Page - 86 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 GL(n)×Rn→Rn (A,x) →Ax (171) ThegroupGL(n) isaLiegroup, isasubgroupof thegeneralaffinegroupGA(n), composedof allpairs (A,υ)whereA∈GL(n)andυ∈Rn, thegroupoperationgivenby: (A1,υ1)(A2,υ2)=(A1A2,A1υ2+υ1) (172) GL(n) isanopensubsetofRn 2 ,andmaybeconsideredasn2-dimensionaldifferentialmanifoldwiththe samedifferentiablestructure thanRn 2 .Multiplicationandinversionare infinitelyoftendifferentiable mappings.Consider thevectorspacegl(n)ofrealn×nmatricesandthecommutatorproduct: gl(n)×gl(n)→ gl(n) (A,B) →AB−BA=[A,B] (173) This isaLieproductmakinggl(n) intoaLiealgebra. Theexponentialmapis thenthemapping definedby: exp:gl(n)→GL(n) A → exp(A)= ∞∑ n=0 An n! (174) RestrictingA tohavepositivedeterminant,oneobtains thepositivegeneralaffinegroupGA+(n) thatacts transitivelyonRnby: ((A,υ) ,x) →Ax+υ (175) IncaseofsymmetricpositivedefinitematricesSym+(n),wecanuse theCholeskydecomposition: R=LLT (176) where L is a lower triangularmatrixwith real andpositive diagonal entries, and LT denotes the transposeofL, todefinethesquarerootofR. Given a positive semidefinite matrix R, according to the spectral theorem, the continuous functional calculus can be applied to obtain a matrix R1/2 such that R1/2 is itself positive and R1/2R1/2=R. TheoperatorR1/2 is theuniquenon-negativesquarerootofR. Nn = {ℵ(μ,Σ)/μ∈Rn,Σ∈Sym+n} the class of regular multivariate normal distributions, whereμ is themeanvectorandΣ is the (symmetricpositivedefinite) covariancematrix, is invariant under the transitiveactionofGA(n). The inducedactionofGA(n)onRn×Sym+n is thengivenby: GA(n)×(Rn×Sym+n)→Rn×Sym+n ((A,υ) ,(μ,Σ)) → (Aμ+υ,AΣAT) (177) and GA(n)×Rn→Rn ((A,υ) ,x) →Ax+υ (178) Asthe isotropygroupof (0, In) isequal toO(n),wecanobserve that: Nn=GA(n)/O(n) (179) Nn is an open subset of the vector spaceTn = {(η,Ω)/η∈Rn,Ω∈Symn} and is adifferentiable manifold,where the tangentspaceatanypointmaybe identifiedwithTn. 86