linearalgebraanditsapplications matrix inequalities

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1、CHAPTER10MatrixInequalitiesInthischapterwestudyself-adjointmappingsofaEuclideanspaceintoitselfthatarepositive.InSection1westateandprovethebasicpropertiesofpositivemappingsandpropertiesoftherelationA

2、sitivematrices.InSection3westudythedependenceoftheeigenvaluesonthematrixinlightofthepartialorderA

3、nofapositivemapping:Definition.Aself-adjointlinearmappingHfromarealorcomplexEuclideanspaceintoitselfiscalledpositiveif(x,Hx)>0forallx#0.(1)PositivityofHisdenotedasH>0or00forallx.(2)N

4、onnegativityofKisdenotedasK>0or0_

5、sitive.(ii)IfMandNarepositive,soistheirsumM+N,aswellasaMforanypositivenumbera.(iii)IfHispositiveandQisinvertible,thenQ*HQ>O.(3)(iv)Hispositiveifallitseigenvaluesarepositive.(v)Everypositivemappingisinvertible.(vi)Everypositivemappinghasapositivesquareroot,

6、uniquelydetermined.(vii)Thesetofallpositivemapsisanopensubsetofthespaceofallself-adjointmaps.(viii)Theboundarypointsofthesetofallpositivemapsarenonnegativemapsthatarenotpositive.Proof.Part(i)isaconsequenceofthepositivityofthescalarproduct;part(ii)isobvious

7、.Forpart(iii)wewritethequadraticformassociatedwithQ"HQas(x,Q`HQx)=(Qx,HQx)=(y,Hy),(3)'wherey=Qx.SinceQisinvertible,ifx#0,y#0,andsoby(1)theright-handsideof(3)'ispositive.Toprove(iv),lethbeaneigenvectorofH,atheeigenvalueHh=ah.Takingthescalarproductwithhweget

8、(h,Hh)=a(h,h);clearly,thisispositiveonlyifa>0.Thisshowsthattheeigenvaluesofapositivemappingarepositive.Toshowtheconverse,weappealtoTheorem4ofChapter8,accordingtowhicheveryself-adjointmappingHhasanorthonormalb