Advanced Linear Algebra, 2nd Edition

Theory of a Single Linear Operator 135

Definition 4.18 The polynomials d

1

(x), d

2

(x), . . . , . . . , d

r

(x) are called the

invariant factors of T .

Definition 4.19 Let V be an n-dimensional vector space and T be a linear

operator on V. The polynomial (of degree n) obtained by multiplying the in-

variant factors of T is called the characteristic polynomial of T . It is

denoted by χ

T

(x).

Note that one of the invariant factors is µ

T

(x) and therefore µ

T

(x) divides

the characteristic polynomial, χ

T

(x). Since µ

T

(T ) = 0

V →V

, we have proved

the following:

Theorem 4.17 χ

T

(T ) = 0

V →V

.

The fact tha t the operator obta ine d when the characteristic polynomial of

T is eva luated at T is the zero operator goes by the name of the Cay-

ley–Hamilto n theorem. In this