文档介绍：南京航空航天大学
硕士学位论文
实对称五对角矩阵的逆特征值问题
姓名:郑永梅
申请学位级别:硕士
专业:计算数学
指导教师:戴华
20090201
南京航空航天大学硕士学位论文
摘要
本文研究实对称五对角矩阵特征值反问题。这些问题出现在振动反问题、结构物理参数识
别和结构设计等领域,其研究具有重要的理论意义和应用价值。
提出了由三组满足分隔条件的特征值构造实对称五对角矩阵的一类特征值反问题,利用正
交相似变换将该问题转化为一类熟知的实对称五对角矩阵特征值反问题,给出了求解这类问题
的广义 Lanczos 方法,并给出了求解这类问题的数值算法和数值例子。
提出了由三组(多组)特征对构造实对称五对角矩阵的一类特征值反问题,利用特征方程
将这类问题转化成线性方程组,给出了问题有唯一解的充分必要条件,并给出了求解这类问题
的数值算法和数值例子。
本文还将实对称五对角矩阵特征值反问题的结果推广到对称带状矩阵特征值反问题。

关键词:实对称五对角矩阵,特征值,特征向量,反问题,广义的 Lanczos 方法

I
实对称五对角矩阵的逆特征值问题

ABSTRACT
This paper deals with the inverse eigenvalue problems for real symmetric pentadiagonal matrices.
These problems arise in inverse problems in vibration, structural parameter identification, structural
design and so on, and are of great importance in sciences and engineering applications.
The problem for reconstructing real symmetric pentadiagonal matrices from three sets of specific
eigenvalues in which every set interlaces each other is considered. Using orthogonally similar
transformations, the problem is converted into the well-known inverse eigenvalue problems for real
symmetric pentadiagonal matrices. A generalized Lanczos method is presented for solving the
problem. A numerical algorithm and some numerical examples are given.
The problem for reconstructing real symmetric pentadiagonal matrices from three eigenpairs is
considered. Based on the characteristic equation, the problem is transformed into a linear system of
equations. The necessary and sufficient conditions under which the problem has a unique solution are
derived. A numerical method for solving the problem is presented. Some numerical examples are
given.
The obtained results for the inverse eigenvalue problems for real symmetric pentadiagonal
matrices are also generalized to the inverse eigenvalue problems for real symmetric band matrices.

Key words: real symmetric pentadiagonal ma