tailieunhanh - Báo cáo toán học: "Eigenvectors and Reconstruction"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Eigenvectors and Reconstruction. | Eigenvectors and Reconstruction Hongyu He Department of Mathematics Louisiana State University Baton Rouge USA hongyu@ Submitted Jul 6 2006 Accepted Jun 14 2007 Published Jul 5 2007 Mathematics Subject Classification 05C88 Abstract In this paper we study the simple eigenvectors of two hypomorphic matrices using linear algebra. We also give new proofs of results of Godsil and McKay. 1 Introduction We start by fixing some notations HE1 . Let A be a n X n real symmetric matrix. Let Aị be the matrix obtaining by deleting the i-th row and i-th column of A. We say that two symmetric matrices A and B are hypomorphic if for each i Bi can be obtained by simultaneously permuting the rows and columns of Ai. Let s be the set of permutations. We write B S A . If M is a symmetric real matrix then the eigenvalues of M are real. We write eigen M A1 M A2 M . An M . If a is an eigenvalue of M we denote the corresponding eigenspace by eigena M . Let 1 be the n-dimensional vector 1 1 . 1 . Put J 11. In HE1 we proved the following theorem. Theorem 1 HE1 Let B and A be two real n X n symmetric matrices. Let s be a hypomorphism such that B S A . Let t be a real number. Then there exists an open interval T such that for t 2 T we have 1. An A tJ An B tJ 2. eigen n A tJ and eigen n B tJ are both one dimensional I would like to thank the referee for his valuable comments. THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 N14 1 3. eigenxn A tJ eigenxn B tJ . As proved in HE1 our result implies Tutte s theorem which says that eigen A tJ eigen B tJ . So det A tJ XI det B tJ XI . In this paper we shall study the eigenvectors of A and B. Most of the results in this paper are not new. Our approach is new. We apply Theorem 1 to derive several well-known results. We first prove that the squares of the entries of simple unit eigenvectors of A can be reconstructed as functions of eigen A and eigen Aị . This yields a proof of a Theorem of Godsil-McKay. We then study how the eigenvectors of A