Scopus
EXPORT DATE:02 Jul 2023

Patrício, P., Hartwig, R.E.
6701711621;7005552138;
From Euclid to Corner Sums – a Trail of Telescoping Tricks
(2021) Filomat, 35 (14), pp. 4613-4636. 
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85126305839&doi=10.2298%2fFIL2114613P&partnerID=40&md5=4e8f7de0896f51eb8d9f7df964b3e4b9

DOI: 10.2298/FIL2114613P
ABSTRACT: Euclid’s algorithm is extended to binomials, geometric sums and corner sums. Two-sided non-commuting, non-constant linear difference equations will be solved, and the solution is applied to corner sums, thereby presenting an explicit formula for the generator of the bi-module spanned by the two starting corner sums. © 2021, University of Nis. All rights reserved.
AUTHOR KEYWORDS: Corner Sum;  Polynomials;  Recurrence Relation;  Telescoping Sum
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze, Green
SOURCE: Scopus

Zhu, H., Wu, L., Peng, F., Patrício, P.
57189272380;57219937007;57214151662;6701711621;
Characterizations and representations of left and right hybrid (b, c)-inverses in rings
(2021) Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas, 115 (1), art. no. 22, . Cited 1 time.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85096177464&doi=10.1007%2fs13398-020-00956-7&partnerID=40&md5=86ade7ae03080f04012b7d4f4d43b2e2

DOI: 10.1007/s13398-020-00956-7
ABSTRACT: Let R be an associative ring with unity 1 and let a, b, c∈ R. In this paper, several characterizations for left and right hybrid (b, c)-inverses of a are derived. Moreover, their formulae are given by regularities of certain elements. Then, we give characterizations of right (b, c)-inverses and right annihilator (b, c)-inverses of the product of three elements. Finally, relations among the right hybrid (b, c)-inverse of paq, the right (qb, c)-inverse of aq and the right annihilator (b, cp)-inverse of aq are given. © 2020, The Royal Academy of Sciences, Madrid.
AUTHOR KEYWORDS: (b, c) -inverses;  Annihilators;  Hybrid (b, c) -inverses;  Ideals;  Rings
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Gao, Y., Chen, J., Patrício, P.
57193950431;55978339700;6701711621;
Continuity of the core-EP inverse and its applications
(2021) Linear and Multilinear Algebra, 69 (3), pp. 557-571. Cited 21 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85065105646&doi=10.1080%2f03081087.2019.1608899&partnerID=40&md5=86f5dabc5c037612b36b3f69cd33b39e

DOI: 10.1080/03081087.2019.1608899
ABSTRACT: In this paper, firstly we study the continuity of the core-EP inverse without explicit error bounds by virtue of two methods. One is the rank equality, followed from the classical generalized inverse. The other one is matrix decomposition. The continuity of the core inverse can be derived as a particular case. Secondly, we study perturbation bounds for the core-EP inverse under prescribed conditions. Perturbation bounds for the core inverse can be derived as a particular case. Also, as corollaries, the sufficient (and necessary) conditions for the continuity of the core-EP inverse are obtained. Thirdly, a numerical example is illustrated to compare derived upper bounds. Finally, an application to semistable matrices is provided. © 2019 Informa UK Limited, trading as Taylor & Francis Group.
AUTHOR KEYWORDS: 15A09;  34D10;  65F35;  continuity;  Core-EP inverse;  perturbation bound;  pseudo core inverse
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Gao, Y., Chen, J., Patrício, P.
57193950431;55978339700;6701711621;
Weighted binary relations involving core-EP inverse
(2020) Frontiers of Mathematics in China, 15 (4), pp. 685-699. Cited 1 time.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85090015197&doi=10.1007%2fs11464-020-0856-7&partnerID=40&md5=a20aa2804c730047dffdc649d524dc12

DOI: 10.1007/s11464-020-0856-7
ABSTRACT: We study a new binary relation defined on the set of rectangular complex matrices involving the weighted core-EP inverse and give its characterizations. This relation becomes a pre-order. Then, one-sided pre-orders associated to the weighted core-EP inverse are given from two perspectives. Finally, we make a comparison for these two sets of one-sided weighted pre-orders. © 2020, Higher Education Press.
AUTHOR KEYWORDS: core-EP inverse;  pre-order;  pseudo core inverse;  Weighted core-EP inverse
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Gao, Y., Chen, J., Patrício, P.
57193950431;55978339700;6701711621;
Representations and properties of the W-weighted core-EP inverse
(2020) Linear and Multilinear Algebra, 68 (6), pp. 1160-1174. Cited 16 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85055564403&doi=10.1080%2f03081087.2018.1535573&partnerID=40&md5=872c0b392a8b927df5d56bf1230c4670

DOI: 10.1080/03081087.2018.1535573
ABSTRACT: In this paper, we investigate the weighted core-EP inverse introduced by Ferreyra, Levis and Thome. Several computational representations of the weighted core-EP inverse are obtained in terms of singular-value decomposition, full-rank decomposition and QR decomposition. These representations are expressed in terms of various matrix powers as well as matrix product involving the core-EP inverse, Moore-Penrose inverse and usual matrix inverse. Finally, those representations involving only Moore-Penrose inverse are compared and analyzed via computational complexity and numerical examples. © 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group.
AUTHOR KEYWORDS: 15A09;  65F20;  68Q25;  complexity;  core-EP inverse;  outer inverse;  pseudo core inverse;  Shmuel Friedland;  Weighted core-EP inverse
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Mary, X., Patrício, P.
12782242200;6701711621;
Characterizations of special clean elements and applications
(2020) Filomat, 34 (14), pp. 4847-4860. Cited 2 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85103646229&doi=10.2298%2fFIL2014847M&partnerID=40&md5=87a44f1f93fb85ebc6c3ae611d3ed475

DOI: 10.2298/FIL2014847M
ABSTRACT: We prove that special clean decompositions of a given element of a ring are in one-to-one correspondence with the set of solutions of a simple equation in a corner ring. We then derive “constructive” proofs that in many rings, regular elements are special clean by solving this equation in specific cases. Other applications, such as uniqueness of decompositions, are given. Many examples of special clean decompositions of 2-2 matrices found by this methodology are also presented. © 2020, University of Nis. All rights reserved.
AUTHOR KEYWORDS: (special) clean elements;  Stable range 1;  Strongly regular (group) elements;  Unit-regular elements
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze, Green
SOURCE: Scopus

Zhu, H., Patrício, P.
57189272380;6701711621;
Several types of one-sided partial orders in rings
(2019) Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas, 113 (4), pp. 3177-3184. Cited 10 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85065569495&doi=10.1007%2fs13398-019-00685-6&partnerID=40&md5=6dfe0b00ac50fb88161d62a50f9dd72b

DOI: 10.1007/s13398-019-00685-6
ABSTRACT: In this paper, we investigate one-sided sharp partial orders and one-sided core and dual core partial orders in rings. Moreover, their characterizations and relations with other partial orders are given. © 2019, The Royal Academy of Sciences, Madrid.
AUTHOR KEYWORDS: Core inverses;  Group inverses;  Moore–Penrose inverses;  One-sided core partial orders;  One-sided sharp partial orders
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Zhu, H., Zou, H., Patrício, P.
57189272380;57192816824;6701711621;
Generalized inverses and their relations with clean decompositions
(2019) Journal of Algebra and its Applications, 18 (7), art. no. 1950133, . Cited 4 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85053043828&doi=10.1142%2fS0219498819501330&partnerID=40&md5=7d9598caae3d32deae9a65229770f370

DOI: 10.1142/S0219498819501330
ABSTRACT: An element a in a ring R is called clean if it is the sum of an idempotent e and a unit u. Such a clean decomposition a = e + u is said to be strongly clean if eu = ue and special clean if aR eR = (0). In this paper, we prove that a is Drazin invertible if and only if there exists an idempotent e and a unit u such that an = e + u is both a strongly clean decomposition and a special clean decomposition, for some positive integer n. Also, the existence of the Moore-Penrose and group inverses is related to the existence of certain - clean decompositions. © 2019 World Scientific Publishing Company.
AUTHOR KEYWORDS: Drazin inverse;  group inverse;  Moore-Penrose inverse;  special clean decomposition;  strongly clean decomposition
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Zhu, H., Patrício, P.
57189272380;6701711621;
Characterizations for pseudo core inverses in a ring with involution
(2019) Linear and Multilinear Algebra, 67 (6), pp. 1109-1120. Cited 11 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85044057293&doi=10.1080%2f03081087.2018.1446506&partnerID=40&md5=0112cec280e2005fa5715811addc7ffe

DOI: 10.1080/03081087.2018.1446506
ABSTRACT: Let R be a unital -ring. In this paper, we give several characterizations for the pseudo core and dual pseudo core inverses of an element by units in R. Moreover, their relations with *-DMP are obtained. As a special case, the existence criteria for EP elements (*-DMP with index 1) are given. Then, the double commutant property of hybrid (b, c)-inverses is considered. Finally, the relations of the inverse along an element and the hybrid (b, c)-inverse, the (dual) pseudo core inverses and the hybrid (b, c)-inverse are investigated. © 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group.
AUTHOR KEYWORDS: -DMP;  15A09;  16U99;  double commutant property;  Drazin inverses;  hybrid (b;  Moore–Penrose inverses;  Pseudo core inverses;   c)-inverses
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Mary, X., Patrício, P.
12782242200;6701711621;
Group-regular rings
(2019) Filomat, 33 (11), pp. 3551-3560. Cited 2 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85077673673&doi=10.2298%2fFIL1911551M&partnerID=40&md5=acd29f158ece875411d7e6b5a8e02a89

DOI: 10.2298/FIL1911551M
ABSTRACT: We propose different generalizations of unit-regularity of elements in general rings (non necessarily unital rings). We then study general rings for which all elements have these properties. We notably compare them with unit-regular ideals and general rings with stable range one. We also prove that these rings are morphic rings. © 2019, University of Nis. All rights reserved.
AUTHOR KEYWORDS: Adjoint semigroup;  General ring;  Morphic ring;  Stable range one;  Strongly regular (group) element;  Unit-regular element
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze, Green
SOURCE: Scopus

Hartwig, R.E., Patrício, P.
7005552138;6701711621;
The group inverse of the nivellateur
(2018) Linear and Multilinear Algebra, 66 (12), pp. 2409-2420. Cited 1 time.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85033674284&doi=10.1080%2f03081087.2017.1397092&partnerID=40&md5=2e1be7ec7d27516916b9d533c061290a

DOI: 10.1080/03081087.2017.1397092
ABSTRACT: We shall derive necessary and sufficient conditions for the nivellateur to have a group inverse over an algebraically closed field. We then extend these results to arbitrary fields. © 2017, © 2017 Informa UK Limited, trading as Taylor & Francis Group.
AUTHOR KEYWORDS: 15A09;  15A21;  group inverse;  matrices over a field;  Nivellateur
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Gao, Y., Chen, J., Patrício, P., Wang, D.
57193950431;55978339700;6701711621;7407075934;
The pseudo core inverse of a companion matrix
(2018) Studia Scientiarum Mathematicarum Hungarica, 55 (3), pp. 407-420. 
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85055136379&doi=10.1556%2f012.2018.55.3.1398&partnerID=40&md5=51967bdd9eb79f729ed32890f44adaf8

DOI: 10.1556/012.2018.55.3.1398
ABSTRACT: The notion of core inverse was introduced by Baksalary and Trenkler for a complex matrix of index 1. Recently, the notion of pseudo core inverse extended the notion of core inverse to an element of an arbitrary index in ∗-rings; meanwhile, it characterized the core-EP inverse introduced by Manjunatha Prasad and Mohana for complex matrices, in terms of three equations. Many works have been done on classical generalized inverses of companion matrices and Toeplitz matrices. In this paper, we discuss existence criteria and formulae of the pseudo core inverse of a companion matrix over a ∗-ring. A {1, 3}-inverse of a Toeplitz matrix plays an important role in that process. © 2018 Akadémiai Kiadó, Budapest
AUTHOR KEYWORDS: Companion matrix;  Core inverse;  Core-EP inverse;  Pseudo core inverse;  Toeplitz matrix
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Zou, H., Chen, J., Patrício, P.
57192816824;55978339700;6701711621;
Characterizations of m-EP elements in rings
(2018) Linear and Multilinear Algebra, 66 (6), pp. 1244-1256. Cited 6 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85022202623&doi=10.1080%2f03081087.2017.1347136&partnerID=40&md5=a5c05a7e9fe24ff8c73915d3982708ef

DOI: 10.1080/03081087.2017.1347136
ABSTRACT: Let R be a ring with involution. In this paper, we extend the notions of m-EP matrices and m-EP operators to an arbitrary ring case. A number of new characterizations of m-EP elements in rings are presented. In particular, the existence criteria for 1-EP (i.e. EP) elements are obtained by means of the group inverse, Moore–Penrose inverse, and core inverse. Some properties of 2-EP are also given. © 2017 Informa UK Limited, trading as Taylor & Francis Group.
AUTHOR KEYWORDS: core inverse;  Drazin inverse;  m-EP;  Moore–Penrose inverse;  ring
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Zou, H., Chen, J., Patrício, P.
57192816824;55978339700;6701711621;
Reverse Order Law for the Core Inverse in Rings
(2018) Mediterranean Journal of Mathematics, 15 (3), art. no. 145, . Cited 15 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85048169359&doi=10.1007%2fs00009-018-1189-6&partnerID=40&md5=d571e59a0207712c29e99a57d45613dc

DOI: 10.1007/s00009-018-1189-6
ABSTRACT: In this paper, necessary and sufficient conditions of the one-sided reverse order law (ab)#=b#a#, the two-sided reverse order law (ab)#=b#a# and (ba)#=a#b# for the core inverse are given in rings with involution. In addition, the mixed-type reverse order laws, such as (ab)#=b#(abb#)#, a#=b(ab)# and (ab)#=b#a#, are also considered. © 2018, Springer International Publishing AG, part of Springer Nature.
AUTHOR KEYWORDS: Core inverse;  group inverse;  reverse order law
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Hartwig, R.E., Patrício, P.
7005552138;6701711621;
Invariance under outer inverses
(2018) Aequationes Mathematicae, 92 (2), pp. 375-383. Cited 7 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85039840125&doi=10.1007%2fs00010-017-0524-5&partnerID=40&md5=76623bbca9885054dbe863baa251ea8b

DOI: 10.1007/s00010-017-0524-5
ABSTRACT: We shall use the minus partial order combined with Pierce’s decomposition to derive the class of outer inverses for idempotents, units and group invertible elements. Subsequently we show, for matrices over a field F, that the triplet BA^ C is invariant under all choices of outer inverses of A if and only if B= 0 or C= 0. © 2017, Springer International Publishing AG, part of Springer Nature.
AUTHOR KEYWORDS: Invariance;  Outer inverses;  Pierce decomposition;  Regularity
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Zhu, H., Patrício, P.
57189272380;6701711621;
A note on clean elements and inverses along an element
(2018) Filomat, 32 (12), pp. 4285-4290. Cited 1 time.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85061335713&doi=10.2298%2fFIL1812285Z&partnerID=40&md5=048af8332214d27ad6e852f8736c998b

DOI: 10.2298/FIL1812285Z
ABSTRACT: Let R be an associative ring with unity 1 and let a, d ∈ R. An element a ∈ R is called invertible along d if there exists unique a ‖d such that a ‖d ad = d = daa ‖d and a ‖d ∈ dR ∩ Rd (see [6, Definition 4]). In this note, we present new characterizations for the existence of a ‖d by clean decompositions of ad and da. As applications, existence criteria for the Drazin inverse and the group inverse are given. © 2018, University of Nis. All rights reserved.
AUTHOR KEYWORDS: Clean elements;  Inverses along an element;  One-sided inverses along an element;  Special clean decompositions;  Strongly clean decompositions
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze, Green
SOURCE: Scopus

Zhu, H., Chen, J., Patrício, P., Mary, X.
57189272380;55978339700;6701711621;12782242200;
Centralizer's applications to the inverse along an element
(2017) Applied Mathematics and Computation, 315, pp. 27-33. Cited 14 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85026345295&doi=10.1016%2fj.amc.2017.07.046&partnerID=40&md5=adb9ccbd3191af0a9b9c1be0b3311120

DOI: 10.1016/j.amc.2017.07.046
ABSTRACT: In this paper, we firstly prove that the absorption law for one-sided inverses along an element holds, and derive the absorption law for the inverse along an element. We then obtain the absorption law for the inverse along different elements. Also, we prove that a left inverse of a along d coincides with a right inverse of a along d, provided that they both exist. Then, the reverse order law and the existence criterion for the inverse along an element are given by centralizers in a ring. Finally, we characterize the Moore–Penrose inverse of a regular element by one-sided invertibilities in a ring with involution. © 2017
AUTHOR KEYWORDS: Absorption laws;  Centralizers;  Inverses along an element;  Left (Right) inverses along an element;  Moore–Penrose inverses;  Reverse order laws
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Chen, J., Zou, H., Zhu, H., Patrício, P.
55978339700;57192816824;57189272380;6701711621;
The One-Sided Inverse Along an Element in Semigroups and Rings
(2017) Mediterranean Journal of Mathematics, 14 (5), art. no. 208, . Cited 8 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85029892081&doi=10.1007%2fs00009-017-1017-4&partnerID=40&md5=2894951a656ceecebe9cfa764191ac5f

DOI: 10.1007/s00009-017-1017-4
ABSTRACT: The concept of the inverse along an element was introduced by Mary in 2011. Later, Zhu et al. introduced the one-sided inverse along an element. In this paper, we first give a new existence criterion for the one-sided inverse along a product and characterize the existence of Moore–Penrose inverse by means of one-sided invertibility of certain element in a ring. In addition, we show that a∈ S†⋂ S# if and only if (a∗a)k is invertible along a if and only if (aa∗)k is invertible along a in a ∗ -monoid S, where k is an arbitrary given positive integer. Finally, we prove that the inverse of a along aa∗ coincides with the core inverse of a under the condition a∈ S{ 1 , 4 } in a ∗ -monoid S. © 2017, Springer International Publishing AG.
AUTHOR KEYWORDS: Inverse along an element;  rings;  semigroups;  Von Neumann regularity
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Chen, J., Zhu, H., Patricio, P., Zhang, Y.
55978339700;57189272380;6701711621;55949740300;
Characterizations and representations of core and dual core inverses
(2017) Canadian Mathematical Bulletin, 60 (2), pp. 269-282. Cited 54 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85021099684&doi=10.4153%2fCMB-2016-045-7&partnerID=40&md5=87fb6402fffd19f29ef20a348b692bf6

DOI: 10.4153/CMB-2016-045-7
ABSTRACT: In this paper, double commutativity and the reverse order law for the core inverse are considered. Then new characterizations of the Moore-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. © Canadian Mathematical Society 2016.
AUTHOR KEYWORDS: Core inverses;  Dedekind-finite rings;  Dual core inverses;  Group inverses;  Moore-penrose inverses;  Regularities
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Zhu, H., Chen, J., Patrício, P.
57189272380;55978339700;6701711621;
Reverse order law for the inverse along an element
(2017) Linear and Multilinear Algebra, 65 (1), pp. 166-177. Cited 16 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84964615087&doi=10.1080%2f03081087.2016.1178209&partnerID=40&md5=64394f433e517e591687eda855c339d4

DOI: 10.1080/03081087.2016.1178209
ABSTRACT: In this paper, we introduce a new concept called left (right) g-MP inverse in a *-monoid. The relations of this type of generalized inverse with left inverse along an element are investigated. Also, the reverse order law for the inverse along an element is studied. Then, the existence criteria and formulae of the inverse of the product of triple elements along an element are investigated in a monoid. Finally, we further study left and right g-MP inverses, the inverse along an element in the context of rings. © 2016 Informa UK Limited, trading as Taylor & Francis Group.
AUTHOR KEYWORDS: (Left;  Green’s preorders;  reverse order law;  right) inverses along an element;  rings;  semigroups
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Mary, X., Patrício, P.
12782242200;6701711621;
The group inverse of a product
(2016) Linear and Multilinear Algebra, 64 (9), pp. 1776-1784. Cited 1 time.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84951291697&doi=10.1080%2f03081087.2015.1120265&partnerID=40&md5=c7e78d0ab42c85f9183cf3cbf0d9c985

DOI: 10.1080/03081087.2015.1120265
ABSTRACT: In this paper, we characterize the existence and give an expression of the group inverse of a product of two regular elements by means of a ring unit. © 2015 Taylor & Francis.
AUTHOR KEYWORDS: Generalized inverse;  group inverse;  rings;  von Neumann inverse
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Zhu, H., Patrício, P., Chen, J., Zhang, Y.
56241401700;6701711621;55978339700;55949740300;
The inverse along a product and its applications
(2016) Linear and Multilinear Algebra, 64 (5), pp. 834-841. Cited 11 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84935417372&doi=10.1080%2f03081087.2015.1059796&partnerID=40&md5=458a3f6d878250b8e616f1a0cdd96395

DOI: 10.1080/03081087.2015.1059796
ABSTRACT: In this paper, we study the recently defined notion of the inverse along an element. An existence criterion for the inverse along a product is given in a ring. As applications, we present the equivalent conditions for the existence and expressions of the inverse along a matrix. © 2015 Taylor & Francis.
AUTHOR KEYWORDS: Green’s relations;  inverse along an element;  matrices over a ring;  von Neumann regularity
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Zhu, H., Chen, J., Patrício, P.
56241401700;55978339700;6701711621;
Further results on the inverse along an element in semigroups and rings
(2016) Linear and Multilinear Algebra, 64 (3), pp. 393-403. Cited 45 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84953357073&doi=10.1080%2f03081087.2015.1043716&partnerID=40&md5=50bcc267f75c89de2c249604a9d80dad

DOI: 10.1080/03081087.2015.1043716
ABSTRACT: In this paper, we introduce a new notion in a semigroup (Formula presented.) as an extension of Mary’s inverse. Let (Formula presented.). An element (Formula presented.) is called left (resp. right) invertible along (Formula presented.) if there exists (Formula presented.) such that (Formula presented.) (resp. (Formula presented.)) and (Formula presented.) (resp. (Formula presented.)). An existence criterion of this type inverse is derived. Moreover, several characterizations of left (right) regularity, left (right) (Formula presented.) -regularity and left (right) (Formula presented.) -regularity are given in a semigroup. Further, another existence criterion of this type inverse is given by means of a left (right) invertibility of certain elements in a ring. Finally, we study the (left, right) inverse along a product in a ring, and, as an application, Mary’s inverse along a matrix is expressed. © 2015 Taylor & Francis.
AUTHOR KEYWORDS: inverse along an element;  left (right);  left (right) -regularity;  left (right) regularity;  rings;  semigroups;  von Neumann regularity
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Zhu, H., Chen, J., Patrício, P.
57189272380;55978339700;6701711621;
The Moore-Penrose inverse of differences and products of projectors in a ring with involution
(2016) Turkish Journal of Mathematics, 40 (6), pp. 1316-1324. Cited 1 time.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-85007189916&doi=10.3906%2fmat-1405-1&partnerID=40&md5=369cd863d4388e75c5686760496c7e85

DOI: 10.3906/mat-1405-1
ABSTRACT: In this paper, we study the Moore-Penrose inverses of differences and products of projectors in a ring with involution. Some necessary and sufficient conditions for the existence of the Moore-Penrose inverse are given. Moreover, the expressions of the Moore-Penrose inverses of differences and products of projectors are presented. © Tübi˙tak.
AUTHOR KEYWORDS: Involutions;  Moore-Penrose inverses;  Normal elements;  Projectors
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Burgos, M., Márquez-García, A.C., Patrício, P.
20733491600;55214710600;6701711621;
On mappings preserving the sharp and star orders
(2015) Linear Algebra and Its Applications, 483, pp. 268-292. Cited 2 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84935002184&doi=10.1016%2fj.laa.2015.05.003&partnerID=40&md5=0def9903539e858a2d78fa6f8ac26802

DOI: 10.1016/j.laa.2015.05.003
ABSTRACT: 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. © 2015 Elsevier Inc. All rights reserved.
AUTHOR KEYWORDS: Banach algebra;  Linear preserver;  Orthogonality;  Sharp order;  Star order;  Zero product
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze, Green
SOURCE: Scopus

Zhu, H., Chen, J., Patrício, P.
56241401700;55978339700;6701711621;
Representations for the pseudo Drazin inverse of elements in a Banach algebra
(2015) Taiwanese Journal of Mathematics, 19 (2), pp. 349-362. Cited 14 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84925351829&doi=10.11650%2ftjm.19.2015.4576&partnerID=40&md5=77f8134cfed60b020269d5b46f0d0be5

DOI: 10.11650/tjm.19.2015.4576
ABSTRACT: In this paper, we investigate the pseudo Drazin invertibility of the sum and the product of elements in a Banach algebra A. Given pseudo Drazin invertible elements a and b such that a2b=abaand b2a=bab, it is shown that ab is pseudo Drazin invertible and a+b is pseudo Drazin invertible if and only if so is 1+a‡b, and the related formulae are provided. © 2015, Mathematical Society of the Rep. of China. All rights reserved.
AUTHOR KEYWORDS: Jacobson radical;  Pseudo Drazin inverse;  Strongly spectral idempotent
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze
SOURCE: Scopus

Patrício, P., Hartwig, R.E.
6701711621;7005552138;
Some regular sums
(2015) Linear and Multilinear Algebra, 63 (1), pp. 185-200. Cited 6 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84908546178&doi=10.1080%2f03081087.2013.860592&partnerID=40&md5=4ed8a0b65c890a4f7135cdcd31e67199

DOI: 10.1080/03081087.2013.860592
ABSTRACT: In this paper, we examine the question of regularity of sums of special elements that appear in the study of orthogonality and invertibility. © 2014, Taylor & Francis.
AUTHOR KEYWORDS: block matrices;  group inverse;  inverse;  outer inverses;  Pierce decomposition;  reflexive inverses
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Zhu, H., Chen, J., Zhang, X., Patrício, P.
56241401700;55978339700;55912092300;6701711621;
The Moore-Penrose inverse of 2 × 2 matrices over a certain *-regular ring
(2014) Applied Mathematics and Computation, 246, pp. 263-267. Cited 5 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84907190553&doi=10.1016%2fj.amc.2014.08.026&partnerID=40&md5=681331cec3792067eb8521a37bae6caf

DOI: 10.1016/j.amc.2014.08.026
ABSTRACT: In this paper, we study representations of the Moore-Penrose inverse of a 2×2 matrix M over a *-regular ring with two term star-cancellation. As applications, some necessary and sufficient conditions for the Moore-Penrose inverse of M to have different types are given. © 2014 Elsevier Inc. All rights reserved.
AUTHOR KEYWORDS: -Regular ring;  Moore-Penrose inverse;  Two term star-cancellation
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Hartwig, R.E., Patrício, P.
7005552138;6701711621;
Two by two units
(2014) Electronic Journal of Linear Algebra, 27, pp. 489-503. Cited 2 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84905440207&doi=10.13001%2f1081-3810.1632&partnerID=40&md5=ff4a26f80134b676b6c8c13605de8401

DOI: 10.13001/1081-3810.1632
ABSTRACT: In this paper, we will use outer inverses and the Brown-McCoy shift to characterize the existence of the inverse and group inverse of a 2 × 2 block matrix.
AUTHOR KEYWORDS: Block matrices;  Generalized inverses;  Group inverse;  Inverse;  Outer inverses;  Pierce decomposition;  Reflexive inverses
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze
SOURCE: Scopus

Lebtahi, L., Patrício, P., Thome, N.
15063014400;6701711621;6603046084;
The diamond partial order in rings
(2014) Linear and Multilinear Algebra, 62 (3), pp. 386-395. Cited 30 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84893557133&doi=10.1080%2f03081087.2013.779272&partnerID=40&md5=6bf097c63689afc0486cecf3db563982

DOI: 10.1080/03081087.2013.779272
ABSTRACT: In this paper we introduce a new partial order on a ring, namely the diamond partial order. This order is an extension of a partial order defined in a matrix setting in [J.K. Baksalary and J. Hauke, A further algebraic version of Cochran's theorem and matrix partial orderings, Linear Algebra and its Applications, 127, 157-169, 1990]. We characterize the diamond partial order on rings and study its relationships with other partial orders known in the literature. We also analyse successors, predecessors and maximal elements under the diamond order. © 2013 © 2013 Taylor & Francis.
AUTHOR KEYWORDS: minus partial order;  principal ideal;  ring;  sharp partial order;  star partial order
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Mary, X., Patrício, P.
12782242200;6701711621;
Generalized inverses modulo ℋ in semigroups and rings
(2013) Linear and Multilinear Algebra, 61 (8), pp. 1130-1135. Cited 42 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84880302365&doi=10.1080%2f03081087.2012.731054&partnerID=40&md5=2f63a3cb8a9abe470a5750effd83a899

DOI: 10.1080/03081087.2012.731054
ABSTRACT: The definition of the inverse along an element was very recently introduced, and it contains known generalized inverses such as the group, Drazin and Moore-Penrose inverses. In this article, we first prove a simple existence criterion for this inverse in a semigroup, and then relate the existence of such an inverse in a ring to the ring units. © 2013 Copyright Taylor and Francis Group, LLC.
AUTHOR KEYWORDS: generalized inverses;  Green's relations;  rings;  semigroups
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
SOURCE: Scopus

Lebtahi, L., Patrício, P., Thome, N.
15063014400;6701711621;6603046084;
Special elements in a ring related to Drazin inverses
(2013) Linear and Multilinear Algebra, 61 (8), pp. 1017-1027. Cited 4 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84880275304&doi=10.1080%2f03081087.2012.728598&partnerID=40&md5=3c8b9380cb80e86f435c34e4974a30d2

DOI: 10.1080/03081087.2012.728598
ABSTRACT: In this article, the existence of the Drazin (group) inverse of an element a in a ring is analysed when amk = kan, for some unit k and m, n ∈ ℕ. The same problem is studied for the case when a* = kamk-1 and for the {k, s + 1}-potent elements. In addition, relationships with other special elements of the ring are also obtained. © 2013 Copyright Taylor and Francis Group, LLC.
AUTHOR KEYWORDS: Drazin inverse;  involutory element;  power;  ring
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Patrício, P., Hartwig, R.E.
6701711621;7005552138;
The (2,2,0) Drazin inverse problem
(2012) Linear Algebra and Its Applications, 437 (11), pp. 2755-2772. Cited 21 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84865675157&doi=10.1016%2fj.laa.2012.07.008&partnerID=40&md5=e60354df6634729054bb55955b400bf9

DOI: 10.1016/j.laa.2012.07.008
ABSTRACT: We consider the additive Drazin problem and we study the existence of the Drazin inverse of a two by two matrix with zero (2,2) entry. © 2012 Elsevier Inc. All rights reserved.
AUTHOR KEYWORDS: Block matrices;  Drazin inverse
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Mary, X., Patrício, P.
12782242200;6701711621;
The inverse along a lower triangular matrix
(2012) Applied Mathematics and Computation, 219 (3), pp. 886-891. Cited 12 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84867336890&doi=10.1016%2fj.amc.2012.06.060&partnerID=40&md5=3f73aae121648e53520158bc62065848

DOI: 10.1016/j.amc.2012.06.060
ABSTRACT: In this paper, we investigate the recently defined notion of inverse along an element in the context of matrices over a ring. Precisely, we study the inverse of a matrix along a lower triangular matrix, under some conditions. © 2012 Elsevier Inc. All rights reserved.
AUTHOR KEYWORDS: Dedekind-finite ring;  Generalized inverse;  Green's relations;  Inverse along an element;  Rings
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Patrício, P.
6701711621;
The Moore-Penrose inverse of a companion matrix
(2012) Linear Algebra and Its Applications, 437 (3), pp. 870-877. Cited 10 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84861101059&doi=10.1016%2fj.laa.2012.03.019&partnerID=40&md5=6f537658300b2120d1250c29f9bf822a

DOI: 10.1016/j.laa.2012.03.019
ABSTRACT: Necessary and sufficient conditions are given for the Moore-Penrose inverse of a companion matrix over an arbitrary ring to exist. © 2012 Elsevier Inc. All rights reserved.
AUTHOR KEYWORDS: Companion matrix;  Moore-Penrose inverse;  Rings;  Von Neumann inverse
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze, Green
SOURCE: Scopus

Hartwig, R.E., Patricio, P.
7005552138;6701711621;
A note on power bounded matrices
(2012) Electronic Journal of Linear Algebra, 23, art. no. 45, pp. 625-645. Cited 1 time.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84908257572&doi=10.13001%2f1081-3810.1546&partnerID=40&md5=173d81edf77b91eb9ef0a09ababee29e

DOI: 10.13001/1081-3810.1546
ABSTRACT: The concept of focal power is used to examine when a nonnegative power bounded matrix is periodic. © 2012, International Linear Algebra Society. All rights reserved.
AUTHOR KEYWORDS: Focal power;  Nonnegative;  Power bounded
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze
SOURCE: Scopus

Hartwig, R.E., Patricio, P.
7005552138;6701711621;
When does the moore-penrose inverse flip?
(2012) Operators and Matrices, 6 (1), pp. 181-192. Cited 11 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84555203695&doi=10.7153%2foam-06-13&partnerID=40&md5=d6aac0307d8ca7817f4fc8cb51f8f649

DOI: 10.7153/oam-06-13
ABSTRACT: In this paper, we give necessary and sufficient conditions for the matrix a *-regular ring, to have a Moore-Penrose inverse of four different types, corresponding to the four cases where the zero element can stand. In particular, we study the case where the Moore-Penrose inverse of the matrix flips. © element, Zagreb.
AUTHOR KEYWORDS: Moore-penrose inverse;  Rings;  Triangular matrices;  Von neumann regularity
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze
SOURCE: Scopus

Patrício, P., Hartwig, R.E.
6701711621;7005552138;
The (2, 2, 0) group inverse problem
(2010) Applied Mathematics and Computation, 217 (2), pp. 516-520. Cited 20 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-77955655795&doi=10.1016%2fj.amc.2010.05.084&partnerID=40&md5=bd5feffe22acce1f601edbd7a0e061e2

DOI: 10.1016/j.amc.2010.05.084
ABSTRACT: We characterize the existence of the group inverse of a two by two matrix with zero (2, 2) entry, over a ring by means of the existence of the inverse of a suitable function of the other three entries. Some special cases are derived. © 2010 Elsevier Inc. All rights reserved.
AUTHOR KEYWORDS: Block matrices;  Group inverse
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Patrício, P., Hartwig, R.E.
6701711621;7005552138;
The link between regularity and strong-pi-regularity
(2010) Journal of the Australian Mathematical Society, 89 (1), pp. 17-22. Cited 15 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-78049265748&doi=10.1017%2fS1446788710001448&partnerID=40&md5=81f5b656084f771e04d90886238338fe

DOI: 10.1017/S1446788710001448
ABSTRACT: It is shown that if all powers of a ring element a are regular, then a is strongly pi-regular exactly when a suitable word in the powers of a and their inner inverses is a unit. Copyright © Australian Mathematical Publishing Association Inc. 2010.
AUTHOR KEYWORDS: Drazin index;  Drazin inverse;  strongly pi-regular
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze, Green
SOURCE: Scopus

Castro-González, N., Mendes-Araújo, C., Patricio, P.
6602240142;8321983200;6701711621;
Generalized inverses of a sum in rings
(2010) Bulletin of the Australian Mathematical Society, 82 (1), pp. 156-164. Cited 39 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-77957342878&doi=10.1017%2fS0004972710000080&partnerID=40&md5=af597aefe9d9cf5e57fca1c65e44c150

DOI: 10.1017/S0004972710000080
ABSTRACT: We study properties of the Drazin index of regular elements in a ring with a unity 1. We give expressions for generalized inverses of 1ba in terms of generalized inverses of 1ab. In our development we prove that the Drazin index of 1ba is equal to the Drazin index of 1ab. © 2010 Australian Mathematical Publishing Association Inc.
AUTHOR KEYWORDS: Drazin index;  Drazin inverse;  EP elements;  reflexive inverse;  regular element
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze, Green
SOURCE: Scopus

Patrício, P., Araújo, C.M.
6701711621;8321983200;
Moore-Penrose invertibility in involutory rings: The case aa†=bb†
(2010) Linear and Multilinear Algebra, 58 (4), pp. 445-452. Cited 27 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-77951908037&doi=10.1080%2f03081080802652279&partnerID=40&md5=9390763043f3667ad78c12756d889c9d

DOI: 10.1080/03081080802652279
ABSTRACT: In this article, we consider Moore-Penrose invertibility in rings with a general involution. Given two von Neumann regular elements a, b in a general ring with an arbitrary involution, we aim to give necessary and sufficient conditions to aa†=bb†. As a special case, EP elements are considered. © 2010 Taylor & Francis.
AUTHOR KEYWORDS: *-regular rings;  Moore-Penrose inverse
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Patrício, P., Hartwig, R.E.
6701711621;7005552138;
Some additive results on Drazin inverses
(2009) Applied Mathematics and Computation, 215 (2), pp. 530-538. Cited 41 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-68949163053&doi=10.1016%2fj.amc.2009.05.021&partnerID=40&md5=6f5cdbbb5a2ebeeaf6e93ea8df7e1ecc

DOI: 10.1016/j.amc.2009.05.021
ABSTRACT: In this paper, some additive results on Drazin inverse of a sum of Drazin invertible elements are derived. Some converse results are also presented. © 2009 Elsevier Inc.
AUTHOR KEYWORDS: Block matrices;  Drazin inverse
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Patrício, P., Costa, A.V.
6701711621;6508035105;
On the Drazin index of regular elements
(2009) Central European Journal of Mathematics, 7 (2), pp. 200-205. Cited 30 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-66249133970&doi=10.2478%2fs11533-009-0015-6&partnerID=40&md5=96c157de3b451cd57748a84d4b355598

DOI: 10.2478/s11533-009-0015-6
ABSTRACT: It is known that the existence of the group inverse a# of a ring element a is equivalent to the invertibility of a2 a- + 1 - aa-, independently of the choice of the von Neumann inverse a- of a. In this paper, we relate the Drazin index of a to the Drazin index of a2a- + 1 - aa-. We give an alternative characterization when considering matrices over an algebraically closed field. We close with some questions and remarks. © Versita Warsaw and Springer-Verlag Berlin Heidelberg 2009.
AUTHOR KEYWORDS: Dedekind finite ring;  Drazin index;  Drazin inverse;  Regular ring
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus

Hartwig, R.E., Patricio, P.
7005552138;6701711621;
On Roth's pseudo equivalence over rings
(2007) Electronic Journal of Linear Algebra, 16, pp. 111-124. Cited 2 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-34248560968&doi=10.13001%2f1081-3810.1187&partnerID=40&md5=b7cb67ab5287439cde8c801bd7963de4

DOI: 10.13001/1081-3810.1187
ABSTRACT: The pseudo-equivalence of a block lower triangular matrix T = [T ij] over a regular ring and its block diagonal matrix D(T) = [T ii] is characterized in terms of suitable Roth consistency conditions. The latter can in turn be expressed in terms of the solvability of certain matrix equations of the form TiiX - YTjj = U ij.
AUTHOR KEYWORDS: Regular rings;  Roth equivalence
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze, Green
SOURCE: Scopus

Hartwig, R.E., Patrício, P., Puystjens, R.
7005552138;6701711621;6506793633;
Diagonalizing triangular matrices via orthogonal Pierce decompositions
(2005) Linear Algebra and Its Applications, 401 (1-3), pp. 381-391. Cited 6 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-16444387623&doi=10.1016%2fj.laa.2004.10.002&partnerID=40&md5=832cf3427497cdf8775f13fd56047bdd

DOI: 10.1016/j.laa.2004.10.002
ABSTRACT: A class of sufficient conditions is given to ensure that the sum a + b in a ring R, is equivalent to a sum x + y, which is an orthogonal Pierce decomposition. This is then used to show that a lower triangular matrix, with a regular diagonal is equivalent to its diagonal iff the matrix admits a lower triangular von Neumann inverse. © 2004 Elsevier Inc. All rights reserved.
AUTHOR KEYWORDS: Diagonalization;  Rings;  Triangular matrices;  Von Neumann regularity
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze, Green
SOURCE: Scopus

Patrício, P., Puystjens, R.
6701711621;6506793633;
Drazin-Moore-Penrose invertibility in rings
(2004) Linear Algebra and Its Applications, 389 (1-3), pp. 159-173. Cited 48 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-4043176373&doi=10.1016%2fj.laa.2004.04.006&partnerID=40&md5=5e93eae8108cb0fa7e3d02d85a45a016

DOI: 10.1016/j.laa.2004.04.006
ABSTRACT: Characterizations are given for elements in an arbitrary ring with involution, having a group inverse and a Moore-Penrose inverse that are equal and the difference between these elements and EP-elements is explained. The results are also generalized to elements for which a power has a Moore-Penrose inverse and a group inverse that are equal. As an application we consider the ring of square matrices of order m over a projective free ring R with involution such that Rm is a module of finite length, providing a new characterization for range-Hermitian matrices over the complexes. © 2004 Published by Elsevier Inc.
AUTHOR KEYWORDS: Core nilpotent decomposition;  Drazin;  EP elements;  Fitting decomposition;  Generalized inverses;  Moore-Penrose
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze, Green
SOURCE: Scopus

Patrício, P., Puystjens, R.
6701711621;6506793633;
Generalized invertibility in two semigroups of a ring
(2004) Linear Algebra and Its Applications, 377 (1-3), pp. 125-139. Cited 12 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-0242654066&doi=10.1016%2fj.laa.2003.08.004&partnerID=40&md5=cff3ced33a94b7d38d5a43b363c4b6ba

DOI: 10.1016/j.laa.2003.08.004
ABSTRACT: In [Linear Multilinear Algebra 43 (1997) 137], Puystjens and Hartwig proved that given a regular element t of a ring R with unity 1, then t has a group inverse if and only if u = t2t- + 1 - tt - is invertible in R if and only if v = t-t2 + 1-t-t is invertible in R. There, Hartwig posed the pertinent question whether the inverse of u and v could be directly related. Similar equivalences appear in the characterization of Moore-Penrose and Drazin invertibility, and therefore analogous questions arise. We present a unifying result to answer these questions not only involving classical invertibility, but also some generalized inverses as well. © 2003 Elsevier Inc. All rights reserved.
AUTHOR KEYWORDS: Corner rings;  Generalized invertibility;  Matrices over rings;  Semigroups
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze, Green
SOURCE: Scopus

Patrício, P.
6701711621;
The Moore-Penrose inverse of a factorization
(2003) Linear Algebra and Its Applications, 370 (SUPP), pp. 227-235. Cited 15 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-0038116166&doi=10.1016%2fS0024-3795%2803%2900391-4&partnerID=40&md5=ccc569217bdc823d7ece4d82c6b83412

DOI: 10.1016/S0024-3795(03)00391-4
ABSTRACT: In this paper, we consider the product of matrices PAQ, where A is von Neumann regular and there exist P′ and Q′ such that P′PA = A = AQQ′. We give necessary and sufficient conditions in order to PAQ be Moore-Penrose invertible, extending known characterizations. Finally, an application is given to matrices over separative regular rings. © 2003 Elsevier Science Inc. All rights reserved.
AUTHOR KEYWORDS: Factorization;  Matrices over rings;  Moore-Penrose invertibility;  Separative regular rings;  Von Neumann regularity
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze, Green
SOURCE: Scopus

Koliha, J.J., Patricio, P.
7003780999;6701711621;
Elements of rings with equal spectral idempotents
(2002) Journal of the Australian Mathematical Society, 72 (1), pp. 137-152. Cited 151 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-0036346975&doi=10.1017%2fS1446788700003657&partnerID=40&md5=e06ee8a5737a8669b8f93b5eff88e513

DOI: 10.1017/S1446788700003657
ABSTRACT: In this paper we define and study a generalized Drazin inverse xD for ring elements x, and give a characterization of elements a, b for which aaD = bbD. We apply our results to the study of EP elements in a ring with involution.
AUTHOR KEYWORDS: Associative ring;  Drazin inverse, Moore-Penrose inverse;  EP element;  Idempotent;  Involution
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze, Green
SOURCE: Scopus

Patrício, P.
6701711621;
The Moore-Penrose inverse of von Neumann regular matrices over a ring
(2001) Linear Algebra and Its Applications, 332-334, pp. 469-483. Cited 20 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-0035427186&doi=10.1016%2fS0024-3795%2801%2900298-1&partnerID=40&md5=a83cbbf6bb1bce56fb9deeddfadd9cf3

DOI: 10.1016/S0024-3795(01)00298-1
ABSTRACT: Necessary and sufficient conditions are given in order that a von Neumann regular matrix A over an arbitrary ring, and also the matrices arising in certain factorizations of A be Moore-Penrose invertible. © 2001 Elsevier Science Inc.
AUTHOR KEYWORDS: Matrices over rings;  Moore-Penrose invertibility;  Von Neumann regularity
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Bronze, Green
SOURCE: Scopus

Patrício, P., Puystjens, R.
6701711621;6506793633;
About the von Neumann regularity of triangular block matrices
(2001) Linear Algebra and Its Applications, 332-334, pp. 485-502. Cited 26 times.
https://www.scopus.com/inward/record.uri?eid=2-s2.0-0035427099&doi=10.1016%2fS0024-3795%2801%2900295-6&partnerID=40&md5=1270ce6ab3e6c085f93ae8f84d592028

DOI: 10.1016/S0024-3795(01)00295-6
ABSTRACT: Necessary and sufficient conditions are given for the von Neumann regularity of triangular block matrices with von Neumann regular diagonal blocks over arbitrary rings. This leads to the characterization of the von Neumann regularity of a class of triangular Toeplitz matrices over arbitrary rings. Some special results and a new algorithm are derived for triangular Toeplitz matrices over commutative rings. Finally, the Drazin invertibility of some companion matrices over arbitrary rings is considered, as an application. © 2001 Elsevier Science Inc.
AUTHOR KEYWORDS: Block matrices over rings;  Triangular Toeplitz matrices;  Von Neumann regularity
DOCUMENT TYPE: Article
PUBLICATION STAGE: Final
OPEN ACCESS: All Open Access, Green
SOURCE: Scopus