D.4 Volume of Invariant Measure for Unitary Matrices
447
and we obtain for the invariant metric of the orthogonal matrix ¯
O
(ds)
2
O = Tr[d ¯
O·d ¯
O
T
] = −Tr
d ¯
K·
2
( ¯
I − ¯
K 2 )
d ¯
K
T
·
2
( ¯
I − ¯
K 2 )
.
(D.84)
We leave it as an exercise to show that the invariant measure of ¯
O can be written
dd O = 2
N(N−1)/2
1
Det[ ¯
1 − ¯
K 2 ]
(N −1)/2
dd K ,
(D.85)
where dd K is the invariant measure of ¯
K. Following the method of Chap. 6, one
can show that the volume of orthogonal matrices is
O = 2
3N(N−1)/4 π
N(N−1)/4
N
ν=2
((ν − 1)/2)
(ν − 1)
(D.86)
(see Hua 1963, page 73).
D.4 Volume of Invariant Measure for Unitary Matrices
The volume of the invariant measure of the three symmetry classes of unitary
matrices can be written (see Eq. (D.81))
U (β)≡
dd U (β) = 2
N(βN+2−β)/2
dd H (β)
Det[ ¯
I + ¯
H 2 ] (βN +2−β)/2 ,
(D.87)
where dd H (β) is the invariant measure for the N×N Hermitian matrix, ¯
H , and
β = 1 for ¯
H real symmetric; β = 2 for ¯
H complex Hermitian; and β = 4 for ¯
H
quaternion real. To determine the volume, (β), we must compute the integral
I
α
N (β) =
dd H (β)
Det[ ¯
I + ¯
H 2 ] α ,
(D.88)
where α = (βN + 2 − β)/2 and ¯
1 is an N×N unit matrix. In this appendix, we
will compute this volume for the case β = 2. The integration for β = 1 and β = 4
proceeds along analogous paths.
For the case β = 2, the integral takes the form
I
α
N (2) = 2
N(N−1)/2
∞
−∞
. . .
∞
−∞
dd H (2)
Det[ ¯
I + ¯
H 2 ] α ,
(D.89)