REPRESENTATION THEORY
one comprisesz, which is unchanged by any of the operations, and the other
comprisesx,y, which change as a pair into linear combinations of themselves.
This is an important observation to which we return in section 29.4.
29.3 Equivalent representationsIfDis ann-dimensional representation of a groupG,andQis any fixed invert-
iblen×nmatrix (|Q|= 0), then the set of matrices defined by the similarity
transformation
DQ(X)=Q−^1 D(X)Q (29.5)also forms a representationDQofG, said to beequivalenttoD.Wecanseefroma
comparison with the definition in section 29.2 that they do form a representation:
(i)DQ(I)=Q−^1 D(I)Q=Q−^1 InQ=In,
(ii)DQ(X)DQ(Y)=Q−^1 D(X)QQ−^1 D(Y)Q=Q−^1 D(X)D(Y)Q
=Q−^1 D(XY)Q=DQ(XY).Since we can always transform between equivalent representations using a non-
singular matrixQ, we will consider such representations to be one and the same.
Despite the similarity of words and manipulations to those of subsection 28.7.1,that two representations are equivalent does not constitute an ‘equivalence re-
lation’ – for example, the reflexive property does not hold for a general fixed
matrixQ. However, ifQwere not fixed, but simply restricted to belonging to
a set of matrices that themselves form a group, then (29.5) would constitute an
equivalence relation.
The general invertible matrixQthat appears in the definition (29.5) of equiv-alent matrices describes changes arising from a change in the coordinate system
(i.e. in the set of basis functions). As before, suppose that the effect of an opera-
tionXon the basis functions is expressed by the action ofM(X)(whichisequal
toDT(X)) on the corresponding basis vector:
u′=M(X)u=DT(X)u. (29.6)A change of basis would be given byuQ=Quandu′Q=Qu′, and we may write
u′Q=Qu′=QM(X)u=QDT(X)Q−^1 uQ. (29.7)This is of the same form as (29.6), i.e.
u′Q=DTQT(X)uQ, (29.8)whereDQT(X)=(QT)−^1 D(X)QTis related toD(X) by a similarity transforma-
tion. ThusDQT(X) represents the same linear transformation asD(X), but with