关于Jacobi矩阵的特征值反问题可解的充分必要条件的一个注记(英).pdfVIP

271  Vol.27, No.1 20072 JOURNAL OF MATHEMATICAL RESEARCH AND EXPOSITION Feb., 2007 Article ID: 1000-341X(2007)01-0107-06 Document code: A A Note on Necessary and Sufficient Conditions for the Jacobi Matrix Inverse Eigenvalue Problem YIN Qing-xiang (Department of Mathematics, Yancheng Teachers College, Jiangsu 224002, China) (E-mail: qx yin@) Abstract: This paper deals with an inverse eigenvalue problem of a specially structured Jacobi matrix, which arises from the discretization of the differential equation governing the axial of a rod with varying cross section. This problem was also studied by Lu L. Z. and Sun W.W.. We give some necessary conditions for such inverse eigenvalue problem to have solutions, and present some numerical examples to show that the sufficient conditions and algorithm presented by Lu is incorrect when the order of matrix is greater than 3. Key words: inverse problem; inverse eigenvalue problem; Jacobi matrix. MSC(2000): 65F10; 15A09 CLC number: O151.2; O241.6 1. Introduction A Jacobi matrix inverse eigenvalue problem is to determine the elements of a Jacobi matrix from prescribed spectrum data. Some good conclusions on this problem have been made [1–5]. In this paper, we study an inverse eigenvalue problem of a specially structured Jacobi matrix as follows: Problem 1 Find n − 1 positive numbers β , β , · · · , β so that the Jacobi matrix 1 2 n−1  1 β1 2 