Let's take the matrix
The characteristic polynomial has to do with finding the eigenvalues of a matrix. The eigenvalues of a matrix are defined as follows:
The eigenvalues of a matrix are the numbers such that for
Here is a vector of the appropriate size so that the matrix multiplication is defined. If you find such an and a , then is called an eigenvector. is, in general, a complex number.
The geometric interpretation of eigenvalues is that when you multiply an eigenvector by the matrix , it's the same thing as multiplying the eigenvector by a simple number. Multiplying by a simple number is basically stretching the eigenvector in some way. It doesn't change the eigenvector's direction. The whole eigenvalue problem, as we call it, has to do with finding a "shortcut" to matrix multiplication.
Why are we interested in the eigenvalue problem? Tons of reasons! You can use eigenvalues and eigenvectors (lump those two concepts together and call it the "eigensystem") to solve systems of ordinary differential equations, which in turn can model the behavior of an electrical circuit or a mass/spring system. This very problem shows up all over the place in Quantum Mechanics.
So, let's rearrange the eigenvalue problem quick, while nobody's looking:
Now, I want to right-factor the out of there. But what's left had better all be of the same type: a matrix. So, I can insert the identity matrix in there to make it all the same:
where is the same size as . You'd agree I haven't changed anything, right? So now I get to factor (technically, right-factor: matrix multiplication is NOT, in general, commutative):
Now, remember: . This system looks like
It's a fact from linear algebra that a homogeneous system can only have nonzero (nontrivial) solutions if the matrix is singular. Another way of saying that is that
is the characteristic equation. The characteristic polynomial is the LHS of this equation.
So let's find this characteristic polynomial for our matrix above. Evidently, we need to form the matrix . Well, that gives us
Now that we've assembled that ingredient, we need to take its determinant and set it equal to zero:
If that step mystifies you, you should review determinants. We can simplify this just a bit:
So the characteristic polynomial associated with our matrix is
To find the eigenvalues of our matrix , you'd set the characteristic polynomial equal to zero and solve:
The solutions are, of course,
Does this all make sense?