非方形模糊线性系统的解
Solution of Non-Square Fuzzy Linear Systems
AI总结：
研究具有清晰系数和模糊右端项的线性方程组，将解视为模糊向量集而非模糊向量，针对满秩系数矩阵的三种情况（方阵、超定、欠定）提出求解方法，并推广到一般情况。
摘要：
本文研究了具有清晰系数和模糊右端项的线性方程组。处理了涉及变量数n和方程数m的所有可能情况。求解不是像通常那样寻求一个模糊向量，而是寻求一个模糊向量集。解集中的每个向量以某种可能性求解给定的模糊线性系统。假设系数矩阵是满秩矩阵，考虑三种情况：对于m=n（方阵系统），解集被证明是坐标空间中的平行六面体，并用显式公式表示。对于m>n（超定系统），解集被证明是凸多面体，并提出了一种新颖的几何方法来计算它。对于m<n（欠定系统），通过确定自由变量的贡献，计算通解。基于上述三种情况的结果，提出了一种处理一般情况的方法，其中系数矩阵不一定是满秩矩阵。提供了全面的例子并深入研究了每种情况和建议的方法。
英文摘要：
In this paper, a linear system of equations with crisp coefficients and fuzzy right-hand sides is investigated. All possible cases pertaining to the number of variables, n, and the number of equations, m, are dealt with. A solution is sought not as a fuzzy vector, as usual, but as a fuzzy set of vectors. Each vector in the solution set solves the given fuzzy linear system with a certain possibility. Assuming that the coefficient matrix is a full rank matrix, three cases are considered: For m = n (square system), the solution set is shown to be a parallelepiped in coordinate space and is expressed by an explicit formula. For m > n (overdetermined system), the solution set is proved to be a convex polyhedron and a novel geometric method is proposed to compute it. For m < n (underdetermined system), by determining the contribution of free variables, general solution is computed. From the results of three cases mentioned above, a method is proposed to handle the general case, in which the coefficient matrix is not necessarily a full rank matrix. Comprehensive examples are provided and investigated in depth to illustrate each case and suggested method.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
