Jim Lambers MAT Summer Session Lecture Notes These - PDF document

Theprecedingexamplecanbegeneralizedasfollows:ifAisannndiagonalmatrixA=266664d10


00d2.........00


0dn377775;thenAis:1.positivedeniteifandonlyifdi�0fori=1;2;:::;n,2.negativedeniteifandonlyifdi0fori=1;2;:::;n,3.positivesemideniteifandonlyifdi0fori=1;2;:::;n,4.negativesemideniteifandonlyifdi0fori=1;2;:::;n,5.indeniteifandonlyifdi&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;
.51; 0 ;&#xTd [;0forsomeindicesi,1in,andnegativeforotherindices.Wenowconsiderageneral22symmetricmatrixA=abbc:ThismatrixinducesthequadraticformQA(x;y)=ax2+2bxy+cy2:Ify=0,thenwehaveQA(x;0)=ax2,sowemustcertainlyhavea&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;
.51; 0 ;&#xTd [;0inorderforAtobepositivedenite.Ify6=0,thenweuseachangeofvariablex=tyandobtainQA(x;y)=QA(ty;y)=at2y2+2bty2+cy2=(at2+2bt+c)y2:ThusQA(x;y)ispositivedeniteifandonlyif'(t)=at2+2bt+cispositive.From'0(t)=2at+2b;'00(t)=2a;weobtainthesinglecriticalpointt=�b=aanddeterminethatitisastrictglobalminimizerof'(t).Theminimumvalueis'(t)=a(�b=a)2+2b(�b=a)+c=c�b2 a=1 adet(A):WeconcludethatAispositivedeniteifandonlyifa�0anddet(A)�0.Thisleadstothefollowingtheorem.TheoremA22symmetricmatrixAis2 whichcanbeconrmedtobepositivedenitebyexamininationoftheprincipalminors:
1=2,
2=3,
3=4.Itfollowsthatthecriticalpoint(0;0;0)isastrictglobalminimizeroff(x;y;z).2ExampleLetf(x;y)=ex�y+ey�x:Wehaverf(x;y)=(ex�y�ey�x;�ex�y+ey�x);whichyieldsthecriticalpoints(x;x)forallx2R.WealsohaveHf(x;y)=ex�y+ey�x�ex�y�ey�x�ex�y�ey�xex�y+ey�x;whichyields
1=ex�y+ey�x�0;
2=(ex�y+ey�x)2�(�ex�y�ey�x)2=0:Thatis,Hf(x;x)ispositivesemidenite,making(x;x)aglobalminimizeroff(x;y).2ExampleLetf(x;y;z)=ex�y+ey�x+ex2+z2:Wehaverf(x;y;z)=(ex�y�ey�x+2xex2;�ex�y+ey�x;2z)whichyieldstheconditionsz=0,x=yandx=0foracriticalpoint.Therefore,(0;0;0)istheonlycriticalpoint.WethenhaveHf(x;y;z)=24ex�y+ey�x+4x2ex2+2ex2�ex�y�ey�x0�ex�y�ey�xex�y+ey�x000235;andthereforeHf(0;0;0)=243�20�22000235:whichispositivednite,inviewof
1=3,
2=2,and
3=4.ItcanalsobeshownbydirectcomputationoftheminorsthatHf(x;y;z)ispositivedeniteonallofR3,usingthefactthatex�0forallx.Therefore,thecriticalpoint(0;0;0)isastrictglobalminimizeroff(x;y;z).2ExampleLetf(x;y)=ex�y+ex+y:4 TheoremIff(x)isafunctionwithcontinuoussecondpartialderivativesonasetDRn,ifxisaninteriorpointofDthatisalsoacriticalpointoff(x),andifHf(x)isindenite,thenxisasaddlepointofx.ExampleLetf(x;y)=x3�12xy+8y3:Wehaverf(x;y)=(3x2�12y;�12x+24y2);whichyieldsthecriticalpoints(0;0)and(2;1).WealsohaveHf(x;y)=6x�12�1248x;andthereforeHf(0;0)=0�12�120;Hf(2;1)=12�12�1248:WeseethatHf(2;1)ispositivedenite,becauseitsprincipalminorsarepositive,butHf(0;0)isnot,as
1=0and
2=�144.Thatis,Hf(0;0)isindenite,so(0;0)isasaddlepoint.Furthermore,limx!1f(x;y)=+1;limx!�1f(x;y)=�1;sof(x;y)hasnoglobalminimizeronR2.Wecanconclude,however,that(2;1)isastrictlocalminimizer.2ItshouldbeemphasizedthatiftheHessianispositivesemideniteornegativesemideniteatacriticalpoint,thenitcannotbeconcludedthatthecriticalpointisnecessarilyaminimizer,maximizerorsaddlepointofthefunction.ExampleLetf(x;y)=x4�y4.Wehaverf(x;y)=(4x3;�4y3);whichyieldsthecriticalpoint(0;0).WethenhaveHf(x;y)=12x200�12y2:ThereforeHf(0;0)isthezeromatrix,whichispositivesemidenite.However,f(x;y)increasesfrom(0;0)alongthex-direction,anddecreasesalongthey-direction,so(0;0)isneitheralocalminimizernormaximizer.2ExampleLetf(x;y)=x4+y4.Asinthepreviousexample,Hf(0;0)isthezeromatrix,butitcanbeseenfromagraphthat(0;0)isastrictglobalminimizer.26