The Chemistry Maths Book, Second Edition

19.3 Properties of the eigenvectors 539

so that, whenλ

k

1 ≠ 1 λ

l

,

x

l

T

x

k

1 = 10 (19.19)

and the vectors are orthogonal.

EXAMPLE 19.6 For the eigenvectors of the symmetric matrix in Example 19.4(ii),

0 Exercises 16 –18

Property 3.For a symmetric matrix, the eigenvectors corresponding to the same

eigenvalue are either orthogonal or can be made so.

EXAMPLE 19.7The eigenvectorsx

2

andx

3

of Example 19.4(ii), belonging to the

degenerate eigenvalueλ 1 = 11 , are notorthogonal. Thus

If the independent eigenvectors x

k

and x

l

correspond to the same eigenvalue

λ

k

1 = 1 λ

l

1 = 1 λ, then

Ax

k

1 = 1 λx

k

and Ax

l

1 = 1 λx

l

(19.20)

so that every linear combination ofx

k

andx

l

is also an eigenvector of Awith the same

eigenvalue. Thus, if

x 1 = 1 ax

k

1 + 1 bx

l

(19.21)

then

Ax 1 = 1 A(ax

k

1 + 1 bx

l

) 1 = 1 a(Ax

k

) 1 + 1 b(Ax

l

)

= 1 a(λx

k

) 1 + 1 b(λx

l

) 1 = 1 λ(ax

k

1 + 1 bx

l

) 1 = 1 λx

(19.22)

xx

2 3 23 23

11 2

1

2

3

126

T

=−

−

























cc() ()=++cc )≠ 0

xx

13 13 12

111

1

2

3

123

T

=

−

























cc() ()=+−cc == 0

xx

1 2 12 12

111

1

1

2

112

T

=

−

























cc() ()=+−cc == 0