Page - 88 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Figure11.AffineLiegroupactionformultivariateGaussian law. Consideringthecurveγ(t)anditsderivative . γ(t): γ(t)= [ R1/2(t) m(t) 0 1 ] and . γ(t)= [ . R 1/2 (t) . m(t) 0 0 ] (185) Wecanconsider thecurvewith thepointγ(0)movedat the identityelementonthe leftoronthe right. Then, the tangentplanat identityelementprovides theLiealgebra: ΓL(t)= LM−1 (γ(t))= [ R−1/2R1/2(t) R−1/2(m(t)−m) 0 1 ] (186) . ΓL(t) ∣∣∣ t=0 = [ R−1/2 . R 1/2 (0) R−1/2 .m(0) 0 1 ] = ddt (LM−1(γ(t))) ∣∣∣ t=0 = dLM−1 . γ(0)= dLM−1 . M (187) Liealgebraontherightandonthe left is thedefinedby: dLM−1 :TM(G)→ gL . M →ΩL= dLM−1 . M=M−1 . M= [ R−1/2 . R 1/2 R−1/2 .m 0 0 ] (188) dRM−1 :TM(G)→ gR . M →ΩR= dRM−1 . M= . MM−1= [ R−1/2 . R 1/2 . m−R−1/2 .R1/2 .m 0 0 ] (189) Wecan thenobserve thevelocities in twodifferentways, either byplacing in afixedoutside frame,eitherbyputting inplaceof theelement in theprocessofmovingbyplacing in thereference frameof theelement.[ X(t) 1 ] =M [ x 1 ] ⇒ [ . X(t) 0 ] =ΩR [ X(t) 1 ] withxfixed (190) [ x(t) 1 ] =M−1 [ X 1 ] ⇒ [ . x(t) 0 ] =−ΩL [ X 1 ] withXfixed (191) In the following,wewill complete the global viewby the operatorswhichwill allow to link algebra (fromthe leftor theright)betweenthemandalsoconnect to theirdual.Wewillfirst consider 88