354 Ordinary Differential Equations
of multiplicity m = 1 and 2 is a n eigenvalue of the matrix A of multiplicity
m=2.
LetX1 = (~~~)
X3 1be the eigenvector of A associated with t he eigenvalue .A 1 = -1. The vector
x 1 must satisfy (A -.A1I)x 1 =(A+ I)x1 = 0 or
C ~: , ;: 1+ D G::) ~ (-: -~ i)(~::) m
Multiplying, we see xu, x 21 , and x3 1 must simulta neously satisfy the system
of equations
2xu - x21 + X31 = 0
(3) -xu + 2x21 + X31 = 0
xu + X21 + 2x31 = 0.
Replacing the first equation in this system by the sum of the first equation and
two times the second equation and also replacing the third equation in this
system by the sum of the second and third equation, we obtain the following
equivalent system of simulta neous equations
3X21 + 3X31 = 0
-xu + 2x 21 + x31 = 0
3x21 + 3x31 = 0.
Since the first and third equations in this system are identical, we h ave a
system which consists of only two independent equ ations in three variables. So
the value of one of the variables-xu , x 21 , or x 31 - may be selected arbitrarily
and the values of the other two variables can b e expressed in terms of that
variable. Solving the first equation for x2 1 in terms of x3 1 , we find x2 1 = -x3 1.
Substituting x2 1 = -x3 1 into the second equation and so lving for xu, we get
Xu = -X31. That is, system (3) is satisfied by any vector x 1 in which the
component X31 is selected arbitrar ily, X21 = -X 31 and xu = -X31. Choosing
X3 1 = 1, we get x2 1 = -1 and xu = -1. So a n eigenvector of the matrix A
associated with the eigenvalue .A 1 = - 1 is