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