# On orderings induced by the Loewner partial ordering

Jan Hauke; Augustyn Markiewicz

Applicationes Mathematicae (1994)

- Volume: 22, Issue: 2, page 145-154
- ISSN: 1233-7234

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topHauke, Jan, and Markiewicz, Augustyn. "On orderings induced by the Loewner partial ordering." Applicationes Mathematicae 22.2 (1994): 145-154. <http://eudml.org/doc/219087>.

@article{Hauke1994,

abstract = {The partial ordering induced by the Loewner partial ordering on the convex cone comprising all matrices which multiplied by a given positive definite matrix become nonnegative definite is considered. Its relation to orderings which are induced by the Loewner partial ordering of the squares of matrices is presented. Some extensions of the latter orderings and their comparison to star orderings are given.},

author = {Hauke, Jan, Markiewicz, Augustyn},

journal = {Applicationes Mathematicae},

keywords = {star ordering; Loewner ordering; spectral radius; matrix norm; cones of matrices; positive matrices; linear estimators; dispersion matrices; convex cone; partial order; generalized inverses; range spaces},

language = {eng},

number = {2},

pages = {145-154},

title = {On orderings induced by the Loewner partial ordering},

url = {http://eudml.org/doc/219087},

volume = {22},

year = {1994},

}

TY - JOUR

AU - Hauke, Jan

AU - Markiewicz, Augustyn

TI - On orderings induced by the Loewner partial ordering

JO - Applicationes Mathematicae

PY - 1994

VL - 22

IS - 2

SP - 145

EP - 154

AB - The partial ordering induced by the Loewner partial ordering on the convex cone comprising all matrices which multiplied by a given positive definite matrix become nonnegative definite is considered. Its relation to orderings which are induced by the Loewner partial ordering of the squares of matrices is presented. Some extensions of the latter orderings and their comparison to star orderings are given.

LA - eng

KW - star ordering; Loewner ordering; spectral radius; matrix norm; cones of matrices; positive matrices; linear estimators; dispersion matrices; convex cone; partial order; generalized inverses; range spaces

UR - http://eudml.org/doc/219087

ER -

## References

top- [1] J. K. Baksalary, E. P. Liski and G. Trenkler, Mean square error matrix improvements and admissibility of linear estimators, J. Statist. Plann. Inference 23 (1989), 313-325. Zbl0685.62052
- [2] J. K. Baksalary and S. K. Mitra, Left-star and right-star partial orderings, Linear Algebra Appl. 149 (1991), 73-89. Zbl0717.15004
- [3] J. K. Baksalary, B. Schipp and G. Trenkler, Some further results on Hermi- tian-matrix inequalities, ibid. 160 (1992), 119-129. Zbl0753.15014
- [4] P. A. Bekker, The positive semidefiniteness of partitioned matrices, ibid. 111 (1988), 261-278. Zbl0658.15021
- [5] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Wiley, New York, 1974. Zbl0305.15001
- [6] M. P. Drazin, Natural structures on semigroups with involution, Bull. Amer. Math. Soc. 84 (1978), 139-141. Zbl0395.20044
- [7] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985. Zbl0576.15001
- [8] S. K. Mitra, The minus partial order and the shorted matrices, Linear Algebra Appl. 83 (1986), 1-27. Zbl0605.15004
- [9] C. R. Rao, Estimation of parameters in a linear model, Ann. Statist. 4 (1976), 1023-1036. Zbl0336.62055
- [10] C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and Its Applications, Wiley, New York, 1971. Zbl0236.15004
- [11] C. Stępniak, Two orderings on a convex cone of nonnegative definite matrices, Linear Algebra Appl. 94 (1987), 263-272. Zbl0626.15008

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