The Chemistry Maths Book, Second Edition

19.6 Complex matrices 551

or, in terms of components,

l

x

1 = 1 I

xx

ω

x

1 + 1 I

xy

ω

y

1 + 1 I

xz

ω

z

l

y

1 = 1 I

yx

ω

x

1 + 1 I

yy

ω

y

1 + 1 I

yz

ω

z

(19.47)

l

z

1 = 1 I

zx

ω

x

1 + 1 I

zy

ω

y

1 + 1 I

zz

ω

z

When the coordinate axes coincide with the principal axes of inertia, Iis diagonal and

equations (19.47) reduce to

l

x

1 = 1 I

xx

ω

x

, l

y

1 = 1 I

yy

ω

y

, l

z

1 = 1 I

zz

ω

z

(19.48)

The kinetic energy of rotation then has the familiar form

(19.49)

that is used in theoretical discussions of the microwave spectroscopy of polyatomic

molecules.

0 Exercise 35

19.6 Complex matrices

Much of the earlier discussion applies equally well to complex matrices as to real

matrices and, in this section, we summarize the more important properties of complex

matrices, and the ways in which these differ from those of real matrices.

A matrix Awhose elements are complex numbers is called a complex matrix, and

can be written in the form

A 1 = 1 B 1 + 1 iC (19.50)

where , and Band Care real matrices.

The complex conjugate matrix A*

The complex conjugate matrixA*is obtained from Aby replacing each element of

Aby its complex conjugate:

if A 1 = 1 (a

ij

)thenA* 1 = 1 (a

ij

*) (19.51)

and

A* 1 = 1 B 1 − 1 iC (19.52)

For a realmatrix,A* 1 = 1 AandC 1 = 10.

i=− 1

T

l

I

l

I

l

I

x

xx

y

yy

z

zz

=++

2

2

2

222