Equivalent representations

REPRESENTATION THEORY
one comprises z, which is unchanged by any of the operations, and the other
comprises x, 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 representations
If D is an n-dimensional representation of a group G, and Q is any fixed invertible n × n matrix (|Q| = 0), then the set of matrices defined by the similarity
transformation
DQ (X) = Q−1 D(X)Q
(29.5)
also forms a representation DQ of G, said to be equivalent to D. We can see from a
comparison with the definition in section 29.2 that they do form a representation:
(i) DQ (I) = Q−1 D(I)Q = Q−1 In Q = 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 nonsingular matrix Q, 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 relation’ – for example, the reflexive property does not hold for a general fixed
matrix Q. However, if Q were 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 matrix Q that appears in the definition (29.5) of equivalent 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 operation X on the basis functions is expressed by the action of M(X) (which is equal
to DT (X)) on the corresponding basis vector:
u = M(X)u = DT (X)u.
(29.6)
A change of basis would be given by uQ = Qu and uQ = Qu , and we may write
uQ = Qu = QM(X)u = QDT (X)Q−1 uQ .
(29.7)
This is of the same form as (29.6), i.e.
uQ = DT QT (X)uQ ,
(29.8)
where DQT (X) = (QT )−1 D(X)QT is related to D(X) by a similarity transformation. Thus DQT (X) represents the same linear transformation as D(X), but with
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29.3 EQUIVALENT REPRESENTATIONS
respect to a new basis vector uQ ; this supports our contention that representations connected by similarity transformations should be considered as the same
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 from u =
(1 i)T to uQ = (3 − i 2i − 5)T . Find the real transformation matrix Q. Show that the
transformed representative matrix for element i, DQT (i), is given by
17 −29
DQT (i) =
10 −17
and verify that DTQT (i)uQ = iuQ .
Firstly, we solve the matrix equation
3−i
2i − 5
=
a
c
b
d
1
i
,
with a, b, c, d real. This gives Q and hence Q−1 as
Q=
−1
2
3
−5
Q−1 =
,
2
5
1
3
.
Following (29.7) we now find the transpose of DQT (i) as
QDT (i)Q−1 =
−1
2
3
−5
0
−1
1
0
2
5
1
3
=
17
−29
10
−17
and hence DQT (i) is as stated. Finally,
10
3−i
1 + 3i
=
−17
2i − 5
−2 − 5i
3−i
=i
= iuQ ,
2i − 5
DT QT (i)uQ =
17
−29
as required. Although we will not prove it, it can be shown that any finite representation
of a finite group of linear transformations that preserve spatial length (or, in
quantum mechanics, preserve the magnitude of a wavefunction) is equivalent to
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