Schur Forms and Normal-Nilpotent Decompositions

LI Zhen

doi:10.21656/1000-0887.450129

Volume 45 Issue 9

Sep. 2024

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LI Zhen. Schur Forms and Normal-Nilpotent Decompositions[J]. Applied Mathematics and Mechanics, 2024, 45(9): 1200-1211. doi: 10.21656/1000-0887.450129

Citation:

LI Zhen. Schur Forms and Normal-Nilpotent Decompositions[J]. Applied Mathematics and Mechanics, 2024, 45(9): 1200-1211. doi: 10.21656/1000-0887.450129

Citation:

LI Zhen. Schur Forms and Normal-Nilpotent Decompositions[J]. Applied Mathematics and Mechanics, 2024, 45(9): 1200-1211. doi: 10.21656/1000-0887.450129

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Schur Forms and Normal-Nilpotent Decompositions

doi: 10.21656/1000-0887.450129

LI Zhen

Beijing Institute of Mathematical Sciences and Applications, Beijing 101400, P.R.China

Received Date: 2024-05-08
Rev Recd Date: 2024-07-03
Publish Date: 2024-09-01

Abstract

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.
- Schur form,
- normal matrix,
- nilpotent matrix,
- tensor decomposition,
- vortex identification

1 A matrix is said to be quasiorthogonal if its columns are mutually orthogonal and so are its rows. But the columns and rows are not required to be normalized to unit length. The accurate name for this class of matrices should be orthogonal, unfortunately, which has been widely accepted for matrices that should have been called orthonormal matrices.
（Recommended by WU Chuijie, M. AMM Editorial Board）

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References(18)

References

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