AI 中文总结
研究奇异模糊线性系统，通过嵌入和列运算转化为清晰线性系统，利用广义逆、满秩分解等方法计算最小范数解等，建立一致性条件和统一框架，开发高效算法并通过实例验证适用性和计算效率。
AI 中文摘要
模糊矩阵为科学与工程问题中的不确定性建模提供了有效框架，尤其是模糊线性系统。本文通过嵌入方法将一般模糊线性系统转化为清晰线性系统，并经列运算将其化为标准块结构形式。在合适范围条件下开发了直接LU分解，能利用广义逆计算矩形模糊线性系统的最小范数解。还提出满秩分解以计算任意矩形矩阵的Moore Penrose逆，并建立这些逆的一致性条件。基于单调性和非负性约束给出了获得强模糊解的统一框架。为块结构矩阵开发了基于LU、QR和SVD分解的高效算法。通过模糊电路方程和马尔可夫链过程证明了所提方法的适用性和计算效率。
英文摘要
Fuzzy matrices provide an effective framework for modeling uncertainty in scientific and engineering problems, particularly fuzzy linear systems. This work transforms a general FLS into a crisp linear system using an embedding approach and reduces it to a standard block structured form via column operations. A direct LU decomposition is developed under suitable range conditions, enabling the computation of minimum norm solutions of rectangular FLS using the generalized inverses. A full rank decomposition is further proposed to compute the Moore Penrose inverse for arbitrary rectangular matrices, and consistency conditions for these inverses are established. A unified framework for obtaining strong fuzzy solutions based on monotonicity and non-negativity constraints are presented. Efficient algorithms based on LU, QR, and SVD decompositions are developed for the block structured matrix. The applicability and computational efficiency of the proposed methods are demonstrated through fuzzy circuit equations and Markov chain processes.
