectors that aren't in R2,
but instead are in C2.
What happens if we feed it vectors
which have complex components?
And in fact we discover that if you
feed it the vector (1, i),
you multiply it out,
2, -1, 1, 2 times (i, 1)—
I said (1, i), I meant (i, 1).
Gives us (2i - 1, 2 + i).
And 2i - 1 is i(2 + i),
and this is 1(2 + i),
so the whole thing gives us
(2 + i)*(i, 1).
In other words, (i, 1) is an—
This is a complex eigenvector,
and that's the complex eigenvalue.
And likewise, if you look at (-i, 1),
then that's also an eigenvector,
only now the eigenvalue is 2 - i
rather than 2 + i.
So the upshot is that there are
no real eigenvalues or eigenvectors
to this matrix,
but there are complex eigenvalues,
namely 2 ± i ,
and there are complex eigenvectors,
namely (± i, 1).
Two eigenvalues, two eigenvectors,
that's what you'd expect
from a 2x2 matrix.
Okay, now the way that you find
complex eigenvalues
is the same as the way that
you find real eigenvalues.
You compute the characteristic polynomial,
and then the eigenvalues
are the roots of the polynomial.
The real eigenvalues are the real roots,
the complex eigenvalues
are the complex roots.
So let's see how that words for
the matrix we've been looking at.
You take λ (lambda) times identity
minus that matrix
and get λ - 2, 0 - (-1),
0 - 1, λ - 2.
Then you take the determinant
of that matrix,
and you get
(λ - 2)(λ - 2) - (1)(-1)—
in other words,
(λ - 2)^2 + 1.
And if you expand that out, that's
λ^2 - (4)λ + 5.
Now if you want to find the roots of this,
you can get it from here,
or you could just use
the quadratic formula.
And you just—you get
-b ± (√(b^2 - 4ac))/2a,
but 4ac is 20, that's bigger than b^2.
And we say that doesn't matter.
We get (4 ± √-4 )/ 2.
√-4 is 2i.
So we get (4 ± 2i)/2.
And that's 2 ± i.
Bingo.
We got our eigenvalues.
Same method as always.
How bout eigenvectors?
Well we also use the
same method as always.
We take the matrix minus λ*I.
Or if you prefer, you could take λ*I
minus the matrix, it doesn't matter.
They're just negatives of each other.
And you row reduce them to find
a basis for the null space.
In this case, this is A - (2 + i)
times the identity.
And if you multiply the top row by i,
(-i)(i) is 1.
(-1)(i) is -i.
And then subtract the
first row from the second.
Bingo, you got yourself
the reduced row echelon form.
And we're looking for the null space,
so each row gives you an equation.
x_1 - ix_2 = 0.
In other words, x_1 = ix_2.
And of course x_2 is itself.
So our vector is a multiple of (i, 1).
That's how you get the eigenvectors.
Just like with real eigenvalues,
you wanna take the A - λ*I,
row reduce it, and find
a basis for the null space.
So in summary,
we already know that real polynomials
can have complex roots,
and real matrices can have complex
eigenvalues and eigenvectors.
Cause the real matrix
will have a real polynomial
as this characteristic polynomial,
and that polynomial may
or may not have complex roots.
Nothing wrong with complex eigenvalues.
Since it's a real polynomial, the roots
come in complex conjugate pairs.
So the complex eigenvalues
of a real matrix...
come in complex conjugate pairs.
And the complex eigenvectors also
come in complex conjugate pairs.
In the next video, we're gonna see
how these two pairs,
you look at the real and imaginary
parts, and that gives us a plane
where something interesting has happened.
We'll examine the geometry of complex—
the real geometry of complex eigenvectors.
