29.3 EQUIVALENT REPRESENTATIONS
respect to a new basis vectoruQ; this supports our contention that representa-
tions connected by similarity transformations should be considered as thesame
representation.
For the four-element Abelian group consisting of the set{ 1 ,i,− 1 ,−i}under ordinary
multiplication, discussed near the end of section 29.2, change the basis vector fromu=
(1 i)TtouQ=(3−i 2 i−5)T. Find the real transformation matrixQ. Show that the
transformed representative matrix for elementi,DQT(i),isgivenbyDQT(i)=(
17 − 29
10 − 17
)
and verify thatDTQT(i)uQ=iuQ.Firstly, we solve the matrix equation
(
3 −i
2 i− 5)
=
(
ab
cd)(
1
i)
,
witha, b, c, dreal. This givesQand henceQ−^1 as
Q=
(
3 − 1
− 52
)
, Q−^1 =
(
21
53
)
.
Following (29.7) we now find the transpose ofDQT(i)as
QDT(i)Q−^1 =(
3 − 1
− 52
)(
01
− 10
)(
21
53
)
=
(
17 10
− 29 − 17
)
and henceDQT(i) is as stated. Finally,
DTQT(i)uQ=(
17 10
− 29 − 17
)(
3 −i
2 i− 5)
=
(
1+3i
− 2 − 5 i)
=i(
3 −i
2 i− 5)
=iuQ,as required.
Although we will not prove it, it can be shown that any finite representationof a finite group of linear transformations that preserve spatial length (or, in
quantum mechanics, preserve the magnitude of a wavefunction) is equivalent to