Generalized Eigenvectors of an Eigenpair
Just as we saw when dealing with eigenvalues, eigenvectors alone won’t cut it. We want to consider the kernel not just of one transformation, but of its powers. Specifically, we will say that is a generalized eigenvector of the eigenpair
if for some power
we have
The same argument as before tells us that the kernel will stabilize by the time we take powers of an operator, so we define the generalized eigenspace of an eigenpair
to be
Let’s look at these subspaces a little more closely, along with the older ones of the form , just to make sure they’re as well-behaved as our earlier generalized eigenspaces are. First, let
be one-dimensional, so
must be multiplication by
. Then the kernel of
is all of
if
, and is trivial otherwise. On the other hand, what happens with an eigenpair
? Well, one application of the operator gives
for any nonzero . But this will always be itself nonzero, since we’re assuming that the polynomial
has no roots. Thus the generalized eigenspace of
will be trivial.
Next, if is two-dimensional, either
has an eigenvalue or it doesn’t. If it does, then this gives a one-dimensional invariant subspace. The argument above shows that the generalized eigenspace of any eigenpair
is again trivial. But if
has no eigenvalues, then the generalized eigenspace of any eigenvalue
is trivial. On the other hand we’ve seen that the kernel of
is either the whole of
or nothing, and the former case happens exactly when
is the trace of
and
is its determinant.
Now if is a real vector space of any finite dimension
we know we can find an almost upper-triangular form. This form is highly non-unique, but there are some patterns we can exploit as we move forward.
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