The Chemistry Maths Book, Second Edition

538 Chapter 19The matrix eigenvalue problem

EXAMPLE 19.5Normalization of eigenvectors

The eigenvectors xof Example 19.3 are normalized (have unit length) if

x

T

x 1 = 1 x

2

1 + 1 y

2

1 + 1 z

2

1 = 11

For example,

and the set of three normalized eigenvectors is

0 Exercises 12–15

Property 2.If Ais a (real) symmetric matrix, the eigenvectors corresponding to

distinct eigenvalues are orthogonal.

Let x

k

and x

l

be eigenvectors of A corresponding to eigenvalues λ

k

and λ

l

,

respectively. Then

Ax

k

1 = 1 λ

k

x

k

(19.13)

and premultiplication of both sides byx

l

T

gives

x

l

T

Ax

k

1 = 1 λ

k

x

l

T

x

k

(19.14)

Also

Ax

l

1 = 1 λ

l

x

l

(19.15)

and premultiplication of both sides byx

k

T

gives

x

k

T

Ax

l

1 = 1 λ

l

x

k

T

x

l

(19.16)

Now the transpose of a product of matrices is the product of the transpose matrices

in reverse order (equation (18.41)). Therefore, taking the transpose of both sides of

(19.16), and remembering thatA

T

1 = 1 Afor a symmetric matrix,

x

l

T

Ax

k

1 = 1 λ

l

x

l

T

x

k

(19.17)

Subtraction of (19.17) from (19.14) then gives

01 = 1 (λ

k

1 − 1 λ

l

)x

l

T

x

k

(19.18)

xx

12

1

2

0

1

1

1

6

1

2

1

=−

























,=

























, xx

3

1

11

1

3

1

=

























xx

22

22 2 2

121 1 16

T

=++=zz()if=