352 Ordinary Differential Equationsx 12 and x22- Hence, x 12 = X22 and X22 is arbitrary. Choosing x22 = 1, we
find an eigenvector associated with the eigenvalue >- 2 = -1 is
The vector kx 2 where k -=f. 0 is an arbitrary constant is also an eigenvector
associated with the eigenvalue .A 2 = -1.
Letting X b e the 2 x 2 matrix whose columns are x 1 and x2-that is, letting
X = (x 1 x 2), we find
det X = det (x 1 x 2 ) = det G D = 3 - 1 = 2 # 0.
Consequently, the eigenvectors x 1 and x 2 are linearly indep endent. Earlier,
we stated that eigenvectors associated with distinct eigenvalues are linearly
independent. If we had remembered this fact, it would not have been necessary
to show det X -=f. 0.
EXAMPLE 2 A 2 x 2 Matrix with Only One Eigenvalue
and One Eigenvector
Find the eigenvalues and associated eigenvectors of the matrixA=(-~ i)·
SOLUTION
The characteristic equation for A isdet (A - .AI) = det ( ( ~ i) -A G ~) ) = det (3 =/ 1 ~ ;_)
= (3 - >-)(1 - >-) + 1 = A^2 - 4.A + 4 = 0.
Solving the characteristic equation .>-^2 - 4 .A + 4 = 0, we find the eigenvalues
of A are .A 1 = .A 2 = 2. Thus, >-= 2 is a root of multiplicity m = 2 and
there will be one or two linearly independent eigenvectors associated with the
eigenvalue A = 2.
An eigenvector of A associated with A = 2 must satisfy Ax = 2x or
(A - 2I)x = 0. Letting x = (~~) and substituting for A and I , we seet hat x 1 and x2 must satisfy
(A - 2I)x =
( (-~ D -^2 G ~)) ( ~~)(3 - 2
-1^1 ~ 2) ( ~~) = ( -i -i) ( ~~) ( ~).