Page 201 - Advanced Linear Algebra

Chapter 8

Eigenvalues and Eigenvectors

Unless otherwise noted, we will assume throughout this chapter that all vector
spaces are finite-dimensional.
Eigenvalues and Eigenvectors

We have seen that for any B ²= ³ , the minimal and characteristic
polynomials have the same set of roots (but not generally the same multiset of
roots). These roots are of vital importance.

Let (~ ´ µ 8 be a matrix that represents . A scalar  - is a root of the
characteristic polynomial ²%³ ~ ²%³ ~ det ²%0 c (³ if and only if

(
det²0 c (³ ~ (8.1 )

that is, if and only if the matrix 0c ( is singular. In particular, if dim ²= ³ ~ ,
(
)
then 8.1 holds if and only if there exists a nonzero vector %- for which
²0 c (³% ~

or equivalently,
( %~ %

If ´#µ ~ % , then this is equivalent to
8

8
8
´ µ ´#µ ~ ´#µ 8
or in operator language,
#~ #
This prompts the following definition.

=
Definition Let be a vector space over a field and let  - B ² = . ³
(
1 A scalar ) is an eigenvalue or characteristic value ) of if there
-
exists a nonzero vector #= for which