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Schur Forms and Normal-Nilpotent Decompositions

LI Zhen

李震. Schur形式与正规-幂零分解[J]. 应用数学和力学, 2024, 45(9): 1200-1211. doi: 10.21656/1000-0887.450129
引用本文: 李震. Schur形式与正规-幂零分解[J]. 应用数学和力学, 2024, 45(9): 1200-1211. doi: 10.21656/1000-0887.450129
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

Schur形式与正规-幂零分解

doi: 10.21656/1000-0887.450129
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    作者简介:
  • 中图分类号: O302; O35

Schur Forms and Normal-Nilpotent Decompositions

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)
  • 摘要: 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。
  • [1] LI Z. Theoretical study on the definition of vortex[D]. Beijing: Tsinghua University, 2010.
    [2] LI Z, ZHANG X, HE F. Evaluation of vortex criteria by virtue of the quadruple decomposition of velocity gradient tensor[J]. Acta Physica Sinica, 2014, 63(5): 054704. doi: 10.7498/aps.63.054704
    [3] MURNAGHAN F D, WINTNER A. A canonical form for real matrices under orthogonal transformations[J]. Proceedings of the National Academy of Sciences, 1931, 17(7): 417-420. doi: 10.1073/pnas.17.7.417
    [4] SCHUR I. Über die charakteristischen wurzeln einer linearen substitution mit einer anwendung auf die theorie der integralgleichungen[J]. Mathematische Annalen, 1909, 66(4): 488-510. doi: 10.1007/BF01450045
    [5] STICKELBERGER L. Ueber Reelle Orthogonale Substitutionen[M]. Bern: Schweizerische Nationalbibliothek, 1877.
    [6] LÉON A M. Sur l’ Hermitien[J]. Rendicontidel Circolo Matematico di Palermo (1884—1940), 1907, 16: 104-128.
    [7] KEYLOCK C J. Synthetic velocity gradient tensors and the identification of statistically significant aspects of the structure of turbulence[J]. Physical Review Fluids, 2017, 2(8): 084607. doi: 10.1103/PhysRevFluids.2.084607
    [8] KEYLOCK C J. The Schur decomposition of the velocity gradient tensor for turbulent flows[J]. Journal of Fluid Mechanics, 2018, 848: 876-905. doi: 10.1017/jfm.2018.344
    [9] DAS R, GIRIMAJI S S. Revisiting turbulence small-scale behavior using velocity gradient triple decomposition[J]. New Journal of Physics, 2020, 22(6): 063015. doi: 10.1088/1367-2630/ab8ab2
    [10] HOFFMAN J. Energy stability analysis of turbulent incompressible flowbased on the triple decomposition of the velocity gradient tensor[J]. Physics of Fluids, 2021, 33(8): 081707. doi: 10.1063/5.0060584
    [11] ZHU J Z. Thermodynamic and vortic structures of real Schur flows[J]. Journal of Mathematical Physics, 2021, 62(8): 083101. doi: 10.1063/5.0052296
    [12] ZHU J Z. Compressible helical turbulence: fastened-structure geometry and statistics[J]. Physics of Plasmas, 2021, 28(3): 032302. doi: 10.1063/5.0031108
    [13] KRONBORG J, SVELANDER F, ERIKSSON-LIDBRINK S, et al. Computational analysis of flow structures in turbulent ventricular blood flow associated with mitral valve intervention[J]. Frontiers in Physiology, 2022, 13: 806534. doi: 10.3389/fphys.2022.806534
    [14] ARUN R, COLONIUS T. Velocity gradient analysis of a head-on vortexring collision[J]. Journal of Fluid Mechanics, 2024, 982: A16. doi: 10.1017/jfm.2024.90
    [15] KOLÁŘ V. 2D Velocity-field analysis using triple decomposition of motion[C]//Proceedings of the Fifteenth Australasian Fluid Mechanics Conference. Sydney, Australia, 2004.
    [16] KOLÁŘ V. Vortex identification: new requirements and limitations[J]. International Journal of Heat and Fluid Flow, 2007, 28(4): 638-652. doi: 10.1016/j.ijheatfluidflow.2007.03.004
    [17] KRONBORG J, HOFFMAN J. The triple decomposition of the velocity gradient tensor as a standardized real Schur form[J]. Physics of Fluids, 2023, 35: 031703. doi: 10.1063/5.0138180
    [18] ZOU W, XU X Y, TANG C X. Spiral streamline pattern around a critical point: its dual directivity and effective characterization by right eigen representation[J]. Physics of Fluids, 2021, 33(6): 067102. doi: 10.1063/5.0050555
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出版历程
  • 收稿日期:  2024-05-08
  • 修回日期:  2024-07-03
  • 刊出日期:  2024-09-01

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