chapter 7The numerical solutionof the matrix eigenvalue problems 1 Import the actual problem$ 2 The power method andinverse power method
chapter 7 The numerical solution of the matrix eigenvalue problem §1 Import the actual problem §2 The power method and inverse power method
s 1 Import the actual problemThere are many vibration problems in engineering, canbe transformed into a matrix eigenvalue and eigenvectorproblem.For example,the problem of stringvibration,which satisfiestheone-dimensional wave equationa?s5-g20's=f(x,t), 0≤x≤l,t≥0ar?at?Whenf (x,t)= O,the available method of separation of variables intothe characteristics of the two order ordinary differential equation value problemu"(x)+\u(x)=0, 0≤x≤l(u(0) = u(l) = 0Then the numerical simulation method, into the matrix eigenvalue problemAU=U. U±0
§1 Import the actual problem There are many vibration problems in engineering, can be transformed into a matrix eigenvalue and eigenvector problem. For example, the problem of string vibration， which satisfies the one-dimensional wave equation 2 2 2 2 2 ( , ), 0 , 0 s s a f x t x l t t x When f ( x,t) 0 ，th e available method of separation of variables into the characteristics of the two order ordinary differential equation value problem ( ) ( ) 0, 0 (0) ( ) 0 u x u x x l u u l Then the numerical simulation method, into the matrix eigenvalue problem AU U, U 0
Then consider the matrix characteristic value problems in generalLet A = (a)be a square matrix of order n, Z is a parameter.l,The characteristicmatrix-an-α12ain元-a22-α21a2n2E-A=a-a,anlan2nn2, The characteristic polynomiala-a11-a12ain-α22- α21a2nf(a)=|aE-A[=2-an-anl-αn2= a" -(au +..+amn)an-l +...+(-1)"|A
n n nn n n a a a a a a a a a E A 1 2 21 22 2 11 12 1 1、 The characteristic matrix 2、The characteristic polynomial Let ( )be a square matrix of order n， is a parameter . ij A a n n nn n n a a a a a a a a a f E A 1 2 21 22 2 11 12 1 ( ) a a A n n nn n ( ) ( 1) 1 11 Then consider the matrix characteristic value problems in general
3, The characteristic eguationE-A=04、TheeigenvalueThe roots of the characteristic eguation is called thecharacteristic roots or eigenvalues ofA,a(A)represents acollection of all the eigenvalues ofA.attention : (1) Eigenvalue of real matrix is not the realnumber, and complex roots are appeared by conjugate(2) There are n eigenvalues for the matrix of order n5、TheeigenvectorLet Z.be the eigenvalues ofA value,then any non-zerosolution vector of (2,E - A)x = O is called A corresponding tothe eigenvalue of a eigenvector 2o, referred to as a eigenvectorofA
Let be the eigenvalues of A value, then any non-zero solution vector of is called A corresponding to the eigenvalue of a eigenvector , referred to as a eigenvector of A (2) There are n eigenvalues for the matrix of order n 3、The characteristic equation E A 0 4、The eigenvalue The roots of the characteristic equation is called the characteristic roots or eigenvalues of A ， represents a collection of all the eigenvalues of A. attention ：(1) Eigenvalue of real matrix is not the real number ，and complex roots are appeared by conjugate 5、The eigenvector ( ) 0 0E A x (A) 0 0 0
6、 If a is the eigenvalue ofA, so ak is the eigenvalue of Akandaa is the eigenvalue of aA ,(a is any real number, k isa natural numberf(2) is the eigenvalue of f(A) , Among themf(a)=ao +aa+...+amamf(A)=a,E+a,A+...+amAmSuppose that a,, 22,... , a, are the eigenvalues for matrix A=(a,)ofordern + 2 +...+a, =au +a22 +..+anna,=A
6、If is the eigenvalue of A ,so is the eigenvalue of m m m m f A a E a A a A f a a a 0 1 0 1 ( ) () , 1 2 n a11 a22 ann 12 n A k k k A and is the eigenvalue of ,( is any real number, k is a natural number a aA a f()is the eigenvalue of f(A), Among them Suppose that are the eigenvalues for matrix of order n 1 2 , , , n ( ) A ij a
8、SimilaritymatrixTwo n X n matrices A and B are said to be similarwheneverthere exists a nonsingular matrix P such thatP-1 AP = BThen B is called the similarity matrix ofA, or matrix Aissimilar to B, The product P- AP is called a similaritytransformation on A.9, the similarity matrices have the same CharacteristicPolynomial So they have the same eigenvalues10,The necessary and sufficient conditions of similarityThe necessary and sufficient conditions of square matrixA of order n is similar to a diagonal matrix is that theremust be a matrix whose columns constitute n linearlyindependenteigenvectorsforA
Two n × n matrices A and B are said to be similar whenever there exists a nonsingular matrix P such that 8、Similarity matrix P AP B 1 9、the similarity matrices have the same Characteristic Polynomial So they have the same eigenvalues 10、The necessary and sufficient conditions of similarity The necessary and sufficient conditions of square matrix A of order n is similar to a diagonal matrix is that there must be a matrix whose columns constitute n linearly independent eigenvectorsf or A. Then B is called the similarity matrix of A, or matrix A is similar to B, The product is called a similarity transformation on A. 1 P AP
1l, The necessary and sufficient conditions of square matrixA of order n is similar to a diagonal matrix is that each k
11、The necessary and sufficient conditions of square matrix A of order n is similar to a diagonal matrix is that each k
12, The eigenvalues of real symmetric matrices is real13, Any real symmetric matrices is similar to adiagonal matrix .14、Suppose Ais a real symmetric matrices,thenthere exist a orthogonal matrices T, make212T-AT=2nAmong them r, 2,... , an, areeigenvalues of A
12、The eigenvalues of real symmetric matrices is real 13、Any real symmetric matrices is similar to a diagonal matrix . 14、Suppose A is a real symmetric matrices ,then there exist a orthogonal matrices T, make n T AT 2 1 1 Among them are eigenvalues of A 1 2 , , , n
Therearetwokinds of method foreigenvalueIt is appliedtothematrixA methodis a direct method,withsmallerorderingeneralThe other method is Iteration method.It is applied to the matrixwithlargerorderingeneral
A method is a direct method， There are two kinds of method for eigenvalue The other method is Iteration method. It is applied to the matrix It is applied to the matrix with smaller order in general with larger order in general
Example 1 Showthe eigenvalueand eigenvectorof matrix A解-1-22E-A=-1-1 =(a-2)(a+1)2-12Sothe eigenvalue ofA is a, =2,a, = a, =-1.when ^=2 ,show the root of Equations(2 E - A)x = 0From2E-A=0
解 2 2 1 1 1 1 1 1 1 ( )( ) E A So the eigenvalue of A is 2, 1. 1 2 3 when ,show the root of Equations(2E A) x 0 . 0 0 0 1 2 1 2 1 1 ~ 1 1 2 1 2 1 2 1 1 2E A From Example 1 Show the eigenvalue and eigenvector of matrix A, 1 1 0 1 0 1 0 1 1 A 12