440
D Invariant Measures
with entries δs
(α)
i,j that are real. The invariant metric for the N×N quaternion real
matrix, ¯
H Q , takes the form
(ds)
2
H Q
= 2
N
j =1
(de j )
2
+ 4
1≤i 3
α=0
(e j − e i )
2 (δs
(α)
i,j )
2 .
(D.46)
The determinant of the metric tensor is Det[ ¯
g] = 2 4N 2 −3N
1≤i Notice that the diagonal elements of δ ¯
S do not contribute. The invariant measure
can be written
dd H Q = 2
N/2
1≤i (e j − e i )
4
de 1 de 2 ×. . .×de N δδ S ,
(D.47)
where
δδ S = 2
2N(N−1)
1≤i 3
α=0
ds
(α)
i,j
(D.48)
is a measure associated with the off-diagonal elements of the matrix, δ ¯
S.
D.2.4 General Formula for Invariant Measure of Hermitian
Matrices
The invariant measure for real symmetric, complex Hermitian, and quaternion real
matrices can be written in a single compact form,
dd H (β) = η N,β
N
i |e j − e i |
β
×de 1 ×. . .×de N δδ V ,β .
(D.49)
For N ×N real symmetric matrices, β = 1, δδ V ,1 = δδ O , and η N,1 = 1. For N×N
complex Hermitian matrices, β = 2, δδ V ,2 = δδ V , and η N,2 = 1. For N ×N
quaternion real matrices (2N×2N matrix), β = 4, δδ V ,4 = δδ S and η N,4 = 2 N/2 .
It is useful to remember that there are a total of N(βN +2−β)/2 independent matrix
elements, N are the diagonal matrix elements and βN(N − 1)/2 off-diagonal.