matrilearning矩阵学习.docx
一、高斯消元法以及（高斯消元法及矩阵）ThreePossibilities UNIQUESO1.UTION:TberViSOneandcmIyonee<“valuw*forbeZJSchatsatfecsallequationssimultaneously. NOSO1.UTION:Then、运DOeofvaluforthrxlVthatatii<'ftallcquatk>!iHinmha!x>i2ilytlwwlutioHetwempty. INFINITE1.YMANYSO1.UTIONS:TiMmart-hhndynuy<iiflcrentHeUofVaJueeforth<tiUlalSat耶allequationsximultAiMMMudy.ItuIKM<iifcdttoPrcyVrthatifwtemhasttnorvtlu<mmM>ht>on.tl>rnithzinfinitelytnAnysolutioic*.R*rxAinplc,itwixnpus*iblcfuraxp*U11UjIiavcrxwtytwodiHcrrtt4ut>o!u<.行阶梯阵及其秩RankofaMatrixSUPPoeCAmxnisreducedbyrowoperationstoanCChCkmformE.Tlrurilofisdefinedtobethenumberrank(=mmbrrOfPiVnU=numberofnonzeroToWxinE=numberofbasiccolumnsinA.wlrvthebtmircolumnsofAarc<k*fimxitolll>eculunmnhitlmt<x>tninth<'j>iu)<dx<iti(H!<.b矩阵的一样性ConsistencyEachOflhOtblk>w*ingfe»equivalenttosayingIIuUbUcotMhunt. Inrwreducing:AlIEa11,<»ftl,t<><m'ingf<11neverpw3<:(OOOI).WhenfoO.(2.3.2) l>hnnilMwi<!<ohuninb.(2.3.3) rnAb=rn（八）.(2.3.4) l>hnCoiiMiiiUKmofthe1nuq<*<x>luutin.(2.3.5)齐次系统A0系统为，假如有段有至少一个非零数字，即系统变成川以，此系统为.假如Av.,的秩是r，和基础列对应的&是，其它的是有r个，有-，个基础解是的线性组合当且仅当其秩为n时有唯解Summan'1.CtAmnbethecoefficientmatrixforaho11>ogriMMnwMyMtrrnofmliiM)arCqiIa【iousinnUUknOWns.anduppobrank（八）=r. Theunknownsthatcorrespondtothepositionsoftheba*iccolumns(i.c.athePiVOulpce4tion»)arccalledthebajtMmriablrtandtheUnknowi4OOrTeHPOndingtotheoe4tionofIbenonbaMcCUIUnUearcEHCdthenvmriablr, Then?areCxaCtlyrbaj>kvanabkal-rfreeVariBl)le. Todescribeallx>!utkjuIredUCCAtoarowechelonformuingGatuvianelimination,andthenusebackMIIEitUtiontouohvforthebasicVariat4internbofIbefreeVariabk».ThibProdUCeHthegrtumlnolutumthatha*theformX=lh+rjh2÷-+"Jiwhen?thetermsl.xjr/Rran*XlwfrwvariablesandWhCmIi1.h2bn.rarrn×Icolumn9thatrepresentPartieUhuBm>1u-tkMsoflivehotnogciwousHystonuTIiollsarh<kp<*ndcntofWhIehrowechelonformMw*cdhitheImm皿IMiIUIiOnproceed.AsIbefreeVnriabIEzlrang?overAllPOWibkVaIUEtheMncTalK)IUtk>ngcn-Crat<*tillpotuiblesolutions.AIxnog<,n<xxi5XyMCmpomm*sm>.aUniqU<x4tion(Chetriviaso)u-tkx)ifandonlyifrni（八）=ni.c.ifandonlyiftherearcnofreeVariabd非齐次系统存在解的条件是伊仍1.,的秩等Fa的秩，即方不是Mi1.的基础列；存在唯一解的条件是其齐次形式只有O解，没有，秩等于未知数X的个数的表达式是齐次解加特别解X；=A+,.X/+.+«,町“Tliesystempossessesauniquesolutionifandonlyifanyofthefollowingistrue.orank（八）nnunl>crofunknowns.>Tlicrcartnofreevariabk.c>TheassociatedhomogeneousSyStelnpossessesonlythetrivialsolution.线性矩阵的应用一一基尔霍夫定律基本的(共枕转置)PropertiesoftheTransposeIfAandBarctwomatricesoftheSameblupe.andifiascalar,theneachofthefollowingstatementsistrue.(A+B)r-A+Brand(A-Bf-A,+B,.(3.2.1)(nA)7-0Aran<l(nA),-7>A>.(3.2.2)满意条件/(x+y)=(x)+(y)fax)=af(x)的系统为线性系统.在平面几何中，线性系统指直线，在高阶系统可表示为多个变量的线性组合f(xl.x2.jcn)=a,.v1+a2x2+.+anxn两个矩阵的枳O代表了两个线性系统的，即两个矩阵的史合。DistributiveandAssociative1.awsFrWafSIuabkIiIalrim<1ithek4kmiyfetnc. A：BC)BAC(Irft-IiAiid<!ix-OUnPbEthatx-O.t11lntivlaw). (D+E)F-DF*EF(血匕kd<lhtrtl*tlvvbu), A(BC)-(AB)C5vwMUti¼vlaw.ExistenceofanInverseFor(Uinxnmatrix.thrGJIovnngKtAtriiwrtfAacCellHvAkik.(3.7.6)(3.7.7)(3.7 A4nguhr).rul（八）11,PropertiesofMatrixInversionEuruoiMinu!ariiutnccvAandB.UtrS>Uw*iuPrUPettk¾bold.(A,),-A.(3.7.13)TbrproductABka!m>nonMinulAr.(3.7.14)<.XB,B1A1(tbccvcnorderlawIcrfnvmx>).(3.7.15)(A,)三(Ar)1al(A,)-(A,.(3.7.16)三、两个矩阵之间的关系 相像矩阵时于矩阵A8当且仅当存在一个可逆方阵P使得产'AP=B则称A相像于B,记为性质：秩相等、行列式相等、相等、相同的特征值，相同个数的特征多项式、相同的初等因子 等价矩阵对矩阵A3,当且仅当存在两个可逆方阵P.Q使得IiAQ=B相像矩阵肯定是等价矩阵，反之不成立。等价矩阵其实就是矩阵经过初等行变换和初等列变换得到的矩阵，两者具有相同的秩 合同矩阵存在可逆矩阵P使得P,AP=B四、四个特别的向量空间Summan'T1m*fofu><lM11mtalslxccsa<jciaUt<luitlAmanairsfolk>u>.R（八）AxRm.(Ar)三yCR".N（八）=xIAx=0)Cr(r)-yI,y=C Therangorcohmnspace: T<r<wmixorl<fl-hndir: Thenulkqiiu'C： Th<l<f-l*u<l11u11jmu<*:1.rtPbeanonsingularmatrixsw*hthatPA=U.whereUinrow<*<'b<k)<>nf<>r11.11><lz"xx*rrank()t=r. Spumingsetfor（八）=tlIW4cCoIulnn5inA Spanningm?<for(A)=thenonzerorowsinI： SjMUining*hfcr.V（八）=th<l,inthe,nrnlMention<>(AxO. S>anningsetforN(A,)=the1j4rn-rrowsofP.IfAandBhavethesameshape,th<n AHBoN（八）=N(B)-(Ar)=A(Br). AB«W（八）=(B)三>.V(A)='(B).五、线性相关或线性独立（）1.inearIndependenceandMatrices1.etAbeannIXnmatrix. EachofthefallowingstatementsisequivalenttosayingthattheCOhHn4ofAformnlinearlyidrMiHl<*tMt.t>2V（八）三0.(4.3.2)t>rank（八）=n.(4.3.3) EachofthefollowingstatementsiscquiaalcnttosayingthattherowsofAforualinearlyindependentset.t>N(A)0.(4.3.4)t>rank（八）三m.(4.3.5)WhenAisftSqHftrOmatrix,eachofthefollowingyitat<kincmsisCqUWUknttosayingtlatistioiiMngidar.c>TheColumn«ofAformalinearlyindependewt«ct.(4.3.6)t>TlicrowsofA(i>nnalinearlyin<lci<l<tt(4.3.7)线性空间的基（）张成这个空间所需的最小数目的线性不相关的向量C
