Schur Forms and Normal-Nilpotent Decompositions

TL;DR

This paper clarifies the mathematical foundations of Schur forms and normal-nilpotent decompositions, addressing confusion and non-uniqueness issues to improve their application in fluid mechanics analysis.

Contribution

It systematically derives and compares complex and real Schur forms and NNDs, proposing conditions for their uniqueness and clarifying their distinctions.

Findings

Constructive derivation of Schur forms from basics.

Proposed conditions for the uniqueness of Schur forms.

Clarified differences between NND and TDM.

Abstract

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.