Page 290 - Advanced engineering mathematics

270 CHAPTER 9 Eigenvalues, Diagonalization, and Special Matrices

To find an eigenvector associated with the eigenvalue 1, put λ = 1 in (2) of the theorem and
solve the system
⎛ ⎞⎛ ⎞ ⎛ ⎞
0 1 0 x 1 0
((1)I 3 − A)X = 0 0 −1 ⎠⎝ x 2 ⎠ = 0 ⎠ .
⎝
⎝
0 0 2 x 3 0
This system of three equations in three unknowns has the general solution
⎛ ⎞
α
⎝ 0 ⎠
0
and this is an eigenvector associated with 1 for any α = 0.
For eigenvectors associated with −1, put λ =−1 in (2) of the theorem and solve
⎛ ⎞
−2 1 0
((−1)I 3 − A)X = ⎝ 0 −2 −1 ⎠ X = O.
0 0 0
This system has the general solution
⎛ ⎞
β
⎝ 2β ⎠
−4β
and this is an eigenvector associated with −1 for any β = 0.

EXAMPLE 9.4

Let

1 −2
A = .
2 0
The characteristic polynomial is

λ 0 1 −2 λ − 1 2 2

− = = λ − λ + 4,
0 λ 2 0 −2 λ
p A (λ) =|λI 2 − A|=
with roots
√ √
1 + 15i 1 − 15i
and
2 2
and these are the eigenvalues of A.
√ √
For an eigenvector corresponding to (1 + 15i)/2solve (((1 + 15i)/2)I 2 − A)X = O,
which is
√

1 + 15i 10 1 −2
− X = O.
2 01 2 0
This is the system
√
⎛ ⎞
−1 + 15i
2
2 √
⎜ ⎟ x 1 0
⎜ ⎟ = .
⎝ 1 + 15i ⎠ x 2 0
−2
2
This 2 × 2 system has general solution

1
α √ .
(1 − 15i)/4
√
This is an eigenvector associated with (1 + 15i)/2 for any α = 0.
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October 14, 2010 14:49 THM/NEIL Page-270 27410_09_ch09_p267-294