1550078481-Ordinary_Differential_Equations__Roberts_

350 Ordinary Differential Equations

or the equiva lent equation Ax - .Ax= (A - .AI)x = 0 if and only if

(2) det (A - .AI) = 0.

If A is an n x n constant matrix, then equation (2) is a polynomial of degree
n in .A- called the characteristic polynomial of A. So each n x n matrix
A has n eigenvalues .A 1 , .A 2 , ... , An, some of which may be repeated. If A is
a root of equation (2) m times, then we say A is an eigenvalue of A with
multiplicity m. Every eigenvalue has at least one associated eigenvector. If

.A is an eigenvalue of multiplicity m > 1, then there are k linearly independent

eigenvectors associated with A where 1 ::;; k ::;; m. When all the eigenvalues

.A 1 , .>- 2 , ... , An of a matrix A have multiplicity one (that is, when the roots

of equation (2) are distinct), then the associated eigenvectors x1, x2, ... , X n

are linearly independent. However, if some eigenvalue of A has multiplicity

m > 1 but fewer than m linearly independent associated eigenvectors, then

there will be fewer than n linearly independent eigenvectors associated with
A. As we shall see, this situation will lead to difficulties when trying to solve
a system of differential equations which involves the matrix A.

EXAMPLE 1 Calculating Eigenvalues and Eigenvectors

of a 2 x 2 Matrix

Find the eigenvalues and associated eigenvectors of the matrix

-3)

-2.

SOLUTION

The characteristic equation of A is det (A - .AI) = 0. For the given matrix

det (A - .AI) = det ( G =~) -.A G ~))

-3 )

-2-.A

= (2 - .A)(-2 - .A) - (1)(-3)

= -4 + .A^2 + 3 = .A^2 - 1 = 0.

Solving the characteristic equation .A^2 - 1 = 0, we see the eigenvalues of A

are >-1 = 1 and >-2 = -1.

An eigenvector x 1 of A associated with .A 1 = 1 must satisfy the equation

Ax1 = A1X1 or equiva lently Ax 1 -.A 1 x 1 = (A-.A 1 I)x 1 = 0. Let x 1 = (Xu).
X21