×

zbMATH — the first resource for mathematics

Further properties of the star, left-star, right-star, and minus partial orderings. (English) Zbl 1048.15016
Authors’ abstract: Certain classes of matrices are indicated for which the star, left-star, right-star, and minus partial orderings, or some of them, are equivalent. Characterizations of the left-star and right-star orderings, similar to those devised by R. E. Hartwig and G. P. H. Styan [Linear Algebra Appl. 82, 145–161 (1986; Zbl 0603.15001)] for the star and minus orderings, are established along with other auxiliary results, which are of independent interest as well. Some inheritance-type properties of matrices are also given. The class of EP matrices appears to be essential in several points of our considerations.

MSC:
15B48 Positive matrices and their generalizations; cones of matrices
15B57 Hermitian, skew-Hermitian, and related matrices
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J.K. Baksalary, Algebraic characterizations and statistical implications of the commutativity of orthogonal projectors, in: T. Pukkila, S. Puntanen (Eds.), Proceedings of the Second International Tampere Conference in Statistics, University of Tampere, Tampere, Finland, 1987, pp. 113-142
[2] Baksalary, J.K.; Baksalary, O.M.; Szulc, T., A property of orthogonal projectors, Linear algebra appl., 354, 35-39, (2002) · Zbl 1025.15039
[3] Baksalary, J.K.; Hauke, J., Inheriting independence and chi-squaredeness under certain matrix orderings, Statist. probab. lett., 2, 35-38, (1984) · Zbl 0584.62075
[4] Baksalary, J.K.; Mitra, S.K., Left-star and right-star partial orderings, Linear algebra appl., 149, 73-89, (1991) · Zbl 0717.15004
[5] Baksalary, J.K.; Pukelsheim, F.; Styan, G.P.H., Some properties of matrix partial orderings, Linear algebra appl., Linear algebra appl., 220, 3-85, (1995), Erratum
[6] Chipman, J.S.; Rao, M.M., Projections, generalized inverses, and quadratic forms, J. math. anal. appl., 9, 1-11, (1964) · Zbl 0142.00401
[7] M.P. Drazin, Natural structures on rings and semigroups with involution, unpublished manuscript, 1977
[8] Drazin, M.P., Natural structures on semigroups with involution, Bull. amer. math. soc., 84, 139-141, (1978) · Zbl 0395.20044
[9] Groß, J.; Hauke, J.; Markiewicz, A., Partial orderings, preordering, and the polar decomposition of matrices, Linear algebra appl., 289, 161-168, (1999) · Zbl 0932.15014
[10] Hartwig, R.E., How to partially order regular elements, Math. japon., 25, 1-13, (1980) · Zbl 0442.06006
[11] Hartwig, R.E.; Spindelböck, K., Some closed form formulae for the intersection of two special matrices under the star order, Linear and multilinear algebra, 13, 323-331, (1983) · Zbl 0575.15009
[12] Hartwig, R.E.; Styan, G.P.H., On some characterizations of the “star” partial ordering for matrices and rank subtractivity, Linear algebra appl., 82, 145-161, (1986) · Zbl 0603.15001
[13] R.E. Hartwig, G.P.H. Styan, Partially ordered idempotent matrices, in: T. Pukkila, S. Puntanen (Eds.), Proceedings of the Second International Tampere Conference in Statistics, University of Tampere, Tampere, Finland, 1987, pp. 361-383
[14] Nambooripad, K.S.S., The natural partial order on a regular semigroup, Proc. Edinburgh math. soc., 23, 249-260, (1980) · Zbl 0459.20054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.