Page - 237 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 383 {Vlk}2≤k<l≤r will play an important role in this paper. Let us define V′ := {V′lk}1≤k<l≤r−1 by V′lk :=Vl+1,k+1. Then,V′ isasystemofrank r−1.Anyx∈ZV iswrittenas: x= ( x11In1 tU U x′ ) (x11∈R,U∈W, x′ ∈ZV′), (3) where: W := ⎧⎪⎨⎪⎩U= ⎛⎜⎝X21... Xr1 ⎞⎟⎠ ;Xl1∈Vl1 (1< l≤ r) ⎫⎪⎬⎪⎭ . (4) Ifx11 =0, thenwehave:( x11In1 tU U x′ ) = ( In1 x−111U In−n1 )( x11In1 x′−x−111UtU )( In1 x −1 11 tU In−n1 ) . (5) Note thatUtU belongs toZV′ thanks to (V1) and (V2). Thus, we deduce the following lemma immediately from(5). Lemma1. (i)Let x∈ZV as in (3). Then, x∈PV if andonly if x11>0andx′−x−111UtU∈PV′. (ii)Forx∈PV, there existuniqueU˜∈W and x˜∈PV′ forwhich: x= ( In1 U˜ In−n1 )( x11In1 x˜′ )( In1 tU˜ In−n1 ) = ( x11In1 x11 tU˜ x11U˜ x˜′+x11U˜tU˜ ) . (6) (iii)TheclosurePV of the conePV isdescribedas: PV := {( x11In1 x11 tU˜ x11U˜ x˜′+x11U˜tU˜ ) ; x11≥0, U˜∈W, x˜′ ∈PV′ } . 2.3. TheDualConeP∗V WedefineaninnerproductonthespaceVlkby(A|B)Vlk :=n−1l trAtB forA,B∈Vlk. Then,wesee from(V1) that: AtB+BtA=2(A|B)Vlk Inl. Gathering these innerproducts (·|·)Vlk, we introduce the standard innerproduct on the spaceZV definedby: (x|x′) := r ∑ k=1 xkkx′kk+2 ∑ 1≤k<l≤r (Xlk|X′lk)Vlk (7) forx,x′ ∈ZV of the form(1).Whenn1=n2= · · ·=nr=1 (andonly in thiscase), thestandard inner productaboveequals the trace innerproduct tr(xx′). 237