Moore-Penrose Inverses and Group Inverses of Block k-Circulant Matrices

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.

REVERSE ORDER LAW FOR GROUP INVERSE OF PRODUCT OF MATRICES ON SKEW FIELDS

Let Kn×n be the set of all n×n matrices and Krn×n the set {A∈Kn×n|rankA=r} on askew field K. Zhuang [1] denotes by A# the group inverse of A∈Kn×n which is the solu-tion of the euqations:AXA=A, XAX=X, AX=AX.

Expression for the Group Inverse of the 2×2 Block Matrix

In this paper, we present a method how to get the expression for the group inverse of 2×2 block matrix and get the explicit expressions of the block matrix (A C B D) under some conditions.

Group Inverse of 2 × 2 Block Matrices over Minkowski Space M

Necessary and sufficient conditions for the existence of the group inverse of the block matrix in Minkowski Space are studied, where are both square and . The representation of this group inverse and some related additive results are also given.

Characterizations of m-weak group inverses

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.

Weighted weak group inverse for Hilbert space operators

We present the weighted weak group inverse,which is a new generalized inverse of operators between two Hilbert spaces,and we extend the notation of the weighted weak group inverse for rectangular matrices.Some characterizations and representations of the weighted weak group inverse are investigated.We also apply these results to define and study the weak group inverse for a Hilbert space operator.Using the weak group inverse,we define and characterize various binary relations.

Group inverses for some 2 × 2 block matrices over rings

Abstract We first consider the group inverses of the block matrices（AB0C）over a weakly finite ring. Then we study the sufficient and necessary conditions for the existence and the representations of the group inverses of the block matrices（ABCD）over a ring with unity 1 under the following conditions respectively： （i） B = C, D = 0,B#and（BπA0#both exist; （ii） B is invertible and m = n;（iii）A#and （D - CA#B）# both exist, C = CAA#, where A and D are m × m and n × n matrices, respectively.

MODIFIED NEWTON＇S ALGORITHM FOR COMPUTING THE GROUP INVERSES OF SINGULAR TOEPLITZ MATRICES

Newton＇s iteration is modified for the computation of the group inverses of singular Toeplitz matrices. At each iteration, the iteration matrix is approximated by a matrix with a low displacement rank. Because of the displacement structure of the iteration matrix, the matrix-vector multiplication involved in Newton＇s iteration can be done efficiently. We show that the convergence of the modified Newton iteration is still very fast. Numerical results are presented to demonstrate the fast convergence of the proposed method.

The Representation of Group Inverse with Affine Combination

In this paper, we first give two equalities in the operation of determinant. Using the expression of group inverse with full-rank factorization Ag = F（GF）^-2G and the Cramer rule of the nonsingular linear system Ax = b, we present a new method to prove the representation of group inverse with arlene combinationAg=∑（I,J）∈N（A） 1/υ^2det（A）IJ ajd AJI.A numerical example is given to demonstrate that the formula is efficient.

On Nonsingularity and Group Inverse of Linear Combinations of Generalized and Hypergeneralized Projectors

In this paper, some conditions for the nonsingularity and group inverses of linear combinations of generalized and hypergeneralized projectors are established. Moreover, some formulae for the inverses and group inverses of them are derived. The work of this paper extends some previous results.

Characterizations of EP, normal and Hermitian elements in rings using generalized inverses

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.

On Characterizations of Drazin Inverse

Let φ be a pre-additive category. Assume that φ： X→X is a morphism of φ. In this paper, we give the necessary and sufficient conditions for φ to have the Drazin inverse by using the von Neumann regular inverse for the φ^k, and extend a result by Puystjens and Hartwig from the group inverse to Drazin inverse.

Centrally clean elements and central Drazin inverses in a ring

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.

ABOUT THE INVERSION FORMULA ON THE LIE GROUP SL (2, R)

Harish-Chandra have got a Fourier inversion formula for C-c(infinity)(SL (2, R)). In this paper, we give a discussion on approximation identity kernels on SL(2, R) and get some properties of their Fourier transforms, and then, making use of these properties and Harish-Chandra's result, we prove that the Fourier inversion formula obtained by Harish-Chandra is also valid for C-o(3)(SL,2, R)).

Representations of Free Groups on Rational Line Q

A.M.W. Glass and S.H.McCleary have given the 2 transitive representation of the countable free l group F η(1<η≤ω 0 ).In this paper we shall give the highly ordered transitive representation of countable free groups on the rational line Q, which generalizes their results. As applications,we obtain the highly ordered transitive representation for the direct product of countable free groups,and the inverse limit of countable free groups must be an action on the set Q.

Generalized Inverses and Units in a Unitary Ring

Let R be a unitary ring and a,b∈R with ab=0.We find the 2/3 property of Drazin invertibility:if any two of a,b and a+b are Drazin invertible,then so is the third one.Then,we combine the 2/3 property of Drazin invertibility to characterize the existence of generalized inverses by means of units.As applications,the need for two invertible morphisms used by You and Chen to characterize the group invertibility of a sum of morphisms is reduced to that for one invertible morphism,and the existence and expression of the inverse along a product of two regular elements are obtained,which generalizes the main result of Mary and Patricio(2016)about the group inverse of a product.

Drazin invertibility for matrices over an arbitrary ring

In order to study the Drazin invertibility of a matrix with the generalized factorization over an arbitrary ring, the necessary and sufficient conditions for the existence of the Drazin inverse of a matrix are given by the properties of the generalized factorization. Let T = PAQ be a square matrix with the generalized factorization, then T has Drazin index k if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible. The formulae to compute the Drazin inverse are also obtained. These results generalize recent results obtained for the Drazin inverse of a matrix with a universal factorization.

ON THE INVERTIBILITY OF HILBERT SPACE IDEMPOTENTS

This note is to present some results on the group invertibility of linear combina- tions of idempotents when the difference of two idempotents is group invertible.

New characterizations for core inverses in rings with involution

The core inverse for a complex matrix was introduced by O.M.Baksalary and G.Trenkler.D.S.Rakic,N.C.Dincic and D.S.Djordjevc generalized the core inverse of a complex matrix to the case of an element in a ring.They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible.It is natural to ask when a group invertible element is core invertible.In this paper,we will answer this question.Let R be a ring with involution,we will use three equations to characterize the core inverse of an element.That is,let a,b∈R.Then a∈R with a=b if and only if(ab)^(*)=ab,ba^(2)=a,and ab^(2)=b.Finally,we investigate the additive property of two core invertible elements.Moreover,the formulae of the sum of two core invertible elements are presented.

Revisitation of the Core Inverse

In this paper, we revisit the core inverse introduced by Baksalary and Trenkler. We first give some new characterizations of the core inverse. Then, we give a new representation of the core inverse, which is related to AT,S^（2）.