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RESOLUTION OF THE CONJECTURE ON STRONG PRESERVERS OF MULTIVARIATE MAJORIZATION

  • Beasley, Leroy-B. (Department of Mathematics And Statistics, Utah State University) ;
  • Lee, Sang-Gu (Department of Mathematics, Sungkyunkwan University) ;
  • Lee, You-Ho (School of Computing, Soongsil University)
  • Published : 2002.05.01

Abstract

In this paper, we will investigate the set of linear operators on real square matrices that strongly preserve multivariate majorisation without any additional conditions on the operator. This answers earlier conjecture on nonnegative matrices in [3] .

Keywords

References

  1. P. M. Alberti and A. Uhlmann, Stochasticity and partial order, Math. and Its Appl. 9, D. Reidel, Berlin, 1982.
  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)90580-6
  3. L. B. Beasley, S.-G. Lee, and Y.-H. Lee, Linear operators strongly preserving multivariate majorization, Kyongpook Math. J. 39 (1999), no. 1, 191-194.
  4. J. V. Bondar, Comments and complements to: Inequalities: Theory of Majorization and Its Appl. by Albert W. Marshall and Ingram Olkin, Linear Algebra Appl. 199 (1994), 115-130. https://doi.org/10.1016/0024-3795(94)90344-1
  5. E. P. Botta, Linear transformations preserving unitary group, Linear and Multilinear Algebra 8 (1979), 8-96. https://doi.org/10.1080/03081087908817304
  6. R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, Cambridge, 1991.
  7. G.-S. Cheon and Y.-H. Lee, The doubly stochastic matrices of a multivariate majorization, J. Korean Math. Soc. 32 (1995), no. 4, 857-867.
  8. G. Dahl, Matrix majorization, Linear Algebra Appl. 288 (1999), 53-73. https://doi.org/10.1016/S0024-3795(98)10175-1
  9. J. Dixon, Rigid embeddings of simple groups in the general linear groups, Canad. J. Math. 29 (1977), 384-391. https://doi.org/10.4153/CJM-1977-041-0
  10. L. Elsner, Matrices leaving invariant a convex set, Linear Algebra Appl. 42 (1982), 103-107. https://doi.org/10.1016/0024-3795(82)90141-0
  11. F. Hiai, Similarity preserving linear maps on matrices, Linear Algebra Appl. 97 (1987), 127-139. https://doi.org/10.1016/0024-3795(87)90145-5
  12. M. Marcus, All linear operators leaving the unitary group invariant, Duke Math. J. 26 (1959), 155-163. https://doi.org/10.1215/S0012-7094-59-02615-8
  13. A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, 1979.
  14. S. Pierce, et al., A Survey of Linear Preserver Problems, Linear and Multilinear Algebra 33 (1992), no. 1 and no. 2. https://doi.org/10.1080/03081089208818176