Up to a point, having complex eigenvalues
works the way you’d probably expect.
There are differences, however.
In this video, we investigate this case.
Suppose that you have a pair of complex eigenvalues,
p plus or minus q i.
The eigenvectors associated with these will
also be complex.
Let’s select one of the eigenvalues, say
P + qi
With it, we have an eigenvector, which we
can write a + bi.
From past experience, you probably expect,
correctly, that we will form a solution as
follows; the eigenvector times e raised to
the eigenvalue times t.
And it will probably come as no surprise when
we rewrite this complex exponential in terms
of sines and cosines,
x = (a+bi)e^(pt) [ cos qt + i sin qt)
The i is different; the last time we did this,
we had an arbitrary constant in front of the
sine, and we absorbed the imaginary unit into
the constant.
Here, we let it be.
We’re going to get two solutions from this
eigenvalue.
To do this, let’s separate these terms so
that the real and imaginary terms are grouped
together.
We’ll pull out this e^(pt) …
Then we’ll FOIL these terms; a times the
cosine of qt plus a times I the sine
of qt + b I cos qt minus b sin qt. these I’s
give us negative one when we multiply them.
Now, we’ll group our terms.
So we have a real part, and an imaginary part.
Let’s go to a new page.
The real part and the imaginary part are both
solutions, taken individually.
How do we know that?
Well, we’re on the complex plane now, we’ve
got the real and the imaginary axis.
And this is a solution on the complex plane.
So since it’s a solution on the complex
plane, it’s a solution on the real number
line, because the real number line is part
of the complex plane.
But on the real number line, this imaginary
part must be 0.
So we must have this purely real solution.
Likewise, if this is a solution on the complex
plane, it’s a solution on the imaginary
axis.
On the imaginary axis, this real part is 0.
These real and imaginary solutions are linearly
independent from each other.
So 
the
complex eigenvalue p + qi gives us two linearly
independent solutions, this and this.
Notice that although this comes from the imaginary
part of the solution, it doesn’t have any
I’s in it.
And we do not need to work with its complex
conjugate eigenvalue, at all.
In this video, we have seen how to use complex
eigenvalues to get solutions.
The process is very similar to the process
we used when we had a single linear differential
equation, with some modifications.
