Mathematical Methods for Physics and Engineering : A Comprehensive Guide

29.2 CHOOSING AN APPROPRIATE FORMALISM

generates a new column matrixu′=(u′ 1 u′ 2 ···u′n)T. Having establisheduandu′

we can determine then×nmatrix,M(X) say, that connects them by

u′=M(X)u. (29.2)

It may seem natural to use the matrixM(X) so generated as the representative

matrix of the elementX; in fact, because we have already chosen the convention

wherebyZ=XYimplies that the effect of applying elementZis the same as that

of first applyingYand then applyingXto the result, one further step has to be

taken. So that the representative matricesD(X) may follow the same convention,

i.e.

D(Z)=D(X)D(Y),

and at the same time respect the normal rules of matrix multiplication, it is

necessary to take thetransposeofM(X) as the representative matrixD(X).

Explicitly,

D(X)=MT(X) (29.3)

and (29.2) becomes

u′=DT(X)u. (29.4)

Thus the procedure for determining the matrixD(X) that represents the group

elementXin a representation based on basis vectoruis summarised by equations

(29.1)–(29.4).§

This procedure is then repeated for each elementXof the group, and the

resulting set ofn×nmatricesD={D(X)}is said to be then-dimensional

representation ofGhavinguas its basis. The need to take the transpose of each

matrixM(X) is not of any fundamental significance, since the only thing that

really matters is whether the matricesD(X) have the appropriate multiplication

properties – and, as defined, they do.

In cases in which the basis functions are labels, the actions of the group

elements are such as to cause rearrangements of the labels. Correspondingly the

matricesD(X) contain only ‘1’s and ‘0’s as entries; each row and each column

contains a single ‘1’.

§An alternative procedure in which a row vector is used as the basis vector is possible. Defining

equations of the formuTX=uTD(X) are used, and no additional transpositions are needed to

define the representative matrices. However, row-matrix equations are cumbersome to write out

and in all other parts of this book we have adopted the convention of writing operators (here the

group element) to the left of the object on which they operate (here the basis vector).