一个特殊矩阵的两类逆问题  

Two Types of Inverse Problems for a Special Matrix

作　　者：雷英杰[1] 吉雁斐 郭雪娟 LEI Yingjie;JI Yanfei;GUO Xuejuan(School of Science, North University of China, Taiyuan 030051, China)

机构地区：[1]中北大学理学院,太原030051

出　　处：《西南师范大学学报（自然科学版）》2021年第10期16-25,共10页Journal of Southwest China Normal University(Natural Science Edition)

基　　金：国家自然科学基金项目(12071444).

摘　　要：研究了一个特殊矩阵A的两类逆问题,其中矩阵A是由图为扫帚形的矩阵推广而来的.问题1利用箭形矩阵和Jacobi矩阵的相关性质,将求解此类矩阵的逆特征值问题转换为求解线性方程组问题,最后得到了该问题有唯一解的充分必要条件.问题2将该矩阵的每个顺序主子式的最小和最大特征值作为其特征数据,利用矩阵顺序主子式之间的递推关系来解决.两个问题最后均给出了解的表达式以及数值模拟实例,验证了结果的准确性,推广了箭形矩阵和Jacobi矩阵的逆特征值问题.In this paper,two kinds of inverse problems have been studied of a special matrix A,where the matrix A is generalized from a matrix whose graph is a broom.The first problem is related to the usage of the related properties of the arrow matrix and the Jacobi matrix to convert the problem of solving the inverse eigenvalue of this type of matrix into the problem of solving a system of linear equations,and finally,the sufficient and necessary conditions for the problem to have a unique solution are obtained.The second problem is related to the minimum and maximum eigenvalues of each sequential principal minors of the matrix as its eigen data and we use the recurrence relations among sequential principal minors to solve it.At the end of the two questions,the understood expressions and numerical simulation examples are given to verify the accuracy of the results,and the inverse eigenvalue problems of the arrow matrix and the Jacobi matrix are generalized.

关 键 词：箭形矩阵 JACOBI矩阵 逆特征值问题 顺序主子式 最大特征值 最小特征值 

分 类 号：O157[理学—数学]