Page 288 - Advanced engineering mathematics

268 CHAPTER 9 Eigenvalues, Diagonalization, and Special Matrices

Because

10 0 0 0
= = 0 ,
00 4 0 4
then 0 is an eigenvalue of A with eigenvector

0
E = .
4
For any nonzero number α,

0
4α
is also an eigenvector. Zero can be an eigenvalue, but an eigenvector must be a nonzero vector
(at least one nonzero component).

EXAMPLE 9.2
Let
⎛ ⎞
1 −1 0
A = 0 1 1 ⎠ .
⎝
0 0 −1
Then
⎛ ⎞ ⎛ ⎞
6 6
.
⎝
⎝
A 0 ⎠ = 0 ⎠
0 0
Therefore 1 is an eigenvalue with eigenvector
⎛ ⎞
6
⎝ 0 ⎠
0
or any nonzero constant times this matrix. Similarly,
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 −1 1
A ⎝ 2 ⎠ = −2 ⎠ = (−1) ⎝ 2 ⎠ .
⎝
−4 4 −4
Therefore −1 is an eigenvalue with eigenvector
⎛ ⎞
1
,
⎝ 2 ⎠
−4
or any nonzero multiple of this vector.

We would like to be able to find all of the eigenvalues of a matrix. We will have AE = λE,
for some number λ and n × 1matrix E, exactly when
λE − AE = O.
This is equivalent to

(λI n − A)E = O,
and this occurs exactly when the system
(λI n − A)X = O

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October 14, 2010 14:49 THM/NEIL Page-268 27410_09_ch09_p267-294