Page 215 - Linear Algebra Done Right

The Characteristic Polynomial
n
Thus u m − u ∈ null p(T) , and hence u m , which equals u + (u m − u),
n
n
is in U + null p(T) . In other words, U m ⊂ U + null p(T) . Therefore
n
n
V = U+U m ⊂ U+null p(T) , and hence U+null p(T) = V, completing 205
the proof.
As we saw in the last chapter, the eigenvalues of an operator on a
complex vector space provide the key to analyzing the structure of the
operator. On a real vector space, an operator may have fewer eigen-
values, counting multiplicity, than the dimension of the vector space.
The previous theorem suggests a definition that makes up for this defi-
ciency. We will see that the definition given in the next paragraph helps
make operator theory on real vector spaces resemble operator theory
on complex vector spaces.
Suppose V is a real vector space and T ∈L(V). An ordered pair
2
(α, β) of real numbers is called an eigenpair of T if α < 4β and Though the word
eigenpair was chosen
2
T + αT + βI
to be consistent with
is not injective. The previous theorem shows that T can have only the word eigenvalue,
finitely many eigenpairs because each eigenpair corresponds to the this terminology is not
characteristic polynomial of a 2-by-2 matrix on the diagonal of 9.10 in widespread use.
and there is room for only finitely many such matrices along that diag-
onal. Guided by 9.9, we define the multiplicity of an eigenpair (α, β)
of T to be
2
dim null(T + αT + βI) dim V .
2
From 9.9, we see that the multiplicity of (α, β) equals the number of
2
times that x +αx+β is the characteristic polynomial of a 2-by-2 matrix
on the diagonal of 9.10.
3
As an example, consider the operator T ∈L(R ) whose matrix (with
respect to the standard basis) equals
 
3 −1 −2
 
 3 2 −3  .
1 2 0
You should verify that (−4, 13) is an eigenpair of T with multiplicity 1;
2
note that T − 4T + 13I is not injective because (−1, 0, 1) and (1, 1, 0)
are in its null space. Without doing any calculations, you should verify
that T has no other eigenpairs (use 9.9). You should also verify that 1 is
an eigenvalue of T with multiplicity 1, with corresponding eigenvector
(1, 0, 1), and that T has no other eigenvalues.