We 64257rst treat the case of 2 2 matrices where the result is simple Then we present the conditions for symmetric matrices to be positive de64257nite Finally we state the corresponding condition for the symmetric matrix to be negative de64257nite o ID: 74735
Download Pdf The PPT/PDF document "TEST FOR POSITIVE AND NEGATIVE DEFINITEN..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
2TESTFORPOSITIVEANDNEGATIVEDEFINITENESS2.POSITIVEDEFINITEQUADRATICFORMSInthegeneralnnsymmetriccase,wewillseetwoconditionssimilartotheseforthe22case.AconditionforQtobepositivedenitecanbegivenintermsofseveraldeterminantsofthe“principal”submatrices.Second,Qispositivedeniteifthepivotsareallpositive,andthiscanbeunderstoodintermsofcompletionofthesquares.LetAbeannnsymmetricmatrix.WeneedtoconsidersubmatricesofA.LetAkbethekksubmatrixformedbydeletingthelastn�krowsandlastn�kcolumnsofA,AkD�ai;j1ik;1jk:ThefollowingtheoremgivesconditionsofthequadraticformbeingpositivedeniteintermsofdeterminantsofAk.Theorem2.LetAbeannnsymmetricmatrixandQ.x/DxTAxtherelatedquadraticform.Thefollowingconditionsareequivalent:(i)Q.x/ispositivedenite.(ii)AlltheeigenvaluesofAarepositive.(iii)Foreach1kn,thequadraticformassociatedtoAkispositivedenite.(iv)Thedeterminants,det.Ak/�0for1kn.(v)Allthepivotsobtainedwithoutrowexchangesorscalarmultiplicationsofrowsarepositive.(vi)Bycompletionofthesquares,Q.x/canberepresentedasasumofsquares,withallpositivecoef-cients,Q.x1;:::;xn/D.x1;:::;xn/UTDU.x1;:::;xn/TDp1.x1Cu1;2x2CCu1;nxn/2Cp2.x2Cu2;3x3CCu2;nxn/2CCpnx2n:Proof.WeassumeAissymmetricsowecanndanorthonormalbasisofeigenvectorsv1,...vnwitheigenvalues1,...n.LetPbetheorthogonalmatrixformedbyputtingthevjasthecolumns.ThenPTAPDDisthediagonalmatrixwithentries1,...n.SettingxDy1v1CCynvnDPy;thequadraticformturnsintoasumofsquares:Q.x/DxTAxDyTPTAPyDyTDyDnXjD1jy2j:Fromthisrepresentation,itisclearthatQispositivedeniteifandonlyifalltheeigenvaluesarepositive,i.e.,conditions(i)and(ii)areequivalent.AssumeQispositivedenite.Thenforany1kn,0Q.x1;:::;xk;0;:::;0/D.x1;:::;xk;0;:::;0/A.x1;:::;xk;0;:::;0/TD.x1;:::;xk/Ak.x1;:::;xk/Tforall.x1;:::;xk/6D0.Thisshowsthat(i)implies(iii). 4TESTFORPOSITIVEANDNEGATIVEDEFINITENESS3.NEGATIVEDEFINITEQUADRATICFORMSTheconditionsforthequadraticformtobenegativedenitearesimilar,alltheeigenvaluesmustbenegative.Theorem4.LetAbeannnsymmetricmatrixandQ.x/DxTAxtherelatedquadraticform.Thefollowingconditionsareequivalent:(i)Q.x/isnegativedenite.(ii)AlltheeigenvaluesofAarenegative.(iii)ThequadraticformsassociatedtoalltheAkarenegativedenite.(iv)Thedeterminants,.-1/kdet.Ak/�0for1kn,i.e.,det.A1/0,det.A2/&#x]TJ/;ཕ ;.9; T; 25;&#x.651;&#x 0 T; [00;0,...,.-1/ndet.An/D.-1/ndet.A/&#x]TJ/;ཕ ;.9; T; 25;&#x.651;&#x 0 T; [00;0.(v)Allthepivotsobtainedwithoutrowexchangesorscalarmultiplicationsofrowsarenegative.(vi)Bycompletionofthesquares,Q.x/canberepresentedasasumofsquares,withallnegativecoef-cients,Q.x1;:::;xn/D.x1;:::;xn/UTDU.x1;:::;xn/TDp1.x1Cu1;2x2CCu1;nxn/2Cp2.x2Cu2;3x3CCu2;nxn/2CCpnx2n:Forcondition(4),theideaisthattheproductofknegativenumbershasthesamesignas.-1/k.4.PROBLEMS1.Decidewhetherthefollowingmatricesarepositivedenite,negativedenite,orneither:(a)0@2-1-1-12-1-1-121A(b)0@2-1-1-121-1121A(c)0@1232543491A(d)0BB@120026-200-25-200-231CCAREFERENCES[1]C.SimonandL.Blume,MathematicsforEconomists,W.W.Norton&Company,NewYork,1994[2]G.Strang,LinearAlgebraanditsApplications,HarcourtBraceJovanovich,Publ.,SanDiego,1976.