Page 179 - Linear Algebra Done Right

Chapter 8. Operators on Complex Vector Spaces
168
where the first and third equalities come from 3.4 and the second equal-
dim V
m
ity comes from 8.6. We already know that range T
⊃ range T .We
m
just showed that dim range T dim V = dim range T , so this implies that
m
range T dim V = range T , as desired.
The Characteristic Polynomial
Suppose V is a complex vector space and T ∈L(V). We know that
V has a basis with respect to which T has an upper-triangular matrix
(see 5.13). In general, this matrix is not unique—V may have many
different bases with respect to which T has an upper-triangular matrix,
and with respect to these different bases we may get different upper-
triangular matrices. However, the diagonal of any such matrix must
contain precisely the eigenvalues of T (see 5.18). Thus if T has dim V
distinct eigenvalues, then each one must appear exactly once on the
diagonal of any upper-triangular matrix of T.
What if T has fewer than dim V distinct eigenvalues, as can easily
happen? Then each eigenvalue must appear at least once on the diag-
onal of any upper-triangular matrix of T, but some of them must be
repeated. Could the number of times that a particular eigenvalue is
repeated depend on which basis of V we choose?
If T happens to have a You might guess that a number λ appears on the diagonal of an
diagonal matrix A with upper-triangular matrix of T precisely dim null(T − λI) times. In gen-
2
respect to some basis, eral, this is false. For example, consider the operator on C whose
then λ appears on the matrix with respect to the standard basis is the upper-triangular matrix
diagonal of A precisely

dim null(T − λI) times, 5 1 .
as you should verify. 0 5
For this operator, dim null(T − 5I) = 1 but 5 appears on the diago-
nal twice. Note, however, that dim null(T − 5I) 2 = 2 for this oper-
ator. This example illustrates the general situation—a number λ ap-
pears on the diagonal of an upper-triangular matrix of T precisely
dim null(T − λI) dim V times, as we will show in the following theorem.
Because null(T − λI) dim V depends only on T and λ and not on a choice
of basis, this implies that the number of times an eigenvalue is repeated
on the diagonal of an upper-triangular matrix of T is independent of
which particular basis we choose. This result will be our key tool in
analyzing the structure of an operator on a complex vector space.