(5.3.1)--5.3Similarmatri.ppt

Linear AlgebraSimilar MatrixDefinition.Let A and B be square matrices with order n.B is said to be similar to A if there exists an invertible matrix P such that P-1AP=B,where P is called similar transformation matrix,and we denote it as AB.1.Similar Matrix1.Similar MatrixThe relationship of similarity has the following properties:()Reflexivity:AA;()Symmetry:If AB,then BA;()Transitivity:If AB and BC,then AC.Theorem.Let A and B be square matrices with order n.If B is similar to A,then the two matrices have the same characteristic polynomial and,consequently,the same eigenvalues.Proof.If matrix A is similar to B,there is an invertible matrix P such that P-1AP=B.Then we haveThus the similar matrices have the same eigenvalues.Example.Note.The converse of this theorem is not true.E B=P1(E)PP1AP=P1(EA)P =P1EAP=EA Characteristic polynomials are both(1)2.However,they are not similar.Corollary.If the square matrix A with order n is similar to the diagonal matrixthen 1,2,n are n eigenvalues of matrix A.Example.Let matrix A be similar to matrix B,then find the values of a and b.Solution.Since similar matrices have the same eigenvalues,they have the same traces and the same determinants,so2+2+1=1+3+b,4+a=3bnamely,a=-1,b=1.。