9.1 Introduction

The standard form of the matrix eigenvalue problem is

(9.1)

where A is a given n n matrix. The problem is to find the scalar ? and the vector x. Rewriting Eq. (9.1) in the form

(9.2)

it becomes apparent that we are dealing with a system of n homogeneous equations. An obvious solution is the trivial one x= 0. A nontrivial solution can exist only if the determinant of the coefficient matrix vanishes; that is, if

(9.3)

Expansion of the determinant leads to the polynomial equation known as the characteristic equation

which has the roots ? _i, i=1, 2, , n, called the eigenvalues of the matrix A. The solutions x _i of ( A ? ? _i I) x= 0 are known as the eigenvectors.

As an example, consider the matrix

(a)

The characteristic...