广义Kronecker积及其应用——处理复杂系统的新思维系列之十一

The Generalized Kronecker Product and Its Application ——New Thinking of Dealing with Complex Systems Series Eleven

本系列论文基于《多边矩阵理论》,由东方整体性思维所启迪,试图提供并完善一套从整体到局部处理复杂系统多指标问题、非均匀性问题、非线性问题的强有力的数学工具,并对其进行严格的理论推导和证明。作为系列论文的第十一篇,首先基于广义Kronecker积的定义,讨论其是否满足三种运算律,介绍其与普通Kronecker积、Kronecker和的关系,及其应用方法,设计Hadamard矩阵、拉丁矩阵和差集矩阵;其次给出了Kronecker积的另一个推广定义,强Kronecker积;最后用SAS语言实现了两种Kronecker积运算方法。

This series of articles, based on ＂Multilateral Matrix Theory＂ and inspired by the Eastern holistic thinking, are trying to provide and improve a whole set of powerful mathematical tools to handle multi-target local issues, non-uniformity problems and nonlinear problems of complex system ranging from the whole to the part with rigorous theoretical analysis and proof. As the eleventh paper of the series, firstly, this paper uses the definition of generalized Kronecker product, discusses whether it meets three operation laws, introduces the re- lationship among generalized Kronecker product, normal Kronecker product and Kronecker sum and its applica- tion methods such as designing Hadamard matrix, Latin matrix and difference matrix. Then another extended definition of Kronecker product, the strong Kronecker product, is given. Finally, two Kronecker product operation methods are programmed in SAS code.