The Chemistry Maths Book, Second Edition

546 Chapter 19The matrix eigenvalue problem

It follows from these invariance properties that, for a square matrix A,

1. The trace of Ais equal to the sum of the eigenvalues of A,

(19.33)

2. The determinant of Ais equal to the product of the eigenvalues of A,

(19.34)

(the determinant of a diagonal matrix is the product of the diagonal elements).

EXAMPLE 19.12For the matrices Aand Dof Example 19.11,

Trace: tr 1 A 1 = 1 − 21 + 141 + 101 = 1 2, tr 1 D 1 = 1 − 11 + 111 + 121 = 12

Determinant:

det 1 D 1 = 1 (− 1 ) 1 × 111 × 121 = 1 − 2

19.5 Quadratic forms

A quadratic form is a polynomial of the second degree in a set of variables. Examples

are

x

2

1 + 1 y

2

,3x

2

1 − 14 xy 1 + 12 y

2

, x

1

2

1 + 12 x

1

x

2

1 + 13 x

2

x

3

1 − 1 x

2

2

1 + 13 x

3

2

The general (real) quadratic form in two variables is

Q(x,y) 1 = 1 ax

2

1 + 1 bxy 1 + 1 byx 1 + 1 cy

2

= 1 ax

2

1 + 12 bxy 1 + 1 cy

2

(19.35)

in which the coefficients a,b, and care real numbers. This can be written in matrix

form as

(19.36)

where Ais the real symmetric matrix of the coefficients, and xis the vector whose

elements are the variables xand y.

Qxy

ab

bc

x

y

() ( )xx= Ax

































=

T

detA=− −

−

−

• 
  

−

−

2 =−−=−

45

10

11 5

10

11 4

11

10 5 7 2

k

n

k

=

∏

=

1

λ detA

k

n

k

=

∑

=

1

λ trA