In this paper, we mainly derive the general solutions of two systems of minus partial ordering equations over von Neumann regular rings. Meanwhile, some special cases are correspondingly presented. As applications, we give some necessary and sufficient conditions for the existence of solutions. It can be seen that some known results can be regarded as the special cases of this paper.
@article{bwmeta1.element.doi-10_1515_math-2015-0022,
author = {Yu Guan and Zhaojia Tong},
title = {Solutions of minus partial ordering equations over von Neumann regular rings},
journal = {Open Mathematics},
volume = {13},
year = {2015},
zbl = {1329.15012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_math-2015-0022}
}
Yu Guan; Zhaojia Tong. Solutions of minus partial ordering equations over von Neumann regular rings. Open Mathematics, Tome 13 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_math-2015-0022/
[1] J.K. Baksalary, A relationship between the star and minus partial orderings, Linear Algebra Appl. 82 (1991), 163–167. | Zbl 0603.15002
[2] J.K. Baksalary, F. Pukelsheim, On the Liner, minus, and star partial orderings of nonnegative definite matrices and their squares, Linear Algebra Appl. 151 (1991), 135–141. | Zbl 0737.15008
[3] J.K. Baksalary, S. Puntanen, Characterizations of the best linear unbiased estimator in the general Gauss–Markov model with the use of matrix partial orderings, Linear Algebra Appl. 127 (1990), 363–370. | Zbl 0695.62152
[4] J.K. Baksalary, F. Pukelsheim, G.P.H. Styan, Some properties of matrix partial orderings, Linear Algebra Appl. 119 (1989), 57–85. | Zbl 0681.15005
[5] R.B. Bapat, S.K. Jain, L.E. Snyder, Nonnegative idempotent matrices and the minus partial order, Linear Algebra Appl. 261 (1997), 143–154. [WoS] | Zbl 0878.15013
[6] B. Blackwood, S.K. Jain, K.M. Prasad, A.K. Srivastava, Shorted operators relative to a partial order in a regular ring, Commun. Algebra 37 (2009), 4141–4152. [WoS] | Zbl 1198.06010
[7] M.P. Drazin, Natural structures on semigroups with involutions, Bull. Amer. Math. Sot. 84 (1978), 139-141.
[8] A. Galantai, A note on the generalized rank reduction, Acta Math. Hung. 116 (2007), 239–246 . [WoS] | Zbl 1135.15300
[9] H. Goller, Shorted operators and rank decomposition matrices, Linear Algebra Appl. 81 (1986), 207–236. | Zbl 0597.15002
[10] K.R. Goodearl, von Neumann Regular Rings, 2nd edn. Krieger, Florida (1991).
[11] J. Grob, A note on the rank-subtractivity ordering, Linear Algebra Appl. 289 (1999), 151–160. | Zbl 0941.15011
[12] A. Guterman, Linear preservers for matrix inequalities and partial orderings, Linear Algebra Appl. 331 (2001), 75–87. | Zbl 0985.15018
[13] R.E. Hartwig, How to partially order regular elements, Math. Jpn. 25 (1980), 1–13. | Zbl 0442.06006
[14] R.E. Hartwig, G.P.H. Styan, On some characterizations of the “star” partial ordering for matrices and rank subtractivity, Linear Algebra Appl. 82 (1986), 145–16l. [Crossref] | Zbl 0603.15001
[15] N. Jacobson, Basic Algebra II, 2nd edn. Freeman, New York (1989). | Zbl 0694.16001
[16] S.K. Jain, S.K. Mitra, H.J. Werner, Extensions of G-based matrix partial orders, Siam J. Matrix Anal. Appl. 17(4) (1996), 834–850. [Crossref] | Zbl 0860.15007
[17] P. Legiša, Automorphisms ofMn, partially ordered by rank subtractivity ordering, Linear Algebra Appl. 389 (2004), 147–158. | Zbl 1080.15017
[18] G. Marsaglia, G.P.H. Styan, Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra 2 (1974), 269–292. [Crossref] | Zbl 0297.15003
[19] S.K. Mitra, The minus partial order and the shorted matrix, Linear Algebra Appl. 83 (1986), 1–27. | Zbl 0605.15004
[20] S.K. Mitra, M.L. Puri, Shorted matrices-an extended concept and some applications, Linear Algebra Appl. 42 (1982), 57–79. | Zbl 0478.15012
[21] S.K. Mitra, Matrix partial orders through generalized inverses unified theory, Linear Algebra Appl. 148 (1991), 237–263. [WoS] | Zbl 0717.15003
[22] K.S.S. Nambooripad, The natural partial order on a regular semigroup. Proc. Edinb. Math. Soc. 23 (1980), 249–260. [Crossref] | Zbl 0459.20054
[23] R. Penrose, A generalized inverse for matrices. Proc. Camb. Philos. Soc. 51 (1955), 406–413. | Zbl 0065.24603
[24] Y. Tian, Solving a minus partial ordering equation over von Neumann regular rings, Revista mathematica complutense. 24(2) (2011), 335-342. [WoS]
[25] Q.W. Wang, A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity, Linear Algebra Appl. 384 (2004) 43-54. | Zbl 1058.15015
[26] H.J. Werner, Generalized inversion and weak bi-complementarity, Linear and Multilinear Algebra 19 (1986), 357–372. | Zbl 0598.15005