Appendix D
Invariant Measures
In this appendix, we will derive invariant measures for the Hermitian and unitary
matrices that determine the dynamics of the three main symmetry classes of
Hamiltonian systems Hua (1963), Porter (1965), Mehta (1991). The invariant
measures that we consider remain invariant under a unitary transformation.
D.1 General Definition of Invariant Measure
We will consider complex square matrices,
¯
W =
⎛
⎜
⎜
⎜
⎝
w 1,1 w 1,2 . . . w 1,N
w 2,1 w 2,2 . . . w 2,N
. . .
. . .
. . .
. . .
w N,1 w N,2 . . . w N,N
⎞
⎟
⎟
⎟
⎠
(D.1)
where w i,j (i = 1, . . ., N, j = 1, . . ., N) is a complex number. The matrix ¯
W can
be diagonalized by a unitary transformation if and only if ¯
W · ¯
W † = ¯
W † · ¯
W Mehta
(1991). These are the types of matrices we consider in this book.
D.1.1 Invariant Metric (Length)
The matrix of differential increments of the matrix ¯
W is denoted d ¯
W and is given by
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https://doi.org/10.1007/978-3-030-63534-3
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