Page 175 - Linear Algebra Done Right

Chapter 8. Operators on Complex Vector Spaces
164
Generalized Eigenvectors
Unfortunately some operators do not have enough eigenvectors to
lead to a good description. Thus in this section we introduce the con-
cept of generalized eigenvectors, which will play a major role in our
description of the structure of an operator.
To understand why we need more than eigenvectors, let’s examine
the question of describing an operator by decomposing its domain into
invariant subspaces. Fix T ∈L(V). We seek to describe T by finding a
“nice” direct sum decomposition

8.1 V = U 1 ⊕· · · U m ,

where each U j is a subspace of V invariant under T. The simplest pos-
sible nonzero invariant subspaces are one-dimensional. A decompo-
sition 8.1 where each U j is a one-dimensional subspace of V invariant
under T is possible if and only if V has a basis consisting of eigenvectors
of T (see 5.21). This happens if and only if V has the decomposition

8.2 V = null(T − λ 1 I) ⊕· · · null(T − λ m I),

where λ 1 ,...,λ m are the distinct eigenvalues of T (see 5.21).
In the last chapter we showed that a decomposition of the form
8.2 holds for every self-adjoint operator on an inner-product space
(see 7.14). Sadly, a decomposition of the form 8.2 may not hold for
more general operators, even on a complex vector space. An exam-
ple was given by the operator in 5.19, which does not have enough
eigenvectors for 8.2 to hold. Generalized eigenvectors, which we now
introduce, will remedy this situation. Our main goal in this chapter is
to show that if V is a complex vector space and T ∈L(V), then
V = null(T − λ 1 I) dim V ⊕· · · null(T − λ m I) dim V ,

where λ 1 ,...,λ m are the distinct eigenvalues of T (see 8.23).
Suppose T ∈L(V) and λ is an eigenvalue of T. A vector v ∈ V is
called a generalized eigenvector of T corresponding to λ if

j
8.3 (T − λI) v = 0
for some positive integer j. Note that every eigenvector of T is a gen-
eralized eigenvector of T (take j = 1 in the equation above), but the
3
converse is not true. For example, if T ∈L(C ) is defined by