Characterizations and Representations of Core and Dual Core Inverses in rings

In this paper, double commutativity and the reverse order law for the core inverse are considered._en new characterizations of theMoore–Penrose inverse of a regular element are given by one-sided invertibilities in a ring. Furthermore, the characterizations and representations of the core and dual core inverses of a regular element are considered.National Natural Science Foundation of China (No.11371089). FCT project UID-MAT-00013/2013info:eu-repo/semantics/publishedVersio

On a matrix group constructed from an {R,s+1,k}-potent matrix

Let R∈C^(n×n) be a {k}-involutory matrix (that is, R^k=I_n) for some integer k≥2, and let s be a nonnegative integer. A matrix A∈C^(n×n) is called an {R,s+1,k}-potent matrix if A satisfies R A = A^(s+1) R. In this paper, a matrix group corresponding to a fixed {R,s+1,k}-potent matrix is explicitly constructed, and properties of this group are derived and investigated. This group is then reconciled with the classical matrix group G_A that is associated with a generalized group invertible matrix A. Let R∈Cn×n be a {k}-involutory matrix (that is, Rk=In) for some integer k≥2, and let s be a nonnegative integer. A matrix A∈Cn×n is called an {R,s+1,k}-potent matrix if A satisfies RA=As+1R. In this paper, a matrix group corresponding to a fixed {R,s+1,k}-potent matrix is explicitly constructed, and properties of this group are derived and investigated. This group is then reconciled with the classical matrix group GA that is associated with a generalized group invertible matrix A

and

The matrix partial orderings considered are: (1) the star ordering and (2) the minus ordering or rank subtractivity, both in the set of m X n complex matrices, and (3) the Lowner ordering, in the set of m X m matrices. The problems discussed are: (1) inheriting certain properties under a given ordering, (2) preserving an ordering under some matrix multiplications, (3) relationships between an ordering among direct (or Kronecker) and Hadamard products and the corresponding orderings between the factors involved, (4) orderings between generalized inverses of a given matrix, and (5) preserving or reversing a given ordering under generalized inversions. Several generalizations of results known in the literature and a number of new results are derived

On a new generalized inverse for matrices of an arbitrary index

[EN] The purpose of this paper is to introduce a new generalized inverse, called DMP inverse, associated with a square complex matrix using its Drazin and Moore-Penrose inverses. DMP inverse extends the notion of core inverse, introduced by Baksalary and Trenkler for matrices of index at most 1 in (Baksalary and Trenkler (2010) [1]) to matrices of an arbitrary index. DMP inverses are analyzed from both algebraic as well as geometrical approaches establishing the equivalence between them. (C) 2013 Elsevier Inc. All rights reserved.This author was partially supported by Ministry of Education of Spain (Grant DGI MTM2010-18228).Malik, SB.; Thome, N. (2014). On a new generalized inverse for matrices of an arbitrary index. Applied Mathematics and Computation. 226:575-580. doi:10.1016/j.amc.2013.10.060S57558022

Some comments on the life and work of Jerzy K. Baksalary (1944-2005)

Following some biographical comments on Jerzy K. Baksalary (1944–2005), this article continues with personal comments by Oskar Maria Baksalary, Tadeusz Cali´nski, R.William Farebrother, Jürgen Groß, Jan Hauke, Erkki Liski, Augustyn Markiewicz, Friedrich Pukelsheim, Tarmo Pukkila, Simo Puntanen, Tomasz Szulc, Yongge Tian, Götz Trenkler, Júlia Volaufová, Haruo Yanai, and Fuzhen Zhang, on the life and work of Jerzy K. Baksalary, and with a detailed list of his publications. Our article ends with a survey by Tadeusz Cali´nski on Jerzy Baksalary’s work in block designs and a set of photographs of Jerzy Baksalary

On mappings preserving the sharp and star orders

The present paper is devoted to the study of linear maps preserving certain relations, such as the sharp partial order and the star partial order in semisimple Banach algebras and C*-algebras.The first author is partially supported by the Spanish Ministry of Science and Innovation, D.G.I. project no. MTM2011-23843, and Junta de Andaluc´ıa grant FQM375. The second author is also supported by a Plan Propio de Investigaci´on grant from University of Almer´ıa, and Junta de Andaluc´ıa grant FQM 3737. The third author is partially supported by FEDER Funds through “Programa Operacional Factores de Competitividade – COMPETE” and by Portuguese Funds through FCT - “Funda¸c˜ao para a Ciˆencia e a Tecnologia”, within the project PEst-OE/MAT/UI0013/2014

On a partial order defined by the weighted Moore Penrose inverse

The weighted Moore-Penrose inverse of a matrix can be used to define a partial order on the set of m x n complex matrices and to introduce the concept of weighted-EP matrices. In this paper we study the weighted star partial order on the set of weighted-EP matrices. In addition, some properties that relate the eigenprojection at zero with the weighted star partial order are obtained. (C) 2013 Elsevier Inc. All rights reserved.This author was partially supported by Ministry of Education of Spain (Grant DGI MTM2010-18228).Hernández, AE.; Lattanzi, MB.; Thome, N. (2013). On a partial order defined by the weighted Moore Penrose inverse. Applied Mathematics and Computation. 219(14):7310-7318. https://doi.org/10.1016/j.amc.2013.02.010S731073182191

On some new pre-orders defined by weighted Drazin inverses

In this paper, we investigate new binary relations defined on the set of rectangular complex matrices based on the weighted Drazin inverse and give some characterizations of them. These relations become pre-orders and improve the results found by the authors in Hernandez et al. (2013) as well as extend those known for square matrices. On the other hand, some new weighted partial orders are also defined and characterized. The advantages of these new relations compared to the ones considered in the mentioned paper are also pointed out.N. Thome was partially supported by Ministerio de Economia y Competitividad of Spain (Grant DGI MTM2013-43678-P and Red de Excelencia MTM2015-68805-REDT).Hernández, AE.; Lattanzi, MB.; Thome Coppo, NJ. (2016). On some new pre-orders defined by weighted Drazin inverses. Applied Mathematics and Computation. 282:108-116. https://doi.org/10.1016/j.amc.2016.01.055S10811628