The Chemistry Maths Book, Second Edition

542 Chapter 19The matrix eigenvalue problem

The secular equations of the problem are

(1) (α 1 − 1 E)c

1

1 + 1 βc

2

1 + βc

4

= 10

(2) βc

1

1 + 1 (α 1 − 1 E)c

2

• 1 βc

3

= 10

(3) βc

2

1 + 1 (α 1 − 1 E)c

3

1 + 1 βc

4

1 = 10

(4) βc

1

1 + 1 βc

3

1 + 1 (α 1 − 1 E)c

4

1 = 10

and the eigenvectors are obtained by solving this system of homogeneous equations

for each eigenvalue Ein turn. We consider first the nondegenerate eigenvalues

E

1

1 = 1 α 1 + 12 βandE

4

1 = 1 α 1 − 12 β.

ForE 1 = 1 E

1

1 = 1 α 1 + 12 β,

(1) − 2 βc

1

1 +βc

2

1 +βc

4

= 10

(2) βc

1

1 − 12 βc

2

• 1 βc

3

1 = 10

(3) βc

2

1 − 12 βc

3

1 +βc

4

= 10

(4) βc

1

1 + 1 βc

3

1 − 12 βc

4

= 10

Only three of the four equations are independent; for example,(1) 1 + 1 (2) 1 + 1 (3) 1 = 1 −(4).

Solving for c

2

, c

3

, and c

4

in terms of c

1

gives c

2

1 = 1 c

1

, c

3

1 = 1 c

1

, c

4

1 = 1 c

1

. Similarly, the

eigenvector corresponding to eigenvalueE

4

1 = 1 α 1 − 12 βhas componentsc

2

1 = 1 −c

1

, c

3

1 = 1 c

1

,

c

4

1 = 1 −c

1

. The eigenvectors corresponding to eigenvaluesE

1

andE

4

are therefore

wherec

1

andc

4

are arbitrary.

ForE

2

1 = 1 E

3

1 = 1 α, the secular equations are

(1) 1 = 1 (3) βc

2

1 + 1 βc

4

1 = 10

(2) 1 = 1 (4) βc

1

1 + 1 βc

3

1 = 10

Only two of the four equations are therefore independent, and have solutionc

3

1 = 1 −c

1

,

c

4

1 = 1 −c

2

. A pair of eigenvectors corresponding to the doubly-degenerate eigenvalue

E 1 = 1 αis therefore

CC

22 33

1

1

1

1

1

1

1

1

=

−

−





























,=

−

−













cc

















CC

11 4 4

1

1

1

1

1

1

1

1

=





























,=

−

−

















cc











