Schur Forms and Normal-Nilpotent Decompositions  

Schur形式与正规-幂零分解

作　　者：LI Zhen 李震(北京雁栖湖应用数学研究院,北京101400)

机构地区：[1]Beijing Institute of Mathematical Sciences and Applications,Beijing 101400,P.R.China

出　　处：《应用数学和力学》2024年第9期1200-1211,共12页Applied Mathematics and Mechanics

摘　　要：Real and complex Schur forms have been receiving increasing attention from the fluid mechanics community recently,especially related to vortices and turbulence.Several decompositions of the velocity gradient tensor,such as the triple decomposition of motion(TDM)and normal-nilpotent decomposition(NND),have been proposed to analyze the local motions of fluid elements.However,due to the existence of different types and non-uniqueness of Schur forms,as well as various possible definitions of NNDs,confusion has spread widely and is harming the research.This work aims to clean up this confusion.To this end,the complex and real Schur forms are derived constructively from the very basics,with special consideration for their non-uniqueness.Conditions of uniqueness are proposed.After a general discussion of normality and nilpotency,a complex NND and several real NNDs as well as normal-nonnormal decompositions are constructed,with a brief comparison of complex and real decompositions.Based on that,several confusing points are clarified,such as the distinction between NND and TDM,and the intrinsic gap between complex and real NNDs.Besides,the author proposes to extend the real block Schur form and its corresponding NNDs for the complex eigenvalue case to the real eigenvalue case.But their justification is left to further investigations.实的和复的Schur形式近年来受到流体力学界(特别是与旋涡和湍流相关)越来越多的关注.几个速度梯度张量分解(例如三元运动分解TDM和正规-幂零分解NND)被提出用于分析流体微元的局部运动.然而,由于Schur形式存在不同类型和非唯一性,以及NND有多种可能定义,一些混淆广泛传播并正在对研究造成危害.该工作旨在清除这种混淆.为此,复的和实的Schur形式由很基本的知识构造性地推导出来,其非唯一性被特别加以考虑,唯一性条件被提出.在对正规性和幂零性加以一般讨论后,一个复NND和几个实NND以及正规-非正规分解被构造出来,并简要地比较了复的和实的分解.在这些基础上,几个混淆点得到澄清,例如NND与TDM的差异以及复的和实的NND之间的内在鸿沟.此外,笔者提议将复本征值情况下实的块Schur形式及其对应的NND拓展到实本征值情况,不过其合理性有待进一步研究.

关 键 词：Schur form normal matrix nilpotent matrix tensor decomposition vortex identification 

分 类 号：O302[理学—力学] O35