Page - 238 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 383 LetW˜k (k=1,. . . ,r)bethevectorspaceofW∈Mat(n,nk,R)of the form: W= ⎛⎜⎜⎜⎜⎜⎜⎝ 0n1+···+nk−1,nk Xkk Xk+1,k ... Xrk ⎞⎟⎟⎟⎟⎟⎟⎠ (Xkk= xkkInk, xkk∈R,Xlk∈Vlk, l> k). Clearly, the space W˜k is isomorphic to R⊕∑l>kVlk, which implies dimW˜k = 1+ qk with qk := ∑l>knlk. Gathering orthogonal bases of Vlk’s, we take a basis of W˜k, so that we have an isomorphismW˜k W →w=vect(W)∈R1+qk,where thefirstcomponentw1 ofw isassumedtobe xkk. Letus introducea linearmapφk :ZV→Sym(1+qk,R)definedinsuchawaythat: (WtW|ξ)= twφk(ξ)w (ξ∈ZV, W∈W˜k,w=vect(W)∈R1+qk). (8) It iseasy tosee thatφr(ξ)= ξrr forξ∈ZV. Theorem1. Thedual coneP∗V ⊂ZV ofPVwithrespect to the standard innerproduct isdescribedas: P∗V={ξ∈ZV ; φk(ξ) ispositivedefinite for all k=1,. . . ,r} ={ξ∈ZV ; detφk(ξ)>0 for all k=1,. . . ,r} . (9) Proof. Weshallprove thestatementby inductionontherank r.When r=1,wehaveφ1(ξ)= ξ11 and ξ= ξ11In1. Thus, (9)holds in thiscase. Letusassumethat (9)holdswhentherank issmaller than r. Inparticular, thestatementholds for P∗V′ ⊂ZV′, that is, P∗V′= { ξ′ ∈ZV′ ; φ′k(ξ′) ispositivedefinite forallk=1,. . . ,r−1 } = { ξ′ ∈ZV′ ; detφ′k(ξ′)>0 forallk=1,. . . ,r−1 } , whereφ′k isdefinedsimilarly to (8) forZV′.Ontheotherhand, if: ξ= ( ξ11In1 tV V ξ′ ) (ξ11∈R,V∈W, ξ′ ∈ZV′), (10) weobserve that: φk(ξ)=φ ′ k−1(ξ ′) (k=2,. . . ,r). Therefore, inorder toprove(9) forP∗V ofrank r, it suffices toshowthat: P∗V= { ξ∈ZV ; ξ′ ∈P∗V′ andφ1(ξ) ispositivedefinite } = { ξ∈ZV ; ξ′ ∈P∗V′ and detφ1(ξ)>0 } . (11) Ifq1=0, thenanyelementξ∈ZV isof the form: ξ= ( ξ11In1 ξ′ ) , whichbelongs toPV if andonly ifξ′ ∈PV′ andφ1(ξ)= ξ11>0, so that (11)holds. 238