# Some remarks on operators preserving partial orders of matrices

Discussiones Mathematicae Probability and Statistics (2008)

- Volume: 28, Issue: 1, page 143-155
- ISSN: 1509-9423

## Access Full Article

top## Abstract

top## How to cite

topJan Hauke. "Some remarks on operators preserving partial orders of matrices." Discussiones Mathematicae Probability and Statistics 28.1 (2008): 143-155. <http://eudml.org/doc/277057>.

@article{JanHauke2008,

abstract = {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.},

author = {Jan Hauke},

journal = {Discussiones Mathematicae Probability and Statistics},

keywords = {star partial ordering; minus partial ordering; löwner partial ordering; sharp partial order; nonnegative definite matrix; Hermitian matrix; EP-matrix; power of a matrix; polynomial of a matrix},

language = {eng},

number = {1},

pages = {143-155},

title = {Some remarks on operators preserving partial orders of matrices},

url = {http://eudml.org/doc/277057},

volume = {28},

year = {2008},

}

TY - JOUR

AU - Jan Hauke

TI - Some remarks on operators preserving partial orders of matrices

JO - Discussiones Mathematicae Probability and Statistics

PY - 2008

VL - 28

IS - 1

SP - 143

EP - 155

AB - 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.

LA - eng

KW - star partial ordering; minus partial ordering; löwner partial ordering; sharp partial order; nonnegative definite matrix; Hermitian matrix; EP-matrix; power of a matrix; polynomial of a matrix

UR - http://eudml.org/doc/277057

ER -

## References

top- [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. Zbl1048.15016
- [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. Zbl1048.15017
- [3] J.K. Baksalary and J. Hauke, Partial orderings of matrices reffering to singular values of matrices, Linear Algebra Appl. 96 (1987), 17-26. Zbl0627.15005
- [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. Zbl0696.15005
- [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. Zbl1061.15018
- [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. Zbl1044.15011
- [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.
- [8] J.K. Baksalary and S.K. Mitra, Left-star and right-star partial orderings, Linear Algebra Appl. 145 (1991), 73-89. Zbl0717.15004
- [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. Zbl0737.15008
- [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).
- [11] M.P. Drazin, Natural structures on semigroups with involution, Bull. Amer. Math. Soc. 84 (1978), 139-141. Zbl0395.20044
- [12] J. Groß, Löwner partial ordering and space preordering of Hermitian non-negative definite matrices, Linear Algebra Appl. 326 (2001), 215-223. Zbl0979.15019
- [13] J. Groß, Remarks on the sharp partial order and the ordering of squares of matrices, Linear Algebra Appl. 417 (2006), 87-93. Zbl1103.15013
- [14] J. Groß, J. Hauke and A. Markiewicz, Some comments on matrix partial orderings, Discuss. Math., Algebra and Stochastic Methods 17 (1997), 203-214. Zbl0907.15016
- [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. Zbl0934.15024
- [16] R.E. Hartwig, How to partially order regular elements, Math. Japon. 25 (1980), 1-13. Zbl0442.06006
- [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. Zbl0603.15001
- [18] J. Hauke and A. Markiewicz, Remarks on partial orderings on the set of rectangular matrices, Discuss. Math. 13 (1993), 149-154. Zbl0807.15013
- [19] J. Hauke and A. Markiewicz, On partial orderings on the set of rectangular matrices and their properties, Discuss. Math. 15 (1995), 5-10. Zbl0840.15013
- [20] J. Hauke and A. Markiewicz, On partial orderings on the set of rectangular matrices, Linear Algebra Apl. 219 (1995), 187-193. Zbl0833.15012
- [21] J. Hauke, A. Markiewicz and T. Szulc, Inter- and extrapolatory properties of matrix partial orderings, Linear Algebra Apl. 332-334 (2001), 437-445. Zbl0982.15027
- [22] K. Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), 177-216. Zbl0008.11301
- [23] R. Mathias, The equivalence of two partial orders on a convex cone of positive semidefinite matrices, Linear Algebra Appl. 151 (1991), 27-55. Zbl0739.15011
- [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. Zbl1131.15022
- [25] S.K. Mitra, On group inverses and the sharp order, Linear Algebra Appl. 92 (1987), 17-37. Zbl0619.15006
- [26] K.S.S. Nambooripad, The natural partial order on a regular semigroup, Proc. Edinburgh Math. Soc. 23 (1980), 249-260. Zbl0459.20054
- [27] Cz. Stępniak, Two orderings on a cone of nonnegative definite matrices, Linear Algebra Appl. 94 (1987), 263-272. Zbl0626.15008