Page - 350 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 396 whereA stands for theconjugateofA. Thepoint iI inHn is identifiedwith thenullmatrixnoted0 in Dn. LetZ∈Dn. ThereexistsPadiagonalmatrixwithdecreasingpositiverealentriesandUaunitary matrixsuchthatZ=UPUt. Letτ1≥ ...≥τnbesuchthat P= ⎛ ⎜⎝ th(τ1) ... th(τn) ⎞ ⎟⎠ . Let A0= ⎛ ⎜⎝ ch(τ1) ... ch(τn) ⎞ ⎟⎠ ,B0= ⎛ ⎜⎝ sh(τ1) ... sh(τn) ⎞ ⎟⎠ and gZ= ( U 0 0 U ) . ( A0 B0 A0 B0 ) . It canbeshownthat gZ∈Sp(n,C)andgZ.0=Z. (3) WeprovidenowacorrespondencebetweentheelementsofDn andtheelementsofpdefinedin Equation(1). Let HZ= ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ τ1 ... τn −τ1 ... −τn ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ∈a, (4) and aZ= ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ eτ1 ... eτn e−τ1 ... e−τn ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ∈A= exp(a). It canbeshownthat thereexistsk∈K suchthat Cexp(Adk(HZ))C−1.0=Z, orequivalently CkaZkC−1.0=Z. Recall thatEquation(2)givesAdk(H)∈pandkak∈ exp(p). ThedistancebetweenZand0 inDn isgivenby d(0,Z)= ( 2∑τ2i )1/2 (5) (seep. 292 in [20] ). 350