Yesterday, we defined eigenvalues, eigenvectors, and
eigenspaces. But we didn’t talk about how to actually find them
(though one commenter decided to jump the gun a bit). It turns
out that determining the eigenspace for any given eigenvalue is
the same sort of problem as determining the kernel. Let’s say
we’ve got a linear endomorphism and a scalar value . We want to
determine the subspace of consisting of those eigenvectors
satisfying the equation First, let’s adjust the right hand side.
Instead of thinking of the scalar product of and , we can write
it as the action of the transformation , where is the identity
transformation on . That is, we have the equation Now we can do
some juggling to combine these two linear transformations being
evaluated at the same vector : And we find that the -eigenspace
of is the kernel . Now, as I stated yesterday most of these
eigenspaces will be trivial, just as the kernel may be trivial.
The interesting stuff happens when is nontrivial. In this case,
we’ll call an eigenvalu