安徽大学学报（自然科学版）
安徽大學學報（自然科學版）
안휘대학학보（자연과학판）
JOURNAL OF ANHUI UNIVERSITY(NATURAL SCIENCES EDITION)
2014年
4期
1-8
,共8页
逆特征值问题的通解%对称双随机矩阵逆特征值问题%特殊正交矩阵
逆特徵值問題的通解%對稱雙隨機矩陣逆特徵值問題%特殊正交矩陣
역특정치문제적통해%대칭쌍수궤구진역특정치문제%특수정교구진
general solution of an inverse eigenvalue problem%symmetric doubly stochastic inverse eigenvalue problem%typical orthogonal matrix
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。
對給定的實或複 n-重Λ={λ1,…,λn},決定是否存在以Λ為譜的非負(隨機)矩陣的問題稱為非負(隨機)矩陣逆特徵值問題,這一直是非負矩陣理論中尚未完全解決的一箇研究熱點。作者曾對 n ∈{2,3,4,5},研究 n 階雙隨機矩陣逆特徵值問題有解的充分條件併給齣相應解的公式。最近,又對任意正整數 n,先給齣行和為常數的對稱矩陣的逆特徵值問題的充要條件和解的公式,後給齣對稱隨機矩陣逆特徵值問題有解的兩種充分條件和解的公式。論文在提齣任意階對稱隨機矩陣逆特徵值問題通解的概唸和3階對稱隨機矩陣逆特徵值問題完全通解的概唸之後,首先給齣3階對稱隨機矩陣逆特徵值問題存在完全通解的充要條件和完全通解的公式；其次給齣3階對稱隨機矩陣逆特徵值問題存在通解的充要條件和通解的公式；最後給齣4階對稱隨機矩陣逆特徵值問題有解的幾種充分條件和相應解的公式。
대급정적실혹복 n-중Λ={λ1,…,λn},결정시부존재이Λ위보적비부(수궤)구진적문제칭위비부(수궤)구진역특정치문제,저일직시비부구진이론중상미완전해결적일개연구열점。작자증대 n ∈{2,3,4,5},연구 n 계쌍수궤구진역특정치문제유해적충분조건병급출상응해적공식。최근,우대임의정정수 n,선급출행화위상수적대칭구진적역특정치문제적충요조건화해적공식,후급출대칭수궤구진역특정치문제유해적량충충분조건화해적공식。논문재제출임의계대칭수궤구진역특정치문제통해적개념화3계대칭수궤구진역특정치문제완전통해적개념지후,수선급출3계대칭수궤구진역특정치문제존재완전통해적충요조건화완전통해적공식；기차급출3계대칭수궤구진역특정치문제존재통해적충요조건화통해적공식；최후급출4계대칭수궤구진역특정치문제유해적궤충충분조건화상응해적공식。
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.
