Schur Forms and Normal-Nilpotent Decompositions

作     者：LI Zhen 李震

作者机构：Beijing Institute of Mathematical Sciences and ApplicationsBeijing 101400P.R.China 

出 版 物：《应用数学和力学》 (Applied Mathematics and Mechanics)

年 卷 期：2024年第45卷第9期

页      面：1200-1211页

学科分类：080704[工学-流体机械及工程] 08[工学] 080103[工学-流体力学] 0807[工学-动力工程及工程热物理] 0801[工学-力学（可授工学、理学学位）] 

主　　题：Schur form normal matrix nilpotent matrix tensor decomposition vortex identification 

摘      要：Real and complex Schur forms have been receiving increasing attention from the fluid mechanics community recently,especially related to vortices and *** 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 ***,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 *** work aims to clean up this *** this end,the complex and real Schur forms are derived constructively from the very basics,with special consideration for their *** of uniqueness are *** 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 *** on that,several confusing points are clarified,such as the distinction between NND and TDM,and the intrinsic gap between complex and real ***,the author proposes to extend the real block Schur form and its corresponding NNDs for the complex eigenvalue case to the real eigenvalue *** their justification is left to further investigations.