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

Abstract

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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.

How to cite

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Hauke, 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

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  1. [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. [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. [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. [4] P. A. Bekker, The positive semidefiniteness of partitioned matrices, ibid. 111 (1988), 261-278. Zbl0658.15021
  5. [5] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Wiley, New York, 1974. Zbl0305.15001
  6. [6] M. P. Drazin, Natural structures on semigroups with involution, Bull. Amer. Math. Soc. 84 (1978), 139-141. Zbl0395.20044
  7. [7] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985. Zbl0576.15001
  8. [8] S. K. Mitra, The minus partial order and the shorted matrices, Linear Algebra Appl. 83 (1986), 1-27. Zbl0605.15004
  9. [9] C. R. Rao, Estimation of parameters in a linear model, Ann. Statist. 4 (1976), 1023-1036. Zbl0336.62055
  10. [10] C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and Its Applications, Wiley, New York, 1971. Zbl0236.15004
  11. [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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