The equations of the same root2xi -X2 -Xg = 0-Xi+2x2-x3 =0We can get the fundamental system of solutions5i = (1,1,1)So, the all eigenvectors of A, = 2 arek,5i(k, ±0)
The equations of the same root 2 0 2 0 1 2 3 1 2 3 x x x x x x We can get the fundamental system of solutions (1,1,1)' 1 ( 0) k1 1 k1 So, the all eigenvectors of are 1 2
When 2,=2,=-l , we can calculate the equation(-E-A)x = OfromE+A=We can get the equation of the same root x, + x, + X, = OWe canget the fundamental system of solutions53 = (-1,0,1)5, =(-1,1,0)'So, the all eigenvectors of 2, = a, = -1 arek,52 +k,5; (k 2 , k , are not O at the same time)
0 x1 x2 x3 We can get the fundamental system of solutions ( 1,1,0)' 2 ( 1,0,1)' 3 0 0 0 0 0 0 1 1 1 ~ 1 1 1 1 1 1 1 1 1 E A from 2 2 3 3 k k When , we can calculate the equation 2 3 1 (E A)x 0 We can get the equation of the same root So, the all eigenvectors of are 2 3 1 2 3 ( k , k are not 0 at the same time)
s 2 The power method and inverse power methodThepowermethodPrincipal eigenvalue and the corresponding eigenvectormatrix calculation. According to the characteristics ofthe maximum modulus value is called the principaleigenvaluesofmatrixA1, The basic idea: First choose an initial vectorX(), Constructingthe following sequence:X(0), X() = AX(0),X(2) = AX(I),., X(k) = AX(k-I)
1、The basic idea：First choose an initial vectorX(0), Constructing the following sequence： Principal eigenvalue and the corresponding eigenvector matrix calculation. According to the characteristics of the maximum modulus value is called the principal eigenvalues of matrix A 一、 The power method §2 The power method and inverse power method (0) (1) (0) (2) (1) ( ) ( 1) , , ,., ,. k k X X AX X AX X AX
2、Example1:supposeamatrix11-51-6141-5Using the characteristic polynomial easy to calculate theeigenvalues ofAfor:2, = 0.41263, 2 = 0.0040196Next we use power method to calculate, choose the initial vectorX(0) =(l,O) The vector sequence isX(0), X(I) = AX(0), X(2) = AX(),..,X(k) = AX(k-1)
Next we use power method to calculate, choose the initial vector 2、Example 1：suppose a matrix Using the characteristic polynomial easy to calculate the eigenvalues of A for： The vector sequence is 1 1 4 5 1 1 5 6 A 1 2 0.41263, 0.0040196 (0) (1) (0) (2) (1) ( ) ( 1) , , ,., ,. k k X X AX X AX X AX (0) X (1,0)