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, also 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 equation is

(b)

The roots of this equation are ? ₁=0, ? ₂=1, ? ₃=3. To compute the eigenvector corresponding the ? ₃, we substitute ?= ? ₃ into Eq. (9.2), obtaining

(c)

We know that the determinant of the coefficient matrix is zero, so that the equations are not linearly independent. Therefore, we can assign an arbitrary value to any one component of x and use two of the equations to compute the other two components. Choosing x