Eigenvalue Problems

Abstract

The problem considered in this chapter is: given an n × n matrix A, find the number(s) λ $$λ$$ and nonzero vectors x that satisfy 4.1Ax=λx. $$\displaystyle{ \mathbf{A}\mathbf{x} =\lambda \mathbf{x}. }$$ This is an eigenvalue problem, where λ $$λ$$ is an eigenvalue and x is an eigenvector. There are a couple of observations worth making about this problem. First, x = 0 is always a solution of (4.1), and so what is of interest are the nonzero solutions. Second, if x is a solution, then αx, for any number α, is also a solution.