448
D Invariant Measures
where α = N,
dd H (2) = dh 1,1 ×···×dh N,N dh
R
1,2 dh
I
1,2 ×···×dh
R
N −1,N dh
I
N −1,N ,
(D.90)
¯
H is an N×N complex Hermitian matrix, h i,j denotes its (i, j )th matrix element,
and dh R
i,j and dh I
i,j are differentials of the real and imaginary parts, respectively, of
the matrix element h i,j .
In order to perform the integration in Eq. (D.89), it is useful to separate out the
dependence of the integral on the element h N,N because we will integrate over it
first. We can write the matrix ¯
H N in the form
¯
H N =
¯
H 1 ¯
v
¯
v † h
,
(D.91)
where h≡h N,N , ¯
H 1 is an (N − 1)×(N − 1)-dimensional complex Hermitian matrix
obtained from ¯
H by removing the Nth column and Nth row of ¯
H . ¯
v denotes the
1×(N − 1)-dimensional column matrix,
¯
v =
⎛
⎜
⎜
⎜
⎝
h 1,N
h 2,N
. . .
h N −1,N
⎞
⎟
⎟
⎟
⎠
,
(D.92)
and ¯
v † denotes its Hermitian adjoint (complex conjugate transpose).
With this decomposition of the Hamiltonian matrix, we can write the integrand
in Eq. (D.88) in the form
Det[ ¯
1 + ¯
H
2
]
−α
= Det
¯
T 2
1 + ¯
v· ¯
v †
¯
H 1 · ¯
v + h ¯
v
¯
v † · ¯
H 1 + ¯
v † h 1 + h 2 + ¯
v † · ¯
v
−α
,
(D.93)
where we have let
¯
T
2
1 ≡ ¯
1 + ¯
H
2
1
(D.94)
( ¯
1 is now an (N − 1)×(N − 1)-dimensional unit matrix).
Useful Properties of Matrices
We introduce a useful property of matrices and their determinants. First note that
¯
1
0
− ¯
u † · ¯
A −1 1
¯
A ¯
u
¯
u † a
¯
1 − ¯
A −1 · ¯
u
0
1
=
¯
A
0
0 a − ¯
u † · ¯
A −1 · ¯
u
,
(D.95)
D Invariant Measures
where α = N,
dd H (2) = dh 1,1 ×···×dh N,N dh
R
1,2 dh
I
1,2 ×···×dh
R
N −1,N dh
I
N −1,N ,
(D.90)
¯
H is an N×N complex Hermitian matrix, h i,j denotes its (i, j )th matrix element,
and dh R
i,j and dh I
i,j are differentials of the real and imaginary parts, respectively, of
the matrix element h i,j .
In order to perform the integration in Eq. (D.89), it is useful to separate out the
dependence of the integral on the element h N,N because we will integrate over it
first. We can write the matrix ¯
H N in the form
¯
H N =
¯
H 1 ¯
v
¯
v † h
,
(D.91)
where h≡h N,N , ¯
H 1 is an (N − 1)×(N − 1)-dimensional complex Hermitian matrix
obtained from ¯
H by removing the Nth column and Nth row of ¯
H . ¯
v denotes the
1×(N − 1)-dimensional column matrix,
¯
v =
⎛
⎜
⎜
⎜
⎝
h 1,N
h 2,N
. . .
h N −1,N
⎞
⎟
⎟
⎟
⎠
,
(D.92)
and ¯
v † denotes its Hermitian adjoint (complex conjugate transpose).
With this decomposition of the Hamiltonian matrix, we can write the integrand
in Eq. (D.88) in the form
Det[ ¯
1 + ¯
H
2
]
−α
= Det
¯
T 2
1 + ¯
v· ¯
v †
¯
H 1 · ¯
v + h ¯
v
¯
v † · ¯
H 1 + ¯
v † h 1 + h 2 + ¯
v † · ¯
v
−α
,
(D.93)
where we have let
¯
T
2
1 ≡ ¯
1 + ¯
H
2
1
(D.94)
( ¯
1 is now an (N − 1)×(N − 1)-dimensional unit matrix).
Useful Properties of Matrices
We introduce a useful property of matrices and their determinants. First note that
¯
1
0
− ¯
u † · ¯
A −1 1
¯
A ¯
u
¯
u † a
¯
1 − ¯
A −1 · ¯
u
0
1
=
¯
A
0
0 a − ¯
u † · ¯
A −1 · ¯
u
,
(D.95)