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