Numerical Methods

4

Eigenvalues and Eigenvectors

The theory of eigenvalues and eigenvectors is a powerful tool to solve the problems in economics, engineering, and physics. These problems revolve around the singularities of A – λI, where λ is a parameter and A is a linear transformation. The aim of this chapter is to study the numerical methods to find eigenvalues of a given matrix A.

4.1 EIGENVALUES AND EIGENVECTORS

Definition 4.1. Let A be a square matrix of dimension n × n. The scalar λ is said to be an eigenvalue for A if there exists a non-zero vector X of dimension n such that AX = λX.

The non-zero vector X is called the eigenvector corresponding to the eigenvalue λ. Thus, if λ is an eigenvalue for a matrix A, then

 

AX = λX, X ≠ 0           (4.1)

or