Xiaoji Liu

• Prof. Dr.
• Lecturer at Guangxi University for Nationalities

In this paper, we study the C-eigenvalue via the tensor C-product. Definitions of the C-eigenvalue and Hermitian tensor are proposed. We also present the C-Jordan canonical form of the third-order F-square tensors and a commutative tensor family based on the C-product. Then, we prove some C-eigenvalues inequalities for Hermitian tensors, including...

In this paper, we present a new concept of the generalized core orthogonality (called the C-S orthogonality) for two generalized core invertible matrices A and B. A is said to be C-S orthogonal to B if ASB=0 and BAS=0, where AS is the generalized core inverse of A. The characterizations of C-S orthogonal matrices and the C-S additivity are also pro...

We define the T-MPWG inverse of third-order F-square tensors by using the T-core EP decomposition of tensors via the T-product. Then, we present some characterizations and properties of the T-MPWG inverse. Moreover, the Cayley-Hamilton theorem of the third-order tensors is extended to T-MPWG inverses. Examples are also given to illustrate these res...

In this paper, we introduce a new generalized inverse, which is called 1WG inverse of complex square matrices. We investigate the existence and uniqueness for the 1WG inverse and give some characterizations, representations, and properties of it. Next, by using the core-EP decomposition, we discuss the relationships between the 1WG inverse and othe...

In this paper, we present a new concept of the generalized core orthogonality (called the C-S orthogonality) for two generalized core invertible matrices $A$ and $B$. $A$ is said to be C-S orthogonal to $B$ if $A^{\tiny\textcircled{S}}B=0$ and $BA^{\tiny\textcircled{S}}=0$, where $A^{\tiny\textcircled{S}}$ is the generalized core inverse of $A$. Th...

In this article, we study the m {\mathfrak{m}} -WG ∘ {}^{\circ } inverse which presents a generalization of the m {\mathfrak{m}} -WG inverse in the Minkowski space. We first show the existence and the uniqueness of the generalized inverse. Then, we discuss several properties and characterizations of the m {\mathfrak{m}} -WG ∘ {}^{\circ } inverse by...

In this paper, we introduce the dual r-rank decomposition of the dual matrix, get its existence conditions and equivalent forms of the decomposition. Then we derive some characteristics of dual Moore–Penrose generalized inverse (DMPGI). Based on DMPGI, we introduce one special dual matrix (dual EP matrix). By applying the dual r-rank composition, w...

This paper studies the issues about the Moore–Penrose inverse of tensors with the M-product. The goal of this paper is threefold. Firstly, we define the Moore–Penrose inverse of tensors under the M-product, and obtain several formulas for the Moore–Penrose inverse of tensors. In addition, we study the least square and minimum-norm solutions of the...

In this paper, we introduce a new generalized inverse, called the G-MPCEP inverse of a complex matrix. We investigate some characterizations, representations, and properties of this new inverse. Cramer’s rule for the solution of a singular equation A x = B is also presented. Moreover, the determinantal representations for the G-MPCEP inverse are st...

In this paper, we introduce one type of matrix, called the weak group-star matrix. We investigate the characterizations, representations, and properties of the matrix. A variant of the successive matrix squaring computational iterative scheme is given for calculating the weak group-star matrix. Moreover, the Cramer?s rule for the solution of a sing...

This paper studies the issues about the generalized inverses of tensors under the C-Product. The aim of this paper is threefold. Firstly, this paper present the definition of the Moore-Penrose inverse, Drazin inverse of tensors under the C-Product. Moreover, the inverse along a tensor is also introduced. Secondly, this paper gives some other expres...

In this article, specific definitions of the Moore-Penrose inverse, Drazin inverse of the quaternion tensor and the inverse along two quaternion tensors are introduced under the T-product. Some characterizations, representations and properties of the defined inverses are investigated. Moreover, algorithms are established for computing the Moore-Pen...

Recently, Ferreyra and Malik (Core and strongly core orthogonal matrices. Linear Multilinear Algebra. 2021;1–16. doi:10.1080/03081087.2021.1902923) have proved that if A is strongly core orthogonal to B, then rk(A+B)=rk(A)+rk(B) and (A+B)◯#=A◯#+B◯#. But whether the reverse holds is an open question. In this paper, we solve the problem completely an...

In this paper, we introduce the dual $r$-rank decomposition of dual matrix, get its existence condition and equivalent form of the decomposition, as well as derive some characterizations of dual Moore-Penrose generalized inverse(DMPGI). Based on DMPGI, we introduce one special dual matrix(dual EP matrix). By applying the dual $r$-rank decomposition...

In this paper, we introduce a new generalized inverse, called MPWG inverse of a complex square matrix. We investigate characterizations, representations, and properties for this new inverse. Then, by using the core-EP decomposition, we discuss the relationships between MPWG inverse and other generalized inverses. A variant of the successive matrix...

In this paper, we introduce the \({{\mathfrak {m}}}\)-core-EP inverse in Minkowski space, consider its properties, and get several sufficient and necessary conditions for the existence of the \({{\mathfrak {m}}}\)-core-EP inverse. We give the \({{\mathfrak {m}}}\)-core-EP decomposition in Minkowski space, and note that not every square matrix has t...

In this paper, we consider the matrix inequality AXA≤?A in the star, sharp and core partial orders, respectively. We get general solutions of those matrix inequalities and prove D*⊆S* and D#⊆S#, although DO#⊈SO#.

The inconsistent or consistent general fuzzy matrix equation are studied in this paper. The aim of this paper is threefold. Firstly, general strong fuzzy matrix solutions of consistent general fuzzy matrix equation are derived, and an algorithm for obtaining general strong fuzzy solutions of general fuzzy matrix equation by Core-EP inverse is also...

In this paper, we introduce the m-core-EP inverse in Minkowski space, consider its properties, and get several sufficient and necessary conditions for the existence of the m-core-EP inverse. We give the m-core-EP decomposition in Minkowski space, and note that not every square matrix has the decomposition. Furthermore, by applying the m-core-EP inv...

On the basis of Löwner partial order and core partial order, we introduce a new partial order: LC partial order. By applying matrix decomposition, core inverse, core partial order, and Löwner partial order, we give some characteristics of LC partial order, study the relationship between LC partial order and Löwner partial order under constraint con...

This paper presents a method for solving fuzzy linear systems, where the coefficient matrix is an \(n\times n\) real matrix, using a block structure of the Core inverse, and we use the Hartwig–Spindelböck decomposition to obtain the Core inverse of the coefficient matrix A. The aim of this paper is twofold. First, we obtain a strong fuzzy solution...

In this article, we will present some results relating sharp ordering (a≤♯b) and investigate the properties of one-sided cyclic ideals in rings. Necessary and sufficient conditions for the invertibility and the group invertibility of the linear combination c1a+c2b are given, where a is below b under the sharp order, a and b are elements of an algeb...

In this paper,we introduce the weak group matrix defined by the one commutable with its weak group inverse, and consider properties and characterizations of the matrix by applying the core-EP decomposition. In particular,the set of weak group matrices is more inclusive than that of group matrices. We also derive some characterizations of p-EP matri...

In this paper, we introduce the m-core inverse in the Minkowski space, and get a sufficient and necessary condition for the existence of the inverse and some other related properties. Furthermore, by using the inverse, we introduce the m-core partial ordering and obtain solutions (or restricted least squares solutions) of some matrix equations in t...

Based on the conditions ab2=0 and bπ(ab)∈Ad, we derive that (ab)n, (ba)n, and ab+ba are all generalized Drazin invertible in a Banach algebra A, where n∈N and a and b are elements of A. By using these results, some results on the symmetry representations for the generalized Drazin inverse of ab+ba are given. We also consider that additive propertie...

The perturbation analysis of the differential for the Drazin inverse of the matrix-value function A(t)∈Cn×n is investigated. An upper bound of the Drazin inverse and its differential is also considered. Applications to the perturbation bound for the solution of the matrix-value function coefficients some matrix equations are given.

In this paper, we present a unique polar-like decomposition theorem for rectangular complex matrices. Applying this decomposition, we define on the set of rectangular matrices a new partial ordering called WL(weak L?wner) partial order ? an extension of the GL(generalized L?wner) partial order, and derive some basic properties of the new partial or...

In this paper, we introduce the notion of the EP-nilpotent decomposition and present some of its applications. By applying the decomposition, we introduce two partial orders (the E-N partial order and the E-S partial order) and characterize their properties. The two partial orders above are non-minus-type partial orders, although the method used to...

We investigate two iterative methods for computing the DMP inverse. The necessary and sufficient conditions for convergence of our schemes are considered and the error estimate is also derived. Numerical examples are given to test the accuracy and effectiveness of our methods.

In this paper, we recall and extend some tensor operations. Then, the generalized inverse of tensors is established by using tensor equations. Moreover, we investigate the least-squares solutions of tensor equations. An algorithm to compute the Moore–Penrose inverse of an arbitrary tensor is constructed. Finally, we apply the obtained results to hi...

In this paper, we use the well-known Löwner order and the core partial order to introduce a new partial order on the class of core matrices which is not dominated by any of the known matrix partial orders. We characterize , study its relations with the Löwner partial order under constraints, and exemplify its differences with other partial orders.

In this paper, we consider perturbation analysis for the generalized Drazin inverse of an operator in Banach space. An necessary and sufficient condition for the generalized Drazin invertible is given. The upper bound is given under some certain conditions, and a relative perturbation bound is also considered.

In this paper, we investigate additive properties of the generalized Drazin inverse in a Banach algebra A. We find explicit expressions for the generalized Drazin inverse of the sum a + b, under new conditions on a; b, ∈ A.

Representations of 1,2,3 -inverses, 1,2,4 -inverses, and Drazin inverse of a partitioned matrix M=ABCD related to the generalized Schur complement are studied. First, we give the necessary and sufficient conditions under which 1,2,3 -inverses, 1,2,4 -inverses, and group inverse of a 2×2 block matrix can be represented in the Banachiewicz-Schur form...

In this article, we characterize the involutiveness of the linear combination of the form a1A1 + a2A2 when a1; a2 are nonzero complex numbers, A1 is a quadratic or tripotent matrix, and A2 is arbitrary, under certain properties imposed on A1 and A2.

In this paper, we study partial orders in terms of the core-nilpotent decomposition. We derive some characterizations of the C-N partial ordering, create a new partial ordering (the G-Drazin partial ordering) which is a generalization of C-N partial ordering, and give some properties and characterizations of the G-Drazin partial ordering.

In this paper, we investigate additive results of the Drazin inverse of elements in a ring R. Under the condition ab = ba, we show that a + b is Drazin invertible if and only if aaD(a+b) is Drazin invertible, where the superscript D means the Drazin inverse. Furthermore we find an expression of (a + b)D. As an application we give some new represent...

We present some new representations for the generalized Drazin inverse of a block matrix in a Banach algebra under conditions weaker than those used in resent papers on the subject.

In this note, we revisit the core inverse and the core partial ordering introduced by Baksalary and Trenkler [Linear Multilinear Algebra. 2010;58:681–697]. We prove that the core inverse of is the unique solution of and , and establish several characterizations of the core inverse, the core partial ordering and the reverse order law for the core in...

In this paper, we derive an interesting result that the mixed-type reverse order laws (Formula presented.) hold if and only if the bounded linear operator products (Formula presented.) is invariant, where (Formula presented.) is taken, for instance, as (Formula presented.), (Formula presented.), (Formula presented.), (Formula presented.), (Formula...

We investigate additive properties of the generalized Drazin inverse in a Banach algebra . We find explicit expressions for the generalized Drazin inverse of the sum , under new conditions on . As an application we give some new representations for the generalized Drazin inverse of an operator matrix.

The main aim of this paper is to compute the generalized inverse over Banach spaces by using semi-iterative method and to present the error bounds of the semi-iterative method for approximating .

Let {a_n} be a sequence of group invertible elements of a unital C*� algebra A that converges to a. We present some equivalent conditions for the group invertibility of a and for the convergence of {a_n^\#} to a#.

We find a simple canonical form for EP complex matrices A and B under simultaneous unitary equivalence.

We investigate a new higher order iterative method for computing the generalized inverse A T, S (2) for a given matrix A. We also discuss how the new method could be applied for finding approximate inverses of nonsingular square matrices. Analysis of convergence is included to show that the proposed scheme has at least fifteenth-order convergence....

We investigate further the invariance properties of the bounded linear operator product and its range with respect to the choice of the generalized inverses and of bounded linear operators. Also, we discuss the range inclusion invariance properties of the operator product involving generalized inverses.

In this paper, we use the Drazin inverse to derive some new equivalences of the reverse order law for the group inverse in unitary rings. Moreover, if the ring has an involution, we present more equivalences when both involved elements are EP.

In this paper, we give explicit expressions of (P ± Q)d of two matrices P and Q, in terms of P; Q; Pd and Qd, under the condition that PQ = P2, and apply the result to finding an explicit representation for the Drazin inverse of a 2 x 2 block matrix (Equation Presented) under some conditions. © 2015, Forum-Editrice Universitaria Udinese SRL. All ri...

We find expressions for many types of generalized inverses of an arbitrary square complex matrix by using two representations given in [Benítez J. A new decomposition for square matrices. Electron. J. Linear Algebra. 2010;20:207–225] and in [Hartwig RE, Spindelböck K. Matrices for which and commute. Linear Multilinear Algebra. 1984;14:241–256].

The main aim of this paper is to provide a higher-order convergent iterative method in order to calculate the generalized inverse of a given matrix. We extend the iterative method proposed in Li et al. [W.G. Li, Z. Li, A family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non-square matrix, Ap...

Cen (Math. Numer. Sin. 29:39–48, 2007) defined a weighted group inverse of rectangular matrices. For given matrices A∈C m×n and W∈C n×m , if X∈C m×n satisfies $$( W_{1} )\ AWXWA=A, \qquad ( W_{2} ) \ XWAWX=X,\qquad ( W_{3} )\ AWX=XWA $$ then X is called the W-weighted group inverse, which is denoted by \(A_{W}^{\#}\). In this paper, for given recta...

In this paper, using some block-operator matrix techniques, we give necessary and sufficient conditions for the reverse order law to hold for {1, 2, 3}-and {1, 2, 4}-inverses of bounded operators on Hilbert spaces. Furthermore, we present some new equivalents of the reverse order law for the Moore-Penrose inverse.

In this article, we consider some representations of {1,3},{1,4},{1,2,3}{1,3},{1,4},{1,2,3} and {1,2,4}{1,2,4}-inverses of the partitioned matrix M which are equivalent to some rank additivity conditions. We present the application of these results to generalizations of the Sherman–Morrison–Woodbury-type formulae.

In this paper, we investigate additive properties for the generalized Drazin inverse in a Banach algebra A . We give some representations for the generalized Drazin inverse of a + b, where a and b are elements of A under some new conditions, extending some known results.

We give a very short proof of the main result of J. Benı´tez, A new decomposition for square matrices, Electron. J. Linear Algebra 20 (2010) 207–225. Also, we present some consequences of this result.

This paper presents a full rank factorization of a 2 × 2 block matrix without any restriction concerning the group inverse. Applying this factorization, we obtain an explicit representation of the group inverse in terms of four individual blocks of the partitioned matrix without certain restriction. We also derive some important coincidence theor...

A higher-order convergent iterative method is provided for calculating the generalized inverse over Banach spaces. We also use this iterative method for computing the generalized Drazin inverse in Banach algebra. Moreover, we estimate the error bounds of the iterative methods for approximating or .

We give explicit expressions of of two matrices and , in terms of , , , and , , under the condition that , and apply the result to finding an explicit representation for the Drazin inverse of some block matrix.

We investigate successive matrix squaring (SMS) algorithms for computing the generalized inverse of a given matrix .

Let {A m } m=1 ∞ be a sequence of complex group invertible matrices that converges to A. We characterize when A is group invertible and {A m # } m=1 ∞ converges to A # in terms of the canonical angles between A m and A m * , where X # denotes the group inverse of the matrix X. We compare this characterization with some known characterizations of th...

By using the principle of simultaneous diagonalization of two commuting K-tripotent matrices and the method of partition of matrix, the group invertibility of the nonlinear combination of two commuting tripotent matrices was mainly discussed and the formula was got. Further, all the cases that the more generalized combination of tripotent matrices...

In this paper, we give necessary and sufficient conditions for the absorption laws in terms of {1},{1,2},{1,3}{1},{1,2},{1,3} and {1,4}{1,4}-inverses. Also, we consider the various types of mixed absorption law for the generalized inverses.

In this paper, we construct a new iterative method for computing the Drazin inverse and deduce the necessary and sufficient condition for its convergence to A(d). Moreover, we present the error bounds of the iterative methods for approximating A(d).

We investigate the relative perturbation bound of the group inverse and also consider the perturbation bound of the generalized Schur complement in a Banach algebra.

A possible type of the operator splitting is studied. Using this operator splitting, we introduce some properties and representations of generalized inverses as well as iterative method for computing various solutions of the restricted linear operator system Ax=b, x∈T, where A∈ℒ(X,Y) and T is an arbitrary but fixed subspace of X.

We consider the perturbation bounds for the Moore-Penrose inverse of a given operator on a Hilbert space and apply these results to the relative errors of the minimum norm least squares solution of the equation Ax=b.

The reverse order laws for {1, 3, 4}-generalized inverses of a product of two operators have been studied by Wang et al. [J. Wang, H. Zhang, G. Ji, A generalized reverse order law for the products of two operators, Journal of Shaanxi Normal University, 38 (4) (2010), 13–17].In this paper using a block-operator matrix technique we study mixed-type r...

In this note, some expressions and characterizations for the weighted group inverses A W# of operator A by using the technique of block operator matrix are given, three iterative methods for computing A W# are established, and the necessary and sufficient conditions for iterative convergence to A W# are discussed.

Let ℛ be a ring and a, b∈ℛ satisfy aba = a and bab = b. We characterize when ab − ba is invertible. This study is specialized when ℛ has an involution and when b is the Moore–Penrose inverse of a.

In this paper, we give explicit expressions of (P+Q)(d) of two matrices P and Q, in terms of P,Q,P-d and Q(d), under the conditions that PQ(2) = -p(2)Q, p(2)Q(2) = 0 and PQ(2) = P(2)Q, QPQ = 0, p(2)Q(2) = 0. AMS classification: 15A09

We consider the various representations of the outer inverse of an operator A on Banach spaces.

We present the explicit expressions of the perturbation of the Drazin inverse under different conditions. Also, we give the upper bounds of ∥(A+E) D -A D ∥ P /∥A D ∥ P for these cases.

We investigate the determinantal representation by exploiting the limiting expression for the generalized inverse 𝐴(2)𝑇,𝑆. We show the equivalent relationship between the existence and limiting expression of 𝐴(2)𝑇,𝑆 and some limiting processes of matrices and deduce the new determinantal representations of 𝐴(2)𝑇,𝑆, based on some analog of the class...

We investigate the generalized Drazin inverse of A − C B over Banach spaces stemmed from the Drazin inverse of a modified matrix and present its expressions under some conditions.

We discuss the following problem: when aP+bQ+cPQ+dQP+ePQP+fQPQ+gPQPQ of idempotent matrices P and Q, where a,b,c,d,e,f,g∈ℂ and a≠0, b≠0, is group involutory.

In this paper, we present the perturbation bounds and the relative error bound of the generalized inverse AT,S(2) with respect to the Frobenius norm, and apply the results to the relative errors of the solution of the general restricted linear equation.

We deduce the explicit expressions for ${(P+Q)}^{D}$ and ${(PQ)}^{D}$ of two matrices $P$ and $Q$ under the conditions ${P}^{2}Q=PQP$ and ${Q}^{2}P=QPQ$ . Also, we give the upper bound of ${||{(P+Q)}^{D}-{P}^{D}||}_{2}$ .

Let T 1 and T 2 be two n × n tripotent matrices and c 1, c 2 two nonzero complex numbers. We mainly study the nonsingularity of combinations T = c 1T 1 + c 2T 2 - c 3T 1T 2 of two tripotent matrices T 1 and T 2, and give some formulae for the inverse of c 1T 1 + c 2T 2 - c 3T 1T 2 under some conditions. Some of these results are given in terms of g...

Let X † denotes the Moore--Penrose pseudoinverse of a matrix X. We study a number of situations when (aA + bB)† = aA + bB provided a, b    {0} and A, B are n × n complex matrices such that A † = A and B † = B.

Let P and Q be two complex matrices satisfying P2=P and Q2=Q. For a,b nonzero complex numbers such that aP+bQ is diagonalizable, we relate the spectrum of aP+bQ to the spectra of P−Q, PQ, PQP and PQ−QP.

In this paper, we derive several representations of {2}-inverses of A using full-rank matrices A *,β and A α,*. Several representations of the generalized inverse A(2)T,S{A^{(2)}_{T,S}} are given under rank conditions using full-rank matrices G *,β and G α,*. A(2)T,S{A^{(2)}_{T,S}} inverse

In this note, additive results are presented for the generalized Drazin inverse in Ba- nach algebra. Necessary and sufficient conditions are given for the generalized Drazin invertibility of the sum of two commuting generalized Drazin invertible elements. These results are a generalization of the results from the paper [C.Y. Deng and Y. Wei. New ad...

In this paper, we construct iterative methods for computing the generalized inverse A over Banach spaces, and also for computing the generalized Drazin inverses ad of Banach algebra element a. Moreover, we estimate the error bounds of the iterative methods for approximating A or ad. Copyright © 2011 John Wiley & Sons, Ltd.

Some formulas are found for the group inverse of aP+bQ, where P and Q are two nonzero group invertible complex matrices satisfying certain conditions and a,b nonzero complex numbers.

Let S1 = A-BD†C and S2 = D-CA†B be the associated Schur complements of. In this paper, we derive necessary and sufficient conditions for S1 = 0 imply S2 = 0 by using generalized inverses of matrices and singular value decompositions.

We derive a very short expression for the group inverse of a 1 + ··· + a n when a 1, … , a n are elements in an algebra having group inverse and satisfying a i a j  = 0 for i < j. We apply this formula in order to find the group inverse of 2 × 2 block operators under some conditions.

In this note, we consider a family of iterative formula for computing the weighted Minskowski inverses A M,N in Min-skowski space, and give two kinds of iterations and the necessary and sufficient conditions of the convergence of iterations. Keywords—iterative method, the Minskowski inverse, A M,N in-verse.

In this paper, we present the perturbation bounds and the relative error bound of the generalized inverse A ð2Þ T;S with respect to the Frobenius norm, and apply the results to the relative errors of the solution of the general restricted linear equation.

We discuss the following problem: when aP+bQ+cPQ+dQP+ePQP+fQPQ+gPQPQ of idempotent matrices P and Q, where a,b,c,d,e,f,g∈ℂ and a≠0, b≠0, is group involutory.

Let S 1 =A-BD + C and S 2 =D-CA + B be the associated Schur complements of M=ABmathbbCD. We derive necessary and sufficient conditions for S 1 =0 imply S 2 =0 by using generalized inverses of matrices and singular value decompositions.