理学线性代数IIchapter

Matrix(matrices)

矩阵

Acolumnvector

行向量Asquarematrix

方阵

Arowvector

列向量Adiagonalmatrix

对角阵

Anidentitymatrix

单位阵Anuppertriangularmatrix

上三角阵Alowertriangularmatrix

下三角阵Asymmetricmatrix

对称阵Askew-symmetricmatrix

反对称阵Row-echelonform

行阶梯型NewWordsarray

排列column

列algebraicoperations代数运算row

行

addition加法

subtraction减法multiplication乘法entry表值,元素thetransposeofamatrix

矩阵的转置commutative

可交换的associative

可结合的

distributive

可分配的NewWordsLinearalgebraisabranchofmathematicsdealingwithmatricesandvectorspaces.Matriceshavebeenintroducedhereasahandytoolforsolvingsystemsoflinearequations,determininglineardependenceofvectorsandsolvingtheproblemofeigenvalueandeigenvector.Theyhavealsomanyapplicationsintheotherfields,suchasstatistics,economics,engineering,physics,chemistry,biologyandbusiness.Contents§1.1Matricesandmatrixoperations§1.2Determinants§1.3Cramerrule

§1.4InversematricesandpartitionedmatricesChapter1MatrixandDeterminants1.TheDefinitionofMatricesThissectionwillconsidermatricesandtheiroperations.Wewilldefinealgebraicoperationssuchasaddition,subtraction,andmultiplicationofmatrices.Definition1§1.1MatricesandmatrixoperationsRemarks(3)ThetermmatrixwasfirstintroducedbyBritishmathematicianJamesJosephSylvesterin1890.Thefollowingsaresomekindsofspecialmatrices:Thediagonalmatricesformanimportantclassofmatricesinthematrixtheory.Matricesthatcanbetransformedtodiagonalmatricesbypremultiplyingandpostmultiplyingwithsuitablematricesareofspecialimportance.Likediagonalmatrices,triangularmatricesformanimportantclassofmatricesinthematrixtheory.(i)Amatrixisinreducedrowechelonformif(1).Itisinrowechelonform;(2)theleading(leftmostnonzero)entryineachnonzerorowis1.(3)allotherelementsofthecolumninwhichtheleadingentry1occursarezero.2.TheOperationofMatrices(1)EqualityofMatrices(2)ScalarMultiplicationofMatrices(3)AdditionofMatricesMatrixadditionisbothcommutativeandassociative,thatis,

Scalarmultiplicationofmatricessatisfythefollowingrules:(4)MultiplicationofMatricesRemarks:(1)NotethatABisdefinedifandonlyifthenumberofcolumnsofAisequaltothenumberofrowsofBExample1Solution(5)PowerofMatricesPowerofmatricessatisfiesthefollowingrules:Example2Supposethatthesame-ordersquarematricesAandBsatisfytheequationProof(6)TransposeofMatricesTransposeofmatricessatisfiesthefollowingrules:Example3SolutionExample4SolutionRemarksExample5SolutionExample6Weclosethissectionwithanexamplethatpointstoanapplicationofmatrixtheorytoreal-lifeproblemConsiderthethreemajorlong-distancetelephonecompaniesAT&T,MCI,andSprint,whichserveagivenareawithatotalpopulationofN(inmillions).Throughtheirconstantadvertisingdirectedtowardattractingcustomers,eachofthemhasbeengainingaswellaslosingcustomers.Assumethatthereisnoincreaseordecreaseinthetotalnumberofcustomersbeingserved.Furthermore,itisobservedthatattheendofeachmonth,thenumberofcustomerschangingcompaniesisasfollows:20%ofAT&TcustomersleaveAT&T,but10%ofMCIcustomersand10%ofSprintcustomersjoinAT&T.MCIloses30%ofi