Matrices - Written Notes

CLASSMALEDatePageMatricesAmatrixisawrittenwithinrectangular[]arrayofnumbersor()Forexample,A2.23=[anj]alla12a13=2x3aazza23(rowxcolumn)Thereareseveraltypesofmatrices-row=has1rowonlycolumn=wasIcolumnonlyzero/null=eachelementiszerosingleton=hasonly1elementequal=eachcorrespondingelementisequal.squarenoofrows=noofcolumnsalla12a13Azx3 =aazza23asA32a333rd orderprincipal diagonalsquar manixIAI=detA=valueofdeterminantsingular = IAI = 01AI0non singular PageThese are several types of square marindiagonalscalaridentityIIIIIIan00a00I0 az2 00a0O0 D azz00a0trangular(upper)(Lower)anO12 a13an 0 00a22 a23ora21 a22 000a32a3, azz azzaij = 0 D i>aij = OV isinote thatdiagonal E upper triang & lower triangandnullEdiagonaluppertri.&lowertiThetraceofamatrixisthesummationoftheelementsinitsprinciplediagonal.For example if-abA =then tr. A= (a + d)edAlgebraicoperationscanbeperformedonmarricesLikeso-CONLYSQUARE)a b+Pq=atpb+qCdrsct r disTheadditionofmatricesiscommutativeandassociative classmateDateTheadditivePageAB=B+inverseofAAis-=cA,andifMultiplyingalike somatrix-byascalarisdonek.a b=ka kbcdALL elements,kc kdunlike determinant.notetr.(A+ B )=tr.A+tr.B&tr.(KA)=ktr.ATOmultiplyAmxnxmxnBnxp=Cmxpwe do,must be samerowInAxBbycolmulti.Aiscalledmultiplier&BisprecalledpostmultiplierForexample,IA =23&B=-02IThe product AB = C0-1+4 0+0+232C ==-2+6-2+0+35tThe productBA=D0-21-42-6:-2-3 -4=0-1-2D =0+0-1+0-2+D0+12+24+3I47InAB#BASometimesA&Bgeneral,cannotbemultiplied(non-conformative).calledcommutative.IfAR=BAtheyare7notediag(abc)dingqr> = diag ap bq 4)(communative) The multiplication of matrices isassociativeaswellasdistributiveHowever,theorderisimportantAB+ Ac = A (B+C)AB + CA = A(B+C)XThemultiplicationofAby0(mule matrix) 1 AO = OA = 0I (identity) AI = IA = Ak (const)KAB = (KA)B = A (kB)noteAAAA2A = AA2 = A3AmAn =Amth&(AM)=AmnIn=I&Amu = Imxn.someidentitiesaredifferentduetothe - order of multiplication :-(A+B)2 = A2 + B2 + AB + BAif AB = BA, thenit will be 2AB(A+B)(A -B) = A2 + BA - AB -B2(AB)2 = AB . AB(A+B)3 = AAA + AAR ABA + BAA + if AB = BA ,ABB + BAB + BBA + BBBthen termscan be combined=A2+3A2B3AB22++B2to give that-(I+A)"="E"CrATnoteIfAR=A&BA=B,thenA2=AandB2=B. CLASSMALEDatePageThetransposeofA'abyorAmatrixItisisrepresentedrowsandthecolumnsinterchangeofMathematicallyinamatrixA=[ay]mxnAT=[aji]nxmForexampleifA=3,45614AT =2536Thepropertiesoftransposeareasfollows--(AT) =(A+B)T = AT + BT(KA)T = k (AT)tr. AT = tr. Ake scalarIATI=IAIandIkAI=km/Al=k"/AlIAB=AllBI=IBAIA&2 mabicesreversal1A1" IAN/law(AB)T = BTAT & (ABC) = CIBTAT(An) = (AT)(SM)AmatrixisareinwhichA=ATsymmeticFor example,a h gh b fAT = AsymmetricA =g f C(SSM)marrixisoneinAskewsymmetricwhichA-ATForexample= Pagea baa C=-aA =AT ==0d=-d=0cdb dcC=toA = - -AT =a - ca 6=-b-dcdthus / A =0k0 -k=-k 0k0Thus,theprinciplediagonalsofSSMarezeronotethatifAisSM&BisarandommatrixAT = AKAKAT-,-A,A"BAB+arealsoSM.alsoifAisSSM&Bisarandommatrix,KA, BABT are also SSMA2nisSMA2nHisSSM.and,ifAandBareSSM,A+BA-B -areSMbooAB + BAisSM,butAB-BAisSSMalsoifAandBareSSMA + B , +A- RareSSMtoo.AB + BAisSM,butAR-BAisSSMnotethatCA+AT)isSM&(A - AT)isSSM.thusA=A+AT+A-AT22SMSSMthedeterminantofoddorderSSM=D .ifBisSSM&CisacolumnmatrixCTBC = 0 CLASSMATEDateThePageadjointofatransposeofsquareaMathematicallymatrixmatrixistheofcofactorsofthematrixanQ12A =Q13azea22G,Q23G2,G3a3,A32azzadj. A =C21C22C23C21C22C32Forexample+ifA=abcds, adj A = -C d-b aThepropertiesofAadjarex(adj,asA)follows-=(adj.A)xA=IA/Itake det on-ladj. AI =An-1communicativeboth sidesadj(adj.A) = LAIN-2 Ato get 2nd ptopladj(adj. A)derived using=IAI1st& 2nd1 atj(adj' adj A));)=A(n-1)m(mthmer)putA=adj(adjx)&use 2ndadj(kA) = kn-1 adj. Ause1stofsubstitutetwiceadj(AT) = (adj. A)Tsimplityinto1=1adj(AB)=(adj.B)x(adj.A)reversallawTheinverseofasquarisdenotedbyA-!Mathematically,A x A= I = A X AcommutativeTherefore,A-'=/IAI)(adj.A)2ThepropertiesofinverseareasfollowsismultiplicativeinvesseItofA.theA-'onlyexistsifAisnon singulari.e AI # 0(A-)=Aand (A-')' = (AT) classmateDatePage(A-') = 1/1A1(AB) = B'A-'reversal law[diagla b c)] = diagl'a 16 1/c)(An) = (A-')noteA=01A10::AB=0*A = 0 or B = 0AB = AC * A = 0 or B = CAB - AC = 0 => A(B 3-C) 0If A is non singular ( IAI # 0) thenifAR=ACB=C(premultiplybothsidesbyA-)detAB=0(A)=0or=0(mustonbothsides)AR=D& A 0B=o& 1B1 - 0 (use A-')AB=OLA#OcontrapositiveBl=0if IRI=0, use 84)AnidempotentmatrixisoneforwhichA2=AandA0Notethatforan idempotentmatrix,An=AForanilpotentmatrixAk=+0whereKEN&theleastvalueofkiscalledindexofthehilpotentmatrixForexampleA =50:]& A2 = [: 00so A is hilpotentof index 2.AninvoluntarymatrikisoneforwhichA2=I,whichimplies-III 1A12 = I1A1==1non singularthus,A-1=A(premium equation by original A-1 ) classmateDatePageanFor A is Orthogonas matrix, AAT = I = ATAifOrthogonal A = ATLeta=andc=air GC + a2j + ask , is = 6, + b,j + b3 k+cj+C3 kThen,as a2 a2A =b,a b, Gb2b2& AT -Ga,b2C2C2C3a.aa b3 a2.6aIA.A =00b.a 5.5 5.2=0I-I2c.a 2.6 C.C00(dot product of perpendicular vectors is zero)Thus if A.AT T = I, la = 151-121= = = I and a, 5,,c are thus orthogonal unit vectorsproduce them via crossproductIf A is a real matrix and t (AAT) o O, AFmust be a null matrix,noteThebinomialexpansionofmatricesiseasier if it is a nilpotent matix.For exampleA =of 2 H = 0 2 2 +=2I+B-B==00Thus, A K = k (C(ZI)"go + + (C,(2I) k-2 B2 +>=2 I+2kmkB+0 Inaoflinearequationssystemax+b,y+43=d,a2x+b2y+C23=d2AX = Ba3x+b3y+Gz=d2a b,4xd,A =a2b,&x=y&B =d2C2asb3}d3GAX = BIAI#0/A) = 0A-1 AX = A+B(adjA) Ax = (adjA) BX = A-BnullIAIIX= (adjA)Bconsistent ,unique solnif # 0it = 0no solutioninfinitesolutionThecharacteristicequationofasquarematrixisI A->Il.ThecayleyhamiltonianStatesthateverysquarematrixsatistiesitscharacteristicequation,if1=AIA - AI = 0Thefollowingelementarytransformationscanbeperformedonmatrices -RiRorCiG.RiKRiorCikCiRiRi + kRjorCiCi + kgj classmateDatePageAny on elementary C row or column transformationthe same can be obtained by applyingon its pre/post factor.elementary transformationFor example1I2,2=I4L34(I. A = A))tR2R2 - 3R,2I2D-3I340-2R2R22I2=3/2-1/2340I1R1R 2R2-21/2=1o3/21/2340 I(A+ A = I)The rank of a marrix is equal tothe number of independent rows in it.For exampleIRsindependent4 56 R,rows = 25 7 R,+R2If/A/ $ 0, the rank is equal to the order.