实对称带状矩阵的广义特征值反问题及最佳逼近  

The Inverse Generalized Eigenvalue Problem and Optimal Approximation of Real Symmetric Band Matrix

作　　者：吴静[1] 丁小丽 王震[1] WU Jing;DING Xiao-li;WANG Zhen(School of Science,Xijing University,Xi'an 710123,China;School of Science,Xi'an Polytechnic University,Xi'an 710048,China)

机构地区：[1]西京学院理学院,陕西西安710123 [2]西安工程大学理学院,陕西西安710048

出　　处：《数学的实践与认识》2020年第22期233-243,共11页Mathematics in Practice and Theory

基　　金：西京学院校级科研基金(XJ200102);西安市科技局自然科学基金(2019218414GXRC020CG021-GXYD20.3);陕西省教育厅专项科研项目(20JK0963)

摘　　要：为了丰富实对称带状矩阵特征值反问题的理论,讨论了如下两类广义特征值反问题:(1)由给定的k个互异的特征对和给定的实对称带状矩阵构造一个实对称带状矩阵;(2)由给定的实对称带状矩阵,在问题(1)的解集合中寻找一个实对称带状矩阵,作为给定矩阵的最佳逼近.根据带宽为两类不同的情形,对问题的可解性分别进行了讨论.利用线性方程组理论,奇异值分解以及投影定理,分别得到了两类反问题存在唯一解的充要条件,并给出了解的显式表达式和数值算法;最后通过数值例子说明了算法的有效性.To enrich the theories of the inverse eigenvalue problems for real symmetric band matrix,two kinds of inverse generalized eigenvalue problems are discussed in this paper:(1) Construct an expected real symmetric band matrix by the given distinct feature pairs and a given real symmetric band matrix;(2)Find an optimal approximation of a given real symmetric band matrix in the solution set of problem(1).We discuss the solvability of the problems separately according to two different types of bandwidth.By using the theories of linear equations,singular value decomposition and projection theorem,the necessary and sufficient conditions to existence of the unique solution of the problems are obtained,explicit solution expression and numerical algorithm are also given.Numerical examples illustrate that the proposed algorithms are effective.

关 键 词：实对称带状矩阵 广义特征值 反问题 最佳逼近 

分 类 号：O241.6[理学—计算数学]