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1、1THE EIGENVALUE PROBLEM4 THE EIGENVALUE PROBLEM2THE EIGENVALUE PROBLEMOverviewlIn section 4.4 we move on to the general case,the eigenvalue problem for(nn)matrices.The general case requires several results from determinant theory,and these are summarized in section 4.2.l The eigenvalue problem is of
2、 great practical importance in mathematics and applications.lIn section 4.1 we introduce the eigenvalue problem for the special case of(22)matrices;this special case can be handled using ideas developed in Chapter 1.3THE EIGENVALUE PROBLEMCore sections The eigenvalue problem for(22)matrices Eigenval
3、ues and the characteristic polynomial Eigenvectors and eigenspaces Similarity transformations and diagonalization 4THE EIGENVALUE PROBLEM21113141111011430212102114.1 The eigenvalue problem for(22)matricesA5THE EIGENVALUE PROBLEM:For an(n n)matrix,find all scalars such Definitionthat the e 4.1.1quaio
4、nA AXXhas a nonzero solution,such a scalar is called an eigenvalue of,and any nonzero(n 1)vector satisfying is called an eigenvector corresponding to.AXAXXAll scalarsNonzero solution/Infinitely many solution 1.The eigenvalue problem6THE EIGENVALUE PROBLEMThe Geometric interpretation of Eigenvalue an
5、d eigenvector AXX00XAXXAX7THE EIGENVALUE PROBLEMThe calculation of Eigenvalue and eigenvector AXX?,?X0AXX0()AI XHomogeneous Systems0()XStep 1:find such thatall scalar is singular.AI0(-)Step 2:given a scalar such that is singular,find such that all nonz ero vectorsXAI XAI12000det()()(),is singular nA
6、Ar AnAXXA AAlinearly dependent8THE EIGENVALUE PROBLEMEigenvalue and eigenvectors for(22)matrices00is singular.abAIcdabcdabAcd9THE EIGENVALUE PROBLEM()(2adadbc)0=?abcdabAcd22det()121112ad=aadbc=aA10THE EIGENVALUE PROBLEMExample:Find all eigenvalues and eigenvectors of A,where 2625Asolution:The matrix
7、 has the form526 2AAII 11THE EIGENVALUE PROBLEM2(1)is singular if and only if52120or 320AI()().212since 3221it follows that is singular if and only if2 or 1()(),AI12THE EIGENVALUE PROBLEM112222222323226400220331(),forAIxxXxxxx122 or 12625A13THE EIGENVALUE PROBLEM21222223142216300110221(),forAIxxXxxx
8、x for a given eigenvalue,there are infinitelymany eigenvectors correspondNointe thag tto.122 or 12625A14THE EIGENVALUE PROBLEM4.2 Determinants and the eigenvalue problem(omit)4.3 Elementary operations and determinants(omit)15THE EIGENVALUE PROBLEM4.4 Eigenvalues and the characteristic polynomial(2)G
9、iven an eigenvalue,find all vectors such that(0.(Such vectors are the eigenvectors corresponding to th ne eigenvalonzeroue.)XAI)X(1)Find such that is singular.(or =0).(Such scalars are tall scalarhe eigenvas lues of)AIAIAThe eigenvalue problem for an(nn)matrix two pahsas:rtA16THE EIGENVALUE PROBLEME
10、xample:Use the singularity test to determine the eigenvalues of the matrix A,where 112330111AIn this section we focus on part 1,finding the eigenvalues.is singular0AIdet(AI)17THE EIGENVALUE PROBLEMsolution:A scalar is an eigenvalue of if and only if is singular.such that0AAIdet(AI)where is the matri
11、x given by111000330021100111033211AIAI112330111A18THE EIGENVALUE PROBLEM32we have 1110332115632det(AI)()()123from the singularity test,we see that is singularif and only if 02 or 3AI,112330111A1231122331235500aaadet(A)19THE EIGENVALUE PROBLEMThe characteristic polynomial111212122212nnnnnnaaaaadeta(A
12、aaaI)Let be an(nn)matrix.TheTheorem:is a polynomial of degree n n i n.det(AIA)20THE EIGENVALUE PROBLEM Let be an(nn)matrix.The nth-degree polynomial,is calledDefinition:characteristi the for.c polyno mialAp()det(AI)A:Let be an(nn)matrix,and let be the characteristic polynomial for.then the eigenvalu
13、es of are precisely the rootTheores of 0m.ApAAp()characteristic polynomialp()det(AI)0p()det(AI)characteristic equation21THE EIGENVALUE PROBLEM(1)an(nn)matrix can have no more than n distinct eigenvalues.(2)an(nn)matrix always has at least one eigenvalue.The number of times the factor()appears in the
14、 factorization of given above is calledalgebraic multiplicity of the rp)r(22THE EIGENVALUE PROBLEMk1 Let be an(nn)matrix,and let be an eigenvalue of.Then (1)is an eigenvalue of;1(2)If is nonsingular,then is an eigenvalue of;(3)If c is any scalar,then+c is an Theorem:k-AAAAAeigenvalue of AcI.Special
15、Results)(Af2aAbAcI)(f2abc111(1)kkkkA XA(AX)A(X)(AX)111(2)or 0111110ndet(AI)det(AA A)det(A(IA)det(A)det(IA)det(IA)()det(AI)(3)(AcI)XAXcXXcX(c)X111(2)AXXXA XXA X23THE EIGENVALUE PROBLEM0TTdet(AI)(det(AI)det(AI)00det(A)det(AI):Let be an(nn)matrix.Then and haveTheoremsame eig thenve.aluesTAAA Let be an(
16、nn)matrix.Then is singTheorem:if and only ifular =0 is an eigenvalue of.AAA24THE EIGENVALUE PROBLEM3232:Let 22 and for anymatrixdefine the matrix polynomialby22where Iis the identity matriprove that if is an eigenvalue of,then thenumber is an ei Examplegxq(t)ttt;(n n)H,q(H)q(H)HHHI,(n n)Hq().envalue of the matrix q(H).25THE EIGENVALUE PROBLEMHXX3232323222222222q(H)X(HHHI)XH XH XHXIXXXXX()Xq()X32 22q(t)ttt 26THE EIGENVALUE PROBLEM4.5 Eigenvectors and EigenspacesEigenspaces and Geometric Multiplic
