Abstract

A normal decomposition (ND) system is an algebraic structure connected with a decomposition statement for vectors of a linear space and with a variational inequality related to the decomposition [see e.g. A. S. Lewis, SIAM J. Matrix Anal. Appl. 17 (1996) 927–947]. The Singular Value Decomposition for complex matrices and the trace inequality of von Neumann provide an example of an ND system. In this paper, we study morphisms and homomorphisms of ND systems. Applications for eigenvalues and to singular values of matrices are given.

Author information

Contact details are reproduced from the original publication and may be historical.

Marek Niezgoda

Department of Applied Mathematics and Computer Science, University of Life Sciences, Akademicka 13, 20-950 Lublin, Poland

marek.niezgoda@up.lublin.pl

Keywords

  • Linear operator
  • weak majorization
  • G-majorization
  • GIC ordering
  • eigenvalue
  • singular value
  • Eaton system
  • normal decomposition system
  • group induced cone ordering
  • morphism
  • homomorphism
  • partial isometry

Mathematics Subject Classification

15A30, 15A39; 15A18, 15A42

References

  1. S. A. Andersson, M. D. Perlman: Group-invariant analogues of Hadamard’s inequality, Linear Algebra Appl. 110 (1988) 91–116.https://doi.org/10.1016/0024-3795(83)90134-9Google Scholar
  2. T. Ando: Majorization, doubly stochastic matrices, and comparison of eigenvalues, Linear Algebra Appl. 118 (1989) 163–243.https://doi.org/10.1016/0024-3795(89)90580-6Google Scholar
  3. T. Ando: Majorization and inequalities in matrix theory, Linear Algebra Appl. 199 (1994) 17–67.https://doi.org/10.1016/0024-3795(94)90341-7Google Scholar
  4. R. B. Bapat: Majorization and singular values III, Linear Algebra Appl. 145 (1991) 59-70.https://doi.org/10.1016/0024-3795(91)90287-7Google Scholar
  5. L. B. Beasley, S.-G. Lee, Y.-H. Lee: A characterization of strong preservers of matrix majorization, Linear Algebra Appl. 367 (2003) 341–346.https://doi.org/10.1016/S0024-3795(02)00657-2Google Scholar
  6. A. M. Dean, J. S. Verducci: Linear transformations that preserve majorization, Schur concavity and exchangeability, Linear Algebra Appl. 127 (1990) 121–138.https://doi.org/10.1016/0024-3795(90)90339-EGoogle Scholar
  7. M. L. Eaton: On group induced orderings, monotone functions, and convolution theorems, in: Inequalities in Statistics and Probability, Y. L. Tong (ed.), Lectures Notes – Monograph Series 5, Institute of Mathematical Statistics, Hayward (1984) 13–25.https://doi.org/10.1214/lnms/1215465625Google Scholar
  8. M. L. Eaton: Group induced orderings with some applications in statistics, CWI Newslett. 16 (1987) 3–31.Google Scholar
  9. M. L. Eaton: Lectures on Topics in Probability Inequalities, CWI Tracts 35, Centrum voor Wiskunde en Informatica, Amsterdam (1987).Google Scholar
  10. M. L. Eaton, M. D. Perlman: Reflection groups, generalized Schur functions, and the geometry of majorization, Ann. Probab. 5 (1977) 829–860.https://doi.org/10.1214/aop/1176995655Google Scholar
  11. R. A. Horn, C. R. Johnson: Matrix Analysis, Cambridge Univ. Press, Cambridge (1985).https://doi.org/10.1017/CBO9780511810817Google Scholar
  12. M. Iwasa: An extension of a convolution inequality for G-monotone functions and an approach to Bartholomew’s conjecture, J. Multivariate Anal. 59 (1996) 249–271.https://doi.org/10.1006/jmva.1996.0064Google Scholar
  13. C. R. Johnson, P. M. Nylen: Converse to a singular value inequality common to the Hadamard and conventional products, Linear Multilinear Algebra 27 (1990) 167–187.https://doi.org/10.1080/03081089008818009Google Scholar
  14. A. S. Lewis: Group invariance and convex matrix analysis, SIAM J. Matrix Anal. Appl. 17 (1996) 927–949.https://doi.org/10.1137/S0895479895283173Google Scholar
  15. A. S. Lewis: Convex analysis on Cartan subspaces, Nonlinear Anal., Theory Methods Appl. 42 (2000) 813–820.https://doi.org/10.1016/S0362-546X(99)00126-1Google Scholar
  16. C.-K. Li, S. Pierce: Linear preserver problems, Amer. Math. Monthly 108 (2001) 591–605.https://doi.org/10.1080/00029890.2001.11919790Google Scholar
  17. A. W. Marshall, I. Olkin: Inequalities: Theory of Majorization and its Applications, Academic Press, New York (1979).Google Scholar
  18. M. Niezgoda: Group majorization and Schur type inequalities, Linear Algebra Appl. 268 (1998) 9–30.https://doi.org/10.1016/S0024-3795(97)89322-6Google Scholar
  19. M. Niezgoda: On Schur-Ostrowski type theorems for group majorizations, J. Convex Analysis 5 (1998) 81–105.https://doi.org/10.68381/jca05-7Google Scholar
  20. M. Niezgoda: Linear maps preserving group majorization, Linear Algebra Appl. 300 (2001) 113–127.https://doi.org/10.1016/S0024-3795(01)00257-9Google Scholar
  21. M. Niezgoda: G-majorization inequalities and canonical forms of matrices, J. Convex Analysis 14 (2007) 35–48.https://doi.org/10.68381/jca14004Google Scholar
  22. M. Niezgoda: G-majorization inequalities for linear maps II, Linear Multilinear Algebra 55 (2007) 1–17.https://doi.org/10.1080/03081080500139702Google Scholar
  23. M. Niezgoda: On linear preservers of normal maps, Linear Algebra Appl. 426 (2007) 149-158.https://doi.org/10.1016/j.laa.2007.04.007Google Scholar
  24. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  25. A. G. M. Steerneman: G-majorization, group-induced cone orderings and reflection groups, Linear Algebra Appl. 127 (1990) 107–119.https://doi.org/10.1016/0024-3795(90)90338-DGoogle Scholar
  26. T.-Y. Tam: An extension of a result of Lewis, Electron. J. Linear Algebra 5 (1999) 1–10.https://doi.org/10.13001/1081-3810.1026Google Scholar
  27. T.-Y. Tam: Group majorization, Eaton triples and numerical range, Linear Multilinear Algebra 47 (2000) 11–28.https://doi.org/10.1080/03081080008818628Google Scholar
  28. T.-Y. Tam: Konstant’s convexity theorem and the compact classical groups, Linear Multilinear Algebra 43 (1997) 87–113.https://doi.org/10.1080/03081089708818518Google Scholar

Suggested citation

M. Niezgoda. “Morphisms of Normal Decomposition Systems.” Journal of Convex Analysis 16 (2009), No. 2, 617–632. https://doi.org/10.68381/jca16034

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.