The weighted Drazin invertibility of rectangular matrixs over an arbitrary ring are studied.Some equivalent conditions and Characterizations are given for existence of the weighted Drazin inverse of a rectangular matr...The weighted Drazin invertibility of rectangular matrixs over an arbitrary ring are studied.Some equivalent conditions and Characterizations are given for existence of the weighted Drazin inverse of a rectangular matrix over an arbitrary ring.Moreover,the weighted Drazin inverse of a rectangular matrices product PAQ can be characterized and computed.This generalizes results obtained for the Drazin inverse of such product of square matrices.The results also apply to morphisms in(additive)categories.展开更多
On the basis of the paoers[3—7],this paper study the monotonicity problems for the positive semidefinite generalized inverses of the positive semidefinite self-conjugate matrices of quaternions in the Lowner partial ...On the basis of the paoers[3—7],this paper study the monotonicity problems for the positive semidefinite generalized inverses of the positive semidefinite self-conjugate matrices of quaternions in the Lowner partial order,give the explicit formulations of the monotonicity solution sets A{1;≥,T_1;≤B^(1)}and B{1;≥,T_2≥A^(1)}for the(1)-inverse,and two results of the monotonicity charac teriaztion for the(1,2)-inverse.展开更多
In this paper we study the properties of homotopy inverses of comultiplications and Mgebraic loops of co-H-spaces based on a wedge of spheres. We also investigate a method to construct new comultiplications out of old...In this paper we study the properties of homotopy inverses of comultiplications and Mgebraic loops of co-H-spaces based on a wedge of spheres. We also investigate a method to construct new comultiplications out of old ones by using a group action. We are primarily interested in the algebraic loops which have inversive, power-associative and Moufang properties for some comultiplications.展开更多
In this paper, we characterize left inverses and right inverses of a strong endomorphism of a graph under some certain Green's equivalence conditions. In addition, the number of them is also given.
Peal[2] shows that a sufficient and necessary condition on the existence of theMoore-Penrose inverse over any fields.Zhuang [3] generalize the result to any divisionrings.In this section we give another sufficient and...Peal[2] shows that a sufficient and necessary condition on the existence of theMoore-Penrose inverse over any fields.Zhuang [3] generalize the result to any divisionrings.In this section we give another sufficient and necessary condition on the existence ofthe Moore-Penrose inverse over any division rings.Our result can be regarded as an im-provement of Theorem lin[1].As a medium result,we also show a characterization ofthe{1,2}-inverse.展开更多
The properties and some equivalent characterizations of equal projection（ EP）, normal and Hermitian elements in a ring are studied by the generalized inverse theory. Some equivalent conditions that an element is EP ...The properties and some equivalent characterizations of equal projection（ EP）, normal and Hermitian elements in a ring are studied by the generalized inverse theory. Some equivalent conditions that an element is EP under the existence of core inverses are proposed. Let a∈R , then a is EP if and only if aa a^# = a^#aa . At the same time, the equivalent characterizations of a regular element to be EP are discussed.Let a∈R, then there exist b∈R such that a = aba and a is EP if and only if a∈R , a = a ba. Similarly, some equivalent conditions that an element is normal under the existence of core inverses are proposed. Let a∈R , then a is normal if and only if a^＊a = a a^＊. Also, some equivalent conditions of normal and Hermitian elements in rings with involution involving powers of their group and Moore-Penrose inverses are presented. Let a∈R ∩R^#, n∈N, then a is normal if and only if a^＊ a^＋（ a^#） n = a^# a＊（ a^＋） ^n. The results generalize the conclusions of Mosiet al.展开更多
The perturbation problem of generalized inverse is studied. And some new stability characteristics of generalized inverses were presented. It was also proved that the stability characteristics of generalized inverses ...The perturbation problem of generalized inverse is studied. And some new stability characteristics of generalized inverses were presented. It was also proved that the stability characteristics of generalized inverses were independent of the choice of the generalized inverse. Based on this result, two sufficient and necessary conditions for the lower semi-continuity of generalized inverses as the set-valued mappings are given.展开更多
Let R be a ring, * be an involutory function of the set of all finite matrices over R. In this paper, necessary and sufficient conditions are given for a matrix to have a (1,3)-inverse, (1,4)-inverse, or Moore-P enros...Let R be a ring, * be an involutory function of the set of all finite matrices over R. In this paper, necessary and sufficient conditions are given for a matrix to have a (1,3)-inverse, (1,4)-inverse, or Moore-P enrose inverse, relative to *. Some results about generalized inverses of matrices over division rings are generalized and improved.展开更多
In this paper,we study the displacement rank of the Core-EP inverse.Both Sylvester displacement and generalized displacement are discussed.We present upper bounds for the ranks of the displacements of the Core-EP inve...In this paper,we study the displacement rank of the Core-EP inverse.Both Sylvester displacement and generalized displacement are discussed.We present upper bounds for the ranks of the displacements of the Core-EP inverse.Numerical experiments are presented to demonstrate the efficiency and accuracy.展开更多
The Moore-Penrose inverse of a block k-circulant matrix whose blocks are arbitrary matrices are obtained when k has unit modulus. In the meantime. explicit formulae for finding group inverses of certain specified k-ci...The Moore-Penrose inverse of a block k-circulant matrix whose blocks are arbitrary matrices are obtained when k has unit modulus. In the meantime. explicit formulae for finding group inverses of certain specified k-circulant matrices are also given.展开更多
To characterize m-weak group inverses,several algebraic methods are used,such as the use of idempotents,one-side principal ideals,and units.Consider an element a within a unitary ring that possesses Drazin invertibili...To characterize m-weak group inverses,several algebraic methods are used,such as the use of idempotents,one-side principal ideals,and units.Consider an element a within a unitary ring that possesses Drazin invertibility and an involution.This paper begins by outlining the conditions necessary for the existence of the m-weak group inverse of a.Moreover,it explores the criteria under which a can be considered pseudo core invertible and weak group invertible.In the context of a weak proper*-ring,it is proved that a is weak group invertible if,and only if,a D can serve as the weak group inverse of au,where u represents a specially invertible element closely associated with a D.The paper also introduces a counterexample to illustrate that a D cannot universally serve as the pseudo core inverse of another element.This distinction underscores the nuanced differences between pseudo core inverses and weak group inverses.Ultimately,the discussion expands to include the commuting properties of weak group inverses,extending these considerations to m-weak group inverses.Several new conditions on commuting properties of generalized inverses are given.These results show that pseudo core inverses,weak group inverses,and m-weak group inverses are not only closely linked but also have significant differences that set them apart.展开更多
Element a in ring R is called centrally clean if it is the sum of central idempotent e and unit u.Moreover,a=e+u is called a centrally clean decomposition of a and R is called a centrally clean ring if every element o...Element a in ring R is called centrally clean if it is the sum of central idempotent e and unit u.Moreover,a=e+u is called a centrally clean decomposition of a and R is called a centrally clean ring if every element of R is centrally clean.First,some characterizations of centrally clean elements are given.Furthermore,some properties of centrally clean rings,as well as the necessary and sufficient conditions for R to be a centrally clean ring are investigated.Centrally clean rings are closely related to the central Drazin inverses.Then,in terms of centrally clean decomposition,the necessary and sufficient conditions for the existence of central Drazin inverses are presented.Moreover,the central cleanness of special rings,such as corner rings,the ring of formal power series over ring R,and a direct product ∏ R_(α) of ring R_(α),is analyzed.Furthermore,the central group invertibility of combinations of two central idempotents in the algebra over a field is investigated.Finally,as an application,an example that lists all invertible,central group invertible,group invertible,central Drazin invertible elements,and centrally clean elements of the group ring Z_(2)S_(3) is given.展开更多
We study a new class of group inverses determined by right c-regular elements.The new concept of right c-group inverses is introduced and studied.It is shown that every right c-group invertible element is group invert...We study a new class of group inverses determined by right c-regular elements.The new concept of right c-group inverses is introduced and studied.It is shown that every right c-group invertible element is group invertible,and an example is given to show that group invertible elements need not be right c-group invertible.The conditions that right c-group invertible elements are precisely group invertible elements are investigated.We also study the strongly clean decompositions of right c-group invertible elements.As applications,we give some new characterizations of abelian rings and directly finite rings from the point of view of right c-groupinverses.展开更多
In this paper, we present an extension of the so-called classical Sherman-Morrison-Woodbury (for short SMW) formula for bounded homogeneous generalized inverse in Banach spaces. Some particular cases and applications ...In this paper, we present an extension of the so-called classical Sherman-Morrison-Woodbury (for short SMW) formula for bounded homogeneous generalized inverse in Banach spaces. Some particular cases and applications will be also considered. Our results generalize the results of many authors for finite dimensional matrices and Hilbert space operators in the literature.展开更多
The group, Drazin and Koliha-Drazin inverses are particular classes of commuting outer inverses. In this note, we use the inverse along an element to study some spectral conditions related to these inverses in the cas...The group, Drazin and Koliha-Drazin inverses are particular classes of commuting outer inverses. In this note, we use the inverse along an element to study some spectral conditions related to these inverses in the case of bounded linear operators on a Banach space.展开更多
Motivated by pseudo strong Drazin inverses and strongly nil G-clean elements,we introduce the concept of pseudo strong Drazin inverses relative to central units in a ring.Basic properties of this new class of generali...Motivated by pseudo strong Drazin inverses and strongly nil G-clean elements,we introduce the concept of pseudo strong Drazin inverses relative to central units in a ring.Basic properties of this new class of generalized inverses are provided.In particular,we give characterizations of elements a,b∈R for which a^(Π)_(u)=b^(Π)_(u)(herein,a^(Π)_(u)=aua^(×)_(u),u∈Uc(R)and a×u is a pseudo strongly Drazin inverse relative to u of a.The definition of b^(Π)_(u) is similar to that of a^(Π)_(u)).Further,weighted pseudo strong Drazin inverses relative to central units are also considered.展开更多
In this paper,the authors derive the existence criteria and the formulae of the weighted Moore-Penrose inverse,the e-core inverse and the f-dual core inverse in rings.Also,new characterizations between weighted Moore-...In this paper,the authors derive the existence criteria and the formulae of the weighted Moore-Penrose inverse,the e-core inverse and the f-dual core inverse in rings.Also,new characterizations between weighted Moore-Penrose inverses and one-sided inverses along an element are given.展开更多
We introduce tensor generalized bilateral inverses(TGBIs)under the Einstein tensor product as an extension of generalized bilateral inverses(GBIs)in the matrix environment.Moreover,the TBGI class includes so far consi...We introduce tensor generalized bilateral inverses(TGBIs)under the Einstein tensor product as an extension of generalized bilateral inverses(GBIs)in the matrix environment.Moreover,the TBGI class includes so far considered composite generalized inverses(CGIs)for matrices and tensors.Applications of TBGIs for solving multilinear systems are presented.The characterizations and representations of the TGBI were studied and verified using a specific algebraic approach.Further,a few characterizations of known CGIs(such as the CMP,DMP,MPD,MPCEP,and CEPMP)are derived.The main properties of the TGBIs were exploited and verified through numerical examples.展开更多
In this paper we explicitly describe all the commuting pseudo inverses of a completely regular strong endomorphism of a graph from a viewpoint of combinatorics. The number of them is also given. In addition, a...In this paper we explicitly describe all the commuting pseudo inverses of a completely regular strong endomorphism of a graph from a viewpoint of combinatorics. The number of them is also given. In addition, a strong endomorphism of a graph, whose commuting pseudo inverse set coincides with its pseudo inverse set, is identified.展开更多
Let X be a Banach space over the complex field C and let T : X→X be a bounded linear operator with Ind(T) = k and R(Tk) closed. Denote the Drazin inverse of T by TD. Let T = T + δT, then TD has the simple expression...Let X be a Banach space over the complex field C and let T : X→X be a bounded linear operator with Ind(T) = k and R(Tk) closed. Denote the Drazin inverse of T by TD. Let T = T + δT, then TD has the simple expression TD = TD(I + δTTD)-1 = (I + TDδT)-1TD under certain hypotheses. The upper bound for the relative error ‖TD-TD‖/‖TD‖and for the solution to the operator equation: Tx = u (u∈R(TD)) is also considered.展开更多
文摘The weighted Drazin invertibility of rectangular matrixs over an arbitrary ring are studied.Some equivalent conditions and Characterizations are given for existence of the weighted Drazin inverse of a rectangular matrix over an arbitrary ring.Moreover,the weighted Drazin inverse of a rectangular matrices product PAQ can be characterized and computed.This generalizes results obtained for the Drazin inverse of such product of square matrices.The results also apply to morphisms in(additive)categories.
文摘On the basis of the paoers[3—7],this paper study the monotonicity problems for the positive semidefinite generalized inverses of the positive semidefinite self-conjugate matrices of quaternions in the Lowner partial order,give the explicit formulations of the monotonicity solution sets A{1;≥,T_1;≤B^(1)}and B{1;≥,T_2≥A^(1)}for the(1)-inverse,and two results of the monotonicity charac teriaztion for the(1,2)-inverse.
基金supported by Basic Science Research Program through the National Research Foundation of Korea (NRF)the Ministry of Education,Science and Technology (2010-0022035)
文摘In this paper we study the properties of homotopy inverses of comultiplications and Mgebraic loops of co-H-spaces based on a wedge of spheres. We also investigate a method to construct new comultiplications out of old ones by using a group action. We are primarily interested in the algebraic loops which have inversive, power-associative and Moufang properties for some comultiplications.
文摘In this paper, we characterize left inverses and right inverses of a strong endomorphism of a graph under some certain Green's equivalence conditions. In addition, the number of them is also given.
基金This work is Supported by NSF of Heilongjiang Province
文摘Peal[2] shows that a sufficient and necessary condition on the existence of theMoore-Penrose inverse over any fields.Zhuang [3] generalize the result to any divisionrings.In this section we give another sufficient and necessary condition on the existence ofthe Moore-Penrose inverse over any division rings.Our result can be regarded as an im-provement of Theorem lin[1].As a medium result,we also show a characterization ofthe{1,2}-inverse.
基金The National Natural Science Foundation of China(No.11371089)the Specialized Research Fund for the Doctoral Program of Higher Education(No.20120092110020)the Natural Science Foundation of Jiangsu Province(No.BK20141327)
文摘The properties and some equivalent characterizations of equal projection（ EP）, normal and Hermitian elements in a ring are studied by the generalized inverse theory. Some equivalent conditions that an element is EP under the existence of core inverses are proposed. Let a∈R , then a is EP if and only if aa a^# = a^#aa . At the same time, the equivalent characterizations of a regular element to be EP are discussed.Let a∈R, then there exist b∈R such that a = aba and a is EP if and only if a∈R , a = a ba. Similarly, some equivalent conditions that an element is normal under the existence of core inverses are proposed. Let a∈R , then a is normal if and only if a^＊a = a a^＊. Also, some equivalent conditions of normal and Hermitian elements in rings with involution involving powers of their group and Moore-Penrose inverses are presented. Let a∈R ∩R^#, n∈N, then a is normal if and only if a^＊ a^＋（ a^#） n = a^# a＊（ a^＋） ^n. The results generalize the conclusions of Mosiet al.
基金Project supported by the National Natural Science Foundation of China (Nos. 10571150 and 10271053)
文摘The perturbation problem of generalized inverse is studied. And some new stability characteristics of generalized inverses were presented. It was also proved that the stability characteristics of generalized inverses were independent of the choice of the generalized inverse. Based on this result, two sufficient and necessary conditions for the lower semi-continuity of generalized inverses as the set-valued mappings are given.
文摘Let R be a ring, * be an involutory function of the set of all finite matrices over R. In this paper, necessary and sufficient conditions are given for a matrix to have a (1,3)-inverse, (1,4)-inverse, or Moore-P enrose inverse, relative to *. Some results about generalized inverses of matrices over division rings are generalized and improved.
基金Supported by Guangxi Natural Science Foundation(2018GXNSFDA281023,2018GXNSFAA138181)High Level Innovation Teams and Distinguished Scholars in Guangxi Universities(GUIJIAOREN201642HAO)+3 种基金Graduate Research Innovation Project of Guangxi University for Nationalities(gxun-chxzs2018041,gxun-chxzs2019026)the Special Fund for Bagui Scholars of Guangxi(2016A17)the National Natural Science Foundation of China(61772006,12061015)the Science and Technology Major Project of Guangxi(AA17204096)。
文摘In this paper,we study the displacement rank of the Core-EP inverse.Both Sylvester displacement and generalized displacement are discussed.We present upper bounds for the ranks of the displacements of the Core-EP inverse.Numerical experiments are presented to demonstrate the efficiency and accuracy.
文摘The Moore-Penrose inverse of a block k-circulant matrix whose blocks are arbitrary matrices are obtained when k has unit modulus. In the meantime. explicit formulae for finding group inverses of certain specified k-circulant matrices are also given.
基金The National Natural Science Foundation of China(No.12171083,12071070)Qing Lan Project of Jiangsu Province and the Postgraduate Research and Practice Innovation Program of Jiangsu Province(No.KYCX22_0231).
文摘To characterize m-weak group inverses,several algebraic methods are used,such as the use of idempotents,one-side principal ideals,and units.Consider an element a within a unitary ring that possesses Drazin invertibility and an involution.This paper begins by outlining the conditions necessary for the existence of the m-weak group inverse of a.Moreover,it explores the criteria under which a can be considered pseudo core invertible and weak group invertible.In the context of a weak proper*-ring,it is proved that a is weak group invertible if,and only if,a D can serve as the weak group inverse of au,where u represents a specially invertible element closely associated with a D.The paper also introduces a counterexample to illustrate that a D cannot universally serve as the pseudo core inverse of another element.This distinction underscores the nuanced differences between pseudo core inverses and weak group inverses.Ultimately,the discussion expands to include the commuting properties of weak group inverses,extending these considerations to m-weak group inverses.Several new conditions on commuting properties of generalized inverses are given.These results show that pseudo core inverses,weak group inverses,and m-weak group inverses are not only closely linked but also have significant differences that set them apart.
基金The National Natural Science Foundation of China(No.12171083,11871145,12071070)the Qing Lan Project of Jiangsu Province。
文摘Element a in ring R is called centrally clean if it is the sum of central idempotent e and unit u.Moreover,a=e+u is called a centrally clean decomposition of a and R is called a centrally clean ring if every element of R is centrally clean.First,some characterizations of centrally clean elements are given.Furthermore,some properties of centrally clean rings,as well as the necessary and sufficient conditions for R to be a centrally clean ring are investigated.Centrally clean rings are closely related to the central Drazin inverses.Then,in terms of centrally clean decomposition,the necessary and sufficient conditions for the existence of central Drazin inverses are presented.Moreover,the central cleanness of special rings,such as corner rings,the ring of formal power series over ring R,and a direct product ∏ R_(α) of ring R_(α),is analyzed.Furthermore,the central group invertibility of combinations of two central idempotents in the algebra over a field is investigated.Finally,as an application,an example that lists all invertible,central group invertible,group invertible,central Drazin invertible elements,and centrally clean elements of the group ring Z_(2)S_(3) is given.
基金Supported by the National Natural Science Foundation of China(Grant No.12161049).
文摘We study a new class of group inverses determined by right c-regular elements.The new concept of right c-group inverses is introduced and studied.It is shown that every right c-group invertible element is group invertible,and an example is given to show that group invertible elements need not be right c-group invertible.The conditions that right c-group invertible elements are precisely group invertible elements are investigated.We also study the strongly clean decompositions of right c-group invertible elements.As applications,we give some new characterizations of abelian rings and directly finite rings from the point of view of right c-groupinverses.
文摘In this paper, we present an extension of the so-called classical Sherman-Morrison-Woodbury (for short SMW) formula for bounded homogeneous generalized inverse in Banach spaces. Some particular cases and applications will be also considered. Our results generalize the results of many authors for finite dimensional matrices and Hilbert space operators in the literature.
文摘The group, Drazin and Koliha-Drazin inverses are particular classes of commuting outer inverses. In this note, we use the inverse along an element to study some spectral conditions related to these inverses in the case of bounded linear operators on a Banach space.
基金Supported by Anhui Provincial Natural Science Foundation(No.2008085MA06)Key Laboratory of Financial Mathematics of Fujian Province University(Putian University)(No.JR202203)the Key project of Anhui Education Committee(No.gxyq ZD2019009)。
文摘Motivated by pseudo strong Drazin inverses and strongly nil G-clean elements,we introduce the concept of pseudo strong Drazin inverses relative to central units in a ring.Basic properties of this new class of generalized inverses are provided.In particular,we give characterizations of elements a,b∈R for which a^(Π)_(u)=b^(Π)_(u)(herein,a^(Π)_(u)=aua^(×)_(u),u∈Uc(R)and a×u is a pseudo strongly Drazin inverse relative to u of a.The definition of b^(Π)_(u) is similar to that of a^(Π)_(u)).Further,weighted pseudo strong Drazin inverses relative to central units are also considered.
基金supported by the National Natural Science Foundation of China(Nos.11971294,11801124)China Postdoctoral Science Foundation(No.2020M671068)the Natural Science Foundation of Anhui Province(No.1808085QA16)。
文摘In this paper,the authors derive the existence criteria and the formulae of the weighted Moore-Penrose inverse,the e-core inverse and the f-dual core inverse in rings.Also,new characterizations between weighted Moore-Penrose inverses and one-sided inverses along an element are given.
基金supported by the Science and Engineering Research Board(SERB),Department of Science and Technology,India(Grant No.EEQ/2022/001065)supported by the Science and Engineering Research Board(SERB),Department of Science and Technology,India(Grant No.SUR/2022/004357)+1 种基金supported by the Science Fund of the Republic of Serbia(Grant No.7750185,Quantitative Automata Models:Fundamental Problems and Applications-QUAM)supported by the Ministry of Science and Higher Education of the Russian Federation(Grant No.075-15-2022-1121).
文摘We introduce tensor generalized bilateral inverses(TGBIs)under the Einstein tensor product as an extension of generalized bilateral inverses(GBIs)in the matrix environment.Moreover,the TBGI class includes so far considered composite generalized inverses(CGIs)for matrices and tensors.Applications of TBGIs for solving multilinear systems are presented.The characterizations and representations of the TGBI were studied and verified using a specific algebraic approach.Further,a few characterizations of known CGIs(such as the CMP,DMP,MPD,MPCEP,and CEPMP)are derived.The main properties of the TGBIs were exploited and verified through numerical examples.
文摘In this paper we explicitly describe all the commuting pseudo inverses of a completely regular strong endomorphism of a graph from a viewpoint of combinatorics. The number of them is also given. In addition, a strong endomorphism of a graph, whose commuting pseudo inverse set coincides with its pseudo inverse set, is identified.
基金Supported by National Natural Science Foundation of China(19871029)
文摘Let X be a Banach space over the complex field C and let T : X→X be a bounded linear operator with Ind(T) = k and R(Tk) closed. Denote the Drazin inverse of T by TD. Let T = T + δT, then TD has the simple expression TD = TD(I + δTTD)-1 = (I + TDδT)-1TD under certain hypotheses. The upper bound for the relative error ‖TD-TD‖/‖TD‖and for the solution to the operator equation: Tx = u (u∈R(TD)) is also considered.
