Stępniak [Linear Algebra Appl. 151 (1991)] considered the problem of equivalence of the Löwner partial order of nonnegative definite matrices and the Löwner partial order of squares of those matrices. The paper was an important starting point for investigations of the problem of how an order between two matrices A and B from different sets of matrices can be preserved for the squares of the corresponding matrices A² and B², in the sense of the Löwner partial ordering, the star partial ordering, the minus partial ordering, and the sharp partial ordering. Many papers have since been published (mostly coauthored by J.K. Baksalary - to whom the present paper is dedicated) that generalize the results in two directions: by widening the class of matrices considered and by replacing the squares by arbitrary powers. In the present paper we make a résumé of some of these results and suggest some further generalizations for polynomials of the matrices considered.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1097,
author = {Jan Hauke},
title = {Some remarks on operators preserving partial orders of matrices},
journal = {Discussiones Mathematicae Probability and Statistics},
volume = {28},
year = {2008},
pages = {143-155},
zbl = {05498880},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1097}
}
Jan Hauke. Some remarks on operators preserving partial orders of matrices. Discussiones Mathematicae Probability and Statistics, Tome 28 (2008) pp. 143-155. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1097/
[000] [1] J.K. Baksalary, O.M. Baksalary and Liu Xiaoji, Further properties of the star, left-star, right-star, and minus partial orderings, Linear Algebra Appl. 375 (2003), 83-94. | Zbl 1048.15016
[001] [2] J.K. Baksalary, O.M. Baksalary and Liu Xiaoji, Further relationships between certain partial orders of matrices and their squares, Linear Algebra Appl. 375 (2003), 171-180. | Zbl 1048.15017
[002] [3] J.K. Baksalary and J. Hauke, Partial orderings of matrices reffering to singular values of matrices, Linear Algebra Appl. 96 (1987), 17-26. | Zbl 0627.15005
[003] [4] J.K. Baksalary and J. Hauke, A further algebraic version of Cochran's theorem and matrix partial ordering, Linear Algebra Appl. 127 (1990), 157-169. | Zbl 0696.15005
[004] [5] J.K. Baksalary and J. Hauke, Characterizations of minus and star orders between the squares of Hermitian matrices, Linear Algebra Appl. 388 (2004), 53-59. | Zbl 1061.15018
[005] [6] J.K. Baksalary, J. Hauke, Liu Xiaoji and Liu Sanyang, Relationships between partial orders of matrices and their powers, Linear Algebra Appl. 379 (2004), 277-287. | Zbl 1044.15011
[006] [7] J.K. Baksalary, J. Hauke and G.P.H. Styan, On some distributional properties of quadratic forms in normal variables and some associated matrix partial orderings, Multivariate analysis and its applications, IMS Lecture Notes - Monograph Series 24 (1994), 111-124.
[007] [8] J.K. Baksalary and S.K. Mitra, Left-star and right-star partial orderings, Linear Algebra Appl. 145 (1991), 73-89. | Zbl 0717.15004
[008] [9] J.K. Baksalary and F. Pukelsheim, On the Löwner, minus, and star partial orderings of nonnegative definite matrices and their squares, Linear Algebra Appl. 151 (1991), 135-141. | Zbl 0737.15008
[009] [10] J.K. Baksalary, F. Pukelsheim and G.P.H. Styan, Some properties of matrix partial orderings of nonnegative definite matrices, Linear Algebra Appl. 119 (1987), 57-85; Addendum, 220:3 (1995).
[010] [11] M.P. Drazin, Natural structures on semigroups with involution, Bull. Amer. Math. Soc. 84 (1978), 139-141. | Zbl 0395.20044
[011] [12] J. Groß, Löwner partial ordering and space preordering of Hermitian non-negative definite matrices, Linear Algebra Appl. 326 (2001), 215-223. | Zbl 0979.15019
[012] [13] J. Groß, Remarks on the sharp partial order and the ordering of squares of matrices, Linear Algebra Appl. 417 (2006), 87-93. | Zbl 1103.15013
[013] [14] J. Groß, J. Hauke and A. Markiewicz, Some comments on matrix partial orderings, Discuss. Math., Algebra and Stochastic Methods 17 (1997), 203-214. | Zbl 0907.15016
[014] [15] J. Groß and S.O. Troschke, Some remarks on matrix partial orderings and of nonnegative definite matrices, Linear Algebra Appl. 264 (1997), 451-467. | Zbl 0934.15024
[015] [16] R.E. Hartwig, How to partially order regular elements, Math. Japon. 25 (1980), 1-13. | Zbl 0442.06006
[016] [17] R.E. Hartwig and G.P.H. Styan, On some characterizations of the 'star' partial ordering for matrices and rank subtractivity, Linear Algebra Appl. 82 (1986), 145-161. | Zbl 0603.15001
[017] [18] J. Hauke and A. Markiewicz, Remarks on partial orderings on the set of rectangular matrices, Discuss. Math. 13 (1993), 149-154. | Zbl 0807.15013
[018] [19] J. Hauke and A. Markiewicz, On partial orderings on the set of rectangular matrices and their properties, Discuss. Math. 15 (1995), 5-10. | Zbl 0840.15013
[019] [20] J. Hauke and A. Markiewicz, On partial orderings on the set of rectangular matrices, Linear Algebra Apl. 219 (1995), 187-193. | Zbl 0833.15012
[020] [21] J. Hauke, A. Markiewicz and T. Szulc, Inter- and extrapolatory properties of matrix partial orderings, Linear Algebra Apl. 332-334 (2001), 437-445. | Zbl 0982.15027
[021] [22] K. Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), 177-216. | Zbl 0008.11301
[022] [23] R. Mathias, The equivalence of two partial orders on a convex cone of positive semidefinite matrices, Linear Algebra Appl. 151 (1991), 27-55. | Zbl 0739.15011
[023] [24] J.K. Merikoski and Liu Xiaoji, On the star partial ordering of normal matrices, J. Ineq. Pure Appl. Math. 7 (1) (2006), Article 17. | Zbl 1131.15022
[024] [25] S.K. Mitra, On group inverses and the sharp order, Linear Algebra Appl. 92 (1987), 17-37. | Zbl 0619.15006
[025] [26] K.S.S. Nambooripad, The natural partial order on a regular semigroup, Proc. Edinburgh Math. Soc. 23 (1980), 249-260. | Zbl 0459.20054
[026] [27] Cz. Stępniak, Two orderings on a cone of nonnegative definite matrices, Linear Algebra Appl. 94 (1987), 263-272. | Zbl 0626.15008