Eigenvalue problems

Lu = λu (5.1)

exists. The resulting nontrivial solution is called an eigenvector of L corresponding to the eigenvalue λ.

An eigenvector corresponding to an eigenvalue of a linear operator is not unique. Let λ be an eigenvalue of a linear operator L, and let u ≠ 0 be a corresponding eigenvector satisfying Equation 5.1. Defi ne v = cu for an arbitrary c ≠ 0. Consider

Lv L u

Lu

u

v

=

=

=

=

( )c

c

cλ

λ

(5.2)

Equation 5.2 illustrates that v is also an eigenvector of L corresponding to the eigenvalue λ.