An Introduction to Numerical Methods and Analysis, 2nd Edition

CHAPTER 8

APPROXIMATE SOLUTION OF THE ALGEBRAIC EIGENVALUE PROBLEM

In this chapter we will discuss, in some detail, some iterative methods for finding single eigenvalue–eigenvector pairs (eigenpairs is a common term) of a given real matrix A; we will also give an overview of more powerful and general methods that are commonly used to find all the eigenpairs of a given real A. As in Chapter 7, our discussion here will depend a fair amount on MATLAB, although we will look at some algorithms in detail.

8.1 EIGENVALUE REVIEW

The algebraic eigenvalue problem is as follows: Given a matrix , find a nonzero vector x ⁿ and the scalar λ such that

Note that this says that the vector Ax is parallel to x, with λ being an amplification factor, or gain. Note also that the above implies that

showing (by Theorem 7.1) that A − λI is a singular matrix. Hence, det(A − λI) = 0; it is easy to show that this determinant is a polynomial (of degree n) in λ, known as the characteristic polynomial of A, p(λ), so ...