The Chemistry Maths Book, Second Edition

534 Chapter 19The matrix eigenvalue problem

The matrix equation (19.3) represents the case of inhomogeneous equations, with

b 1 ≠ 10. In the homogeneous case, we have

Ax 1 = 10 (19.5)

and the solution isx 1 = 10 if Ais nonsingular. When Ais singular,x 1 = 10 is the trivial

solution but, as discussed in Section 17.4, other solutions may exist. In the following

section we consider these other solutions for the most important case of the

homogeneous system, the matrix eigenvalue problem.

19.2 The eigenvalue problem

A matrix equation of type

Ax 1 = 1 λx (19.6)

where Ais a square matrix, xis a column vector, and λis a number, is a linear

transformation in which the matrix Atransforms the vector xinto a multiple of x.

The equation can be written as

(A 1 − 1 λI)x 1 = 10 (19.7)

(sinceIx 1 = 1 x), and therefore represents the system of homogeneous linear equations

(a

11

1 − 1 λ)x

1

• 1 a

12

x

2

• a

13

x

3

+1-1+ a

1 n

x

n

= 10

a

21

x

1

• 1 (a

22

1 − 1 λ)x

2

1 + a

23

x

3

+1-1+ a

2 n

x

n

= 10

a

31

x

1

• a

32

x

2

+(a

33

1 − 1 λ)x

3

+1-1+ a

3 n

x

n

= 10 (19.8)

a

n 1

x

1

• a

n 2

x

2

• a

n 3

x

3

+1-1+(a

nn

1 − 1 λ)x

n

= 10

(these are the secular equations discussed in Section 17.4). The equations have the

trivial solutionx 1 = 10 for all values of λ. The equations also have a nonzero solution

if the value of λcan be chosen to make the matrix (A 1 − 1 λI) singular; that is, if λis

chosen to make the determinant of (A 1 − 1 λI) zero:

det 1 (A 1 − 1 λI) 1 = 10 (19.9)

or

(19.10)

()

()

(

aaa a

aa a a

aaa

n

n

11 12

13

1

21 22 23 2

31 32 3

−

−

λ

λ





33

3

1

23

0

−

−

=

λ

λ

)

()





a

aaa a

n

n

nn

nn