Page - 236 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 383 decomposition. Basedonsuchresults,ourmatrixrealizationmethod[15,17,18]hasbeendeveloped for thepurposeof theefficient studyofhomogeneouscones. In thispaper,wepresentageneralization ofmatrix realizationdealingwith awide class of convex cones,which turns out to include cones associatedwithchordalgraphs.Actually, itwasanenigmafor theauthor thatsomeformulas in[11,19] for thechordalgraphresembletheformulas in[8,17] forhomogeneousconessomuch,andthemystery isnowsolvedbytheunifiedmethodinthispaper toget the formulas. Furthermore, the techniques andideas in the theoryofhomogeneouscones, suchasRieszdistributions [8,20,21]andhomogeneous Hessianmetrics [4,18,22],willbeapplied tovariouscones toobtainnewresults inour futureresearch. Here,wefix somenotationused in thispaper. WedenotebyMat(p,q,R) thevector spaceof p×q realmatrices. ForamatrixA,wewrite tA for the transposeofA. The identitymatrixof size p is denotedby Ip. 2.NewConesPV andP∗V 2.1. Setting Wefixapartitionn=n1+n2+ · · ·+nrofapositive integern. LetV={Vlk}1≤k<l≤rbeasystem ofvectorspacesVlk⊂Mat(nl,nk,R) satisfying (V1)A∈Vlk⇒AtA∈RInl (1≤ k< l≤ r), (V2)A∈Vlj,B∈Vkj⇒AtB∈Vlk (1≤ j< k< l≤ r). The integer r is called therankof thesystemV.Wedenotebynlk thedimensionofVlk.Note that somenlk canbezero. LetZV bethespaceof real symmetricmatricesx∈Sym(n,R)of the form: x= ⎛⎜⎜⎜⎜⎝ X11 tX21 . . . tXr1 X21 X22 tXr2 ... ... Xr1 Xr2 . . . Xrr ⎞⎟⎟⎟⎟⎠ ( Xkk= xkkInk, xkk∈R, k=1,. . . ,r Xlk∈Vlk, 1≤ k< l≤ r ) , (1) andPV thesubsetofZV consistingofpositivedefinitematrices. Then,PV isaregularopenconvex cone inZV. Example 1. Let r= 3, and setV21 := {( a 0 ) ; a∈R } ,V31 := {( 0 a ) ; a∈R } , andV32 :=R. Then,ZV is the spaceof symmetricmatrices x of the form: x= ⎛⎜⎜⎜⎝ x1 0 x4 0 0 x1 0 x5 x4 0 x2 x6 0 x5 x6 x3 ⎞⎟⎟⎟⎠ . (2) Weshall see later that the conePV=ZV∩P4 isnothomogeneous in this case, but admitsvarious integral formulas, aswell as explicit expressionof theKoszul–Vinbergcharacteristic function. 2.2. InductiveDescriptionofPV If the systemV = {Vlk}1≤k<l≤r satisfies (V1) and (V2), any subsystemVI := {Vlk}k,l∈I with I⊂{1,. . . ,r}alsosatisfiesthesameconditions. Inparticular, theconecorrespondingtothesubsystem 236