Theorem 18.1 [1, 2] A representation of any finite group can be converted to a
unitary representation by a similarity transformation.
Proof Let ℊ ¼ {g 1 , g 2 , Á Á Á, g n } be a finite group of order n. Let D(g ν ) be a
representation matrix of g ν of dimension d. Here we suppose that D(g ν ) is not
unitary, but non-singular. Using D(g ν ), we construct the following matrix H such
that
H ¼
X n
i¼1
D g i
ð Þ
{ D g i
ð Þ:
ð18:7Þ
Then, for an arbitrarily chosen group element g j we have
D g j
À Á { HD g j
À Á ¼
X n
i¼1
D g j
À Á { D g i
ð Þ
{ D g i
ð ÞD g j
À Á
¼
X n
i¼1
D g i
ð ÞD g j
À Á
Â
à { D g i
ð ÞD g j
À Á
¼
X n
i¼1
D g i g j
À Á
Â
à { D g i g j
À Á
¼
X n
k¼1
D g k
ð Þ
{ D g k
ð Þ
¼ H,
ð18:8Þ
where the third equality comes from homomorphism of the representation and the
second last equality is due to rearrangement theorem (see Sect. 16.1).
Note that each matrix D(g i )
{ D(g i ) is an Hermitian Gram matrix constructed by a
non-singular matrix and that H is a summation of such Gram matrices. Consequently, on the basis of the argument of Sect. 13.2, H is positive definite and all
the eigenvalues of H are positive. Then, using an appropriate unitary matrix U, we
can get a diagonal matrix Λ such that
U
{ HU ¼ Λ:
ð18:9Þ
Here, the diagonal matrix Λ is given by
Λ ¼
λ 1
λ 2
⋱
λ d
0
B
B
B
@
1
C
C
C
A
,
ð18:10Þ
with λ i > 0 (1 i d). We define Λ
1/2 such that
18.1 Definition of Representation
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