436
D Invariant Measures
The determinant of the metric tensor is Det[ ¯
g] = 2 N(N−1) . The invariant measure
for an N×N complex Hermitian matrix can be written
dd H C = (2)
N(N−1)/2 dh 1,1 ×. . .×dh N,N dh
R
1,2 ×. . .×dh
R
N −1,N
×dh
I
1,2 ×. . .×dh
I
N −1,N .
(D.21)
D.2.2.1 Polar Form of Measure for Complex Hermitian Matrix
An N×N complex Hermitian matrix can be diagonalized by an N ×N unitary
matrix, ¯
V C . The unitary matrix ¯
V C has the property that its Hermitian adjoint is
equal to its inverse. Thus,
¯
V
†
C = ¯
V
−1
C
and ¯
V
†
C · ¯
V C = ¯
V C · ¯
V
†
C = ¯
1,
(D.22)
where ¯
1 is an N ×N unit matrix. The N ×N complex Hermitian matrix can be
written
¯
H C = ¯
V C · ¯
E C · ¯
V
†
C ,
(D.23)
where ¯
E C is the diagonal N×N matrix containing the real eigenvalues, e j (j =
1, . . ., N), of ¯
H C ,
¯
E C =
⎛
⎜
⎜
⎜
⎝
e 1 0 . . . 0
0 e 2 . . . 0
. . .
. . .
. . .
. . .
0 0 . . . e N
⎞
⎟
⎟
⎟
⎠
,
(D.24)
and ¯
V C is composed of the eigenvectors of the ¯
H C .
To obtain the polar form of the measure, dd H C , we first take the differential of
Eq. (D.23),
d ¯
H C = d ¯
V C · ¯
E C · ¯
V
†
C + ¯
V C ·d ¯
E C · ¯
V
†
C + ¯
V C · ¯
E C ·d ¯
V
†
C ,
(D.25)
and introduce the differential
δ ¯
V C = ¯
V
†
C ·d ¯
V C = −d ¯
V
†
C · ¯
V C = −δ ¯
V
†
C ,
(D.26)
where we have used the fact that ¯
V
†
C · ¯
V C = ¯
1 so its differential gives d ¯
V
†
C · ¯
V C +
¯
V
†
C ·d ¯
V C = 0.