Page 291 - Advanced engineering mathematics

9.1 Eigenvalues and Eigenvectors 271

√
For eigenvectors associated with (1 − 15i)/2, solve the 2 × 2system
√
(((1 − 15i)/2)I 2 − A)X = O
to obtain
1

β √ .
(1 + 15i)/4
√
This is an eigenvector associated with (1 − 15i)/2 for any β = 0.
If A has real numbers as elements and λ = α + iβ is an eigenvalue, then the conjugate λ =
α − iβ is also an eigenvalue. This is because the characteristic polynomial of A has real coeffi-
cients in this case, so complex roots (eigenvalues of A) occur in conjugate pairs. Furthermore, if
E is an eigenvector corresponding to λ, then E is an eigenvector corresponding to λ, where we
take the conjugate of a matrix by taking the conjugate of each of its elements. This can be seen
by taking the conjugate of AE = λE to obtain
AE = λE.
Because A has real elements, A = A so
AE = λE.

This observation can be seen in Example 9.4.
There is a general expression for the eigenvalues of a matrix that will be used soon to draw
conclusions about eigenvalues of matrices having special properties.

LEMMA 9.1

Let A be an n × n matrix of numbers. Let λ be an eigenvalue of A, with eigenvector E. Then
t
E AE
λ = t . (9.2)
E E
Before giving the one line proof of this expression, examine what the right side means. Let

⎛ ⎞
e 1
⎜ ⎟
e 2
E = ⎜ . ⎟.
⎜ ⎟
.
⎝ . ⎠
e n
Then
⎛ ⎞⎛ ⎞
a 11 a 12 ··· a 1n e 1
e 2
⎜
t
a 21 a 22 ··· a 2n ⎟⎜ ⎟
⎟⎜ ⎟
⎜
E AE = e 1 e 2 ··· e n ⎜ . . . . ⎟⎜ . ⎟.
⎝ . . . . . . . . ⎠⎝ . ⎠
.
a n1 a n2 ··· a nn e n
This is a product of a 1 × n matrix with an n × n matrix, then an n × 1 matrix, hence is a 1 × 1
matrix, which we think of as a number. If we carry out this matrix product we obtain the number
n n
t
E AE = a ij e i e j .
i=1 j=1
For the denominator of equation (9.2) we have a 1 × n matrix multiplied by an n × 1 matrix,
which is also a 1 × 1 matrix, or number. Specifically,

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 14:49 THM/NEIL Page-271 27410_09_ch09_p267-294