There's this little theory called Galois Theory. And what it tells us is that
you cannot come up with a closed-form
formula for the roots of a polynomial of
degree 5 or larger.  Well, it turns out
that given any polynomial of degree m
you can construct a matrix that has that
polynomial as its characteristic
polynomial.  Well, you'd have to scale the
matrix a little bit to ensure that
lambda can have a coefficient. So in
other words any polynomial that has an,
you know, the coefficient associated with
the first term equal to 1, you can
construct a matrix that has that
polynomial as its characteristic
polynomial.  And therefore, if we could
come up with a formula for finding the
eigenvalues of a matrix as a closed form,
then we would have a formula for finding
the roots of a polynomial of degree 5 or
greater.  And that would contradict what
Galois Theory tells us.  So we
need to go about our business in some
other way.  And what we're going to find
out is that it's easier to find eigenvectors
and from those eigenvectors find
the eigenvalues than the other way
around.
