低阶对称双随机矩阵逆特征值问题的通解

摘要

Given an n-tuple Λ of numbers, real or complex, the problem of deciding the existence of a nonnegative (stochastic) matrix with spectrum Λ is called the nonnegative (stochastic) inverse eigenvalue problem. This problem has long time been one of the problems of main interest in the theory of matrices. Other reference gave the sufficient conditions for doubly stochastic inverse eigenvalue problem of order two to five to have a solution and the formulas of the corresponding solution, and firstly gave the sufficient conditions for constant row sums symmetric inverse eigenvalue problem (of any order) to have a solution and the formula of corresponding solution, and then gave the sufficient conditions for the symmetric stochastic inverse eigenvalue problem to have a solution and the corresponding solution. In this paper, after presenting the concept of general solution of an inverse eigenvalue problem ( of any order) and the concept of totally general solution of a 3 ×3 symmetric doubly stochastic inverse eigenvalue problem, we firstly gave the sufficient and necessary conditions for a 3 × 3 symmetric doubly stochastic inverse eigenvalue problem to had the totally general solution with the formula of the totally general solution, secondly gave the sufficient and necessary conditions for a 3 ×3 symmetric doubly stochastic inverse eigenvalue problem to had the general solution with the formula of the totally general solution, and finally gave several sufficient conditions for a 4×4 symmetric doubly stochastic inverse eigenvalue problem to had a solution with the formula of the general solution.%对给定的实或复 n-重Λ={λ1,…,λn},决定是否存在以Λ为谱的非负(随机)矩阵的问题称为非负(随机)矩阵逆特征值问题,这一直是非负矩阵理论中尚未完全解决的一个研究热点。作者曾对 n ∈{2,3,4,5},研究 n 阶双随机矩阵逆特征值问题有解的充分条件并给出相应解的公式。最近,又对任意正整数 n,先给出行和为常数的对称矩阵的逆特征值问题的充要条件和解的公式,后给出对称随机矩阵逆特征值问题有解的两种充分条件和解的公式。论文在提出任意阶对称随机矩阵逆特征值问题通解的概念和3阶对称随机矩阵逆特征值问题完全通解的概念之后,首先给出3阶对称随机矩阵逆特征值问题存在完全通解的充要条件和完全通解的公式；其次给出3阶对称随机矩阵逆特征值问题存在通解的充要条件和通解的公式；最后给出4阶对称随机矩阵逆特征值问题有解的几种充分条件和相应解的公式。