Linear Systems of First-Order Differential Equations 355Letbe an eigenvector of the matrix A associated with the eigenvalue .A = 2. The
vector z must satisfy (A - 2I)z = 0 orc~: !~~ 1-D G:) (=: =: j) G:) m
Multiplying, we see z 1 , z2, and z3 must simultan eously satisfy the system of
equations
-z1 - Z2 + Z3 = 0
- Z1 - Z2 + Z3 = 0
Z1 + Z2 - Z3 = 0.
Observe that the first two equations are identical and the third equation is
-1 t imes the first equation. Thus, this system reduces to the single equation
(4)
Hence, the values of two of the three variables z 1 , z 2 , and z 3 may be cho-
sen arbitrarily and the third is t hen determined by equation ( 4). Since two
components of t he vector z may be chosen arbitrarily, we make two different
choices of two components in such a manner that the resulting two vectors
are linearly independent. For example, choosing z2 = 1, choosing z3 = 0,
substituting these values into equation (4) and solving for z 1 , we get z 1 = -1.
So one eigenvector of the matrix A associated with the eigenvalue A = 2 is
Next, choosing z 2 = 0, choosing z 3 = 1, substituting these values into equ a-
tion ( 4), and solving for z 1 , we get z 1 = 1. Thus, a second eigenvector of the
matrix A associated with t he eigenvalue .A = 2 is
The following computation shows that the set of eigenvectors {x 1 ,x2,x3}
is linearly independent. Let X be the 3 x 3 matrix X = ( x 1 x 2 x 3 ). Then
detX det ( =: -~ D ~ -3" 0
Thus, in this example there are two linearly independent eigenvectors associ-