Page 294 - Advanced engineering mathematics

274 CHAPTER 9 Eigenvalues, Diagonalization, and Special Matrices

THEOREM 9.3 Eigenvalues of Real Symmetric Matrices
The eigenvalues of a real symmetric matrix are real.

Proof By Lemma 9.1 (equation (9.2)), for any eigenvalue λ of A, with eigenvector E =
(e 1 ,··· ,e n ),
t
E AE
λ = t .
E E
As noted previously, the denominator is
n
t 2
E E = |e j |
j=1
and this is real. All we have to do is show that the numerator real, which we will do by showing
t
that E AE equals its complex conjugate. First, because elements of A are real, each equals its
t
own conjugate, so A = A. Further, because A is symmetric, A = A. Therefore
t t t
t
E AE = E AE = E AE = E AE.
But the last quantity is a 1 × 1 matrix, which equals its own transpose. Thus, continuing the last
equation,
t
t
t
t
t t
t
E AE = (E AE) = (E )A(E ) = E AE.
t
The last two equations together show that E AE is its own conjugate, hence is real, proving the
theorem.
If the eigenvalues of a real matrix are all real, then associated eigenvectors will have real
elements as well. In the case that A is also symmetric, we claim that eigenvectors associated
with distinct eigenvalues must be orthogonal.

THEOREM 9.4 Orthogonality of Eigenvectors

Let A be a real symmetric matrix. Then eigenvectors associated with distinct eigenvalues are
orthogonal.
Proof We can derive this result by a useful interplay between matrix and vector notation. Let
λ and μ be distinct eigenvalues of A, with eigenvectors, respectively,

⎛ ⎞ ⎛ ⎞
e 1 g 1
e 2 g 2
⎜ ⎟ ⎜ ⎟
E = ⎜ . ⎟ and G = ⎜ . ⎟.
⎜ ⎟
⎜ ⎟
.
.
⎝ . ⎠ ⎝ . ⎠
e n g n
We have seen that
t
E · G = e 1 g 1 + e 2 g 2 + ··· + e n g n = E G.
t
Now use the facts that AE = λE, AG = μG, and A = A to write
t
t
t
t
λE G = (AE) G = (E A )G
t
t
t
t
= (E A)G = E (AG) = E (μG) = μE G.
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-274 27410_09_ch09_p267-294