Abstract

An Eaton system is connected with a decomposition statement for vectors of a linear space and with a scalar inequality related to the decomposition. The Singular Value Decomposition for the space of complex matrices associated with von Neumann's trace inequality is a typical example. We present a G-majorization inequality involving two orthoprojectors related to an Eaton system. The inequality generalizes a variety of majorization results on eigenvalues and singular values of matrices. A relationship between the inequality and canonical form theorems for certain spaces of matrices is shown. G-doubly stochastic operators are discussed.

Author information

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

Keywords

  • G-majorization
  • Eaton system
  • normal decomposition system
  • finite reflection group
  • G-doubly stochastic operator
  • eigenvalue
  • singular value

Mathematics Subject Classification

15A18, 15A21; 15A42, 15A30

References

  1. S. A. Anderson, 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) 163–248.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. M. L. Eaton: On group induced orderings, monotone functions, and convolution theorems, in: Inequalities in Statistics and Probability, Y. L. Tong (ed.), IMS Lectures Notes – Monograph Series 5, IMS, Hayward (1984) 13–25.https://doi.org/10.1214/lnms/1215465625Google Scholar
  5. M. L. Eaton: Group induced orderings with some applications in statistics, CWI Newsletter 16 (1987) 3–31.Google Scholar
  6. 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
  7. K. Fan: On a theorem of Weyl concerning eigenvalues of linear transformations, Proc. Natl. Acad. Sci. USA 35 (1949) 652–655.https://doi.org/10.1073/pnas.35.11.652Google Scholar
  8. A. Giovagnoli, H. P. Wynn: G-majorization with applications to matrix orderings, Linear Algebra Appl. 67 (1985) 111–135.https://doi.org/10.1016/0024-3795(85)90190-9Google Scholar
  9. R. A. Horn, C. R. Johnson: Matrix Analysis, Cambridge University Press, Cambridge (1985).https://doi.org/10.1017/CBO9780511810817Google Scholar
  10. R. A. Horn, C. R. Johnson: Topics in Matrix Analysis, Cambridge University Press, Cambridge (1991).https://doi.org/10.1017/CBO9780511840371Google Scholar
  11. J. E. Humphreys: Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge (1990).https://doi.org/10.1017/CBO9780511623646Google Scholar
  12. J. Karro, C.-K. Li: A unified elementary approach to canonical forms of matrices, SIAM Rev. 39 (1997) 305–309.https://doi.org/10.1137/S0036144595294862Google Scholar
  13. A. S. Lewis: Von Neumann’s Lemma and a Chevalley-type theorem for convex functions on Cartan subspaces, Technical Report of Combinatorics and Optimization, University of Waterloo, Canada (1995).Google 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. A. W. Marshall, I. Olkin: Inequalities: Theory of Majorization and its Applications, Academic Press, New York (1979).Google Scholar
  17. H. F. Miranda, R. C. Thompson: Group majorization, the convex hull of sets of matrices, and the diagonal elements – singular values inequalities, Linear Algebra Appl. 199 (1994) 131–141.https://doi.org/10.1016/0024-3795(94)90345-XGoogle 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(1) (1998) 81–105.https://doi.org/10.68381/jca05-7Google Scholar
  20. M. Niezgoda: G-majorization inequalities for linear maps, Linear Algebra Appl. 292 (1999) 207–231.https://doi.org/10.1016/S0024-3795(99)00034-8Google Scholar
  21. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  22. 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
  23. T.-Y. Tam: Partial superdiagonal elements and the singular values of a complex skew symmetric matrix, SIAM J. Matrix Anal. Appl. 19 (1998) 737–754.https://doi.org/10.1137/S0895479896312559Google Scholar
  24. 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
  25. C. M. Theobald: An inequality for the trace of the product of two symmetric matrices, Math. Proc. Camb. Philos. Soc. 77 (1975) 265–267.https://doi.org/10.1017/S0305004100051070Google Scholar

Suggested citation

M. Niezgoda. “G-Majorization Inequalities and Canonical Forms of Matrices.” Journal of Convex Analysis 14 (2007), No. 1, 35–48. https://doi.org/10.68381/jca14004

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2007. Rights now held by Banach Press LLC.