Page 93 - Linear Algebra Done Right

Chapter 5. Eigenvalues and Eigenvectors
80
Polynomials Applied to Operators
The main reason that a richer theory exists for operators (which
map a vector space into itself) than for linear maps is that operators
can be raised to powers. In this section we define that notion and the
key concept of applying a polynomial to an operator.
If T ∈L(V), then TT makes sense and is also in L(V). We usually
2
write T instead of TT. More generally, if m is a positive integer, then
T m is defined by
T m = T ...T .

m times
0
For convenience we define T to be the identity operator I on V.
Recall from Chapter 3 that if T is an invertible operator, then the
inverse of T is denoted by T −1 .If m is a positive integer, then we define
m
T −m to be (T −1 ) .
You should verify that if T is an operator, then
m n
m
T T n = T m+n and (T ) = T mn ,
where m and n are allowed to be arbitrary integers if T is invertible
and nonnegative integers if T is not invertible.
If T ∈L(V) and p ∈P(F) is a polynomial given by
2
p(z) = a 0 + a 1 z + a 2 z +· · ·+ a m z m
for z ∈ F, then p(T) is the operator defined by

2
m
p(T) = a 0 I + a 1 T + a 2 T + ··· + a m T .
2
For example, if p is the polynomial defined by p(z) = z for z ∈ F, then
2
p(T) = T . This is a new use of the symbol p because we are applying
it to operators, not just elements of F. If we fix an operator T ∈L(V),
then the function from P(F) to L(V) given by p p(T) is linear, as
you should verify.
If p and q are polynomials with coefficients in F, then pq is the
polynomial defined by

(pq)(z) = p(z)q(z)

for z ∈ F. You should verify that we have the following nice multiplica-
tive property: if T ∈L(V), then