Narrow Results

We generalize the notion of the left-sharp and the right-sharp partial orders to ${𝒢}({ℛ})$ where ${ℛ}$ is a ring with identity and ${𝒢}({ℛ})$ the subset of elements in ${ℛ}$ which have the group inverse. We connect these orders to well-known sharp and minus partial orders. Properties of one-sided sharp partial orders in ${𝒢}({ℛ})$ are studied and some known results are generalized.

Let $S$ be a semigroup and $b,c\in S$. The concept of $(b,c)$-inverses was introduced by Drazin in 2012. It is well known that the Moore–Penrose inverse, the Drazin inverse, the Bott–Duffin inverse, the inverse along an element, the core inverse and dual core inverse are all special cases of the $(b,c)$-inverse. In this paper, a new relationship between the $(b,c)$-inverse and the Bott–Duffin $(e,f)$-inverse is established. The relations between the $(b,c)$-inverse of $paq$ and certain classes of generalized inverses of $pa$ and $aq$, and the $(b^{\prime },c^{\prime })$-inverse of $a$ are characterized for some $b^{\prime },c^{\prime }\in S$, where $p,a,q\in S$. Necessary and sufficient conditions for the existence of the $(B,C)$-inverse of a lower triangular matrix over an associative ring $R$ are also given, and its expression is derived, where $B,C$ are regular triangular matrices.

Let $G_1$ and $G_2$ be two graphs on disjoint sets of $n_1$ and $n_2$ vertices, respectively. The corona of graphs $G_1$ and $G_2$, denoted by $G_1\circ G_2$, is the graph formed from one copy of $G_1$ and $n_1$ copies of $G_2$ where the $i$th vertex of $G_1$ is adjacent to every vertex in the $i$th copy of $G_2$. The neighborhood corona of $G_1$ and $G_2$, denoted by $G_1◇G_2$, is the graph obtained by taking one copy of $G_1$ and $n_1$ copies of $G_2$ and joining every neighbor of the $i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$ by a new edge. In this paper, the Laplacian generalized inverse for the graphs $G_1\circ G_2$ and $G_1◇G_2$ is investigated, based on which the resistance distances of any two vertices in $G_1\circ G_2$ and $G_1◇G_2$ can be obtained. Moreover, some examples as applications are presented, which illustrate the correction and efficiency of the proposed method.

In this paper, we define the direct sum relation on a module and prove that this is a partial order when the module is regular. We also focus on a regular support and a strong regular support of an element in a module, which are extensions of a generalized inverse and a reflexive generalized inverse of an element in a ring, respectively. Using regular supports and strong regular supports, we present various characterizations of the direct sum order as an application. We obtain some decompositions of a module via the direct sum order.

In this paper, the concept of “Inverse Complemented Matrix Method”, introduced by Eagambaram (2018), has been reestablished with the help of minus partial order and several new properties of complementary matrices and the inverse of complemented matrix are discovered. Class of generalized inverses and outer inverses of given matrix are characterized by identifying appropriate inverse complement. Further, in continuation, we provide a condition equivalent to the regularity condition for a matrix to have unique shorted matrix in terms of inverse complemented matrix. Also, an expression for shorted matrix in terms of inverse complemented matrix is given.

This paper approaches some universal-algebraic properties of the two kinds of multilinear functions $f(x_1,\dots ,x_k)=(a_1+b_1{x}_1{c}_1)\cdots (a_k+b_k{x}_k{c}_k)-a$ and $g(x_1,y_1,\dots ,x_k,y_k)=(a_1+b_1{x}_1{c}_1+d_1{y}_1{e}_1)\cdots (a_k+b_k{x}_k{c}_k+d_k{y}_k{e}_k)-a$ in a prime ring $R$, where $x_i,y_i\in R$ are variable elements, $i=1,\dots ,k$. We shall demonstrate an algebraic procedure of deriving necessary and sufficient conditions for the two multilinear functional identities $f(x_1,\dots ,x_k)\equiv 0$ and $g(x_1,y_1,\dots ,x_k,y_k)\equiv 0$ to hold for all $x_i,y_i\in R$, $i=1,\dots ,k$. Subsequently, we use these multilinear functional identities to describe the invariance properties of the products $a^{(i,\dots ,j)}{b}^{(s,\dots ,t)},$$ba{a}^{(i,\dots ,j)}{b}^{(s,\dots ,t)}$, $a^{(i,\dots ,j)}{b}^{(s,\dots ,t)}ba$, $ba{a}^{(i,\dots ,j)}{b}^{(s,\dots ,t)}ba$ with respect to the eight commonly-used types of generalized inverses of two MP-invertible elements $a$ and $b$ in a prime ring $R$ with an identity element 1 and ∗-involution.

In this paper, the notion of “strongly unit regular element”, for which every reflexive generalized inverse is associated with an inverse complement, is introduced. Noting that every strongly unit regular element is unit regular, some characterizations of unit regular elements are obtained in terms of inverse complements and with the help of minus partial order. Unit generalized inverses of given unit regular element are characterized as sum of reflexive generalized inverses and the generators of its annihilators. Surprisingly, it has been observed that the class of strongly regular elements and unit regular elements are the same. Also, several classes of generalized inverses are characterized in terms of inverse complements.

The Drazin inverse is connected with the notion of index and core-nilpotent decomposition whenever it is discussed in the context of ring of matrices over complex field. In the absence of Drazin inverse for a given element from an arbitrary associative ring (not necessarily with unity), in this paper, the notion of right (left) core-nilpotent decomposition has been introduced and established its relations with right (left) $\pi$-regular property. In fact, the class of such decomposition has been characterized. In case of regular ring, observed that an element is right (left) $\pi$-regular if and only if it has a right (left) core-nilpotent decomposition. In the process, several properties of sharp order in an associative ring are studied and with the help of the same, new characterizations of Drazin inverse over an associative ring are obtained and the relation between core-nilpotent decomposition and the Drazin inverse is obtained.

Using a derivative decomposition of the Hochschild differential complex we define a generalized inverse of the Hochschild coboundary operator. It can be applied for systematic computations of star products on Poisson manifolds.

In this paper, for a consistent quaternion matrix equation AXB = C, the formulas are established for maximal and minimal ranks of real matrices X₁, X₂, X₃, X₄ in solution X = X₁ + X₂i + X₃j + X₄k. A necessary and sufficient condition is given for the existence of a real solution of the quaternion matrix equation. The expression is also presented for the general solution to this equation when the solvability conditions are satisfied. Moreover, necessary and sufficient conditions are given for this matrix equation to have a complex solution or a pure imaginary solution. As applications, the maximal and minimal ranks of real matrices E, F, G, H in a generalized inverse (A +Bi + Cj + Dk)^- = E + Fi + Gj + Hk of a quaternion matrix A + Bi + Cj + Dk are also considered. In addition, a necessary and sufficient condition is derived for the quaternion matrix equations A₁XB₁ = C₁ and A₂XB₂ = C₂ to have a common real solution.

In this paper, we deal with the generalized inverse of upper triangular infinite dimensional Hamiltonian operators. Based on the structure operator matrix J in infinite dimensional symplectic spaces, it is shown that the generalized inverse of an infinite dimensional Hamiltonian operator is also Hamiltonian. Further, using the decomposition of spaces, an upper triangular Hamiltonian operator can be written as a new operator matrix of order 3, and then an explicit expression of the generalized inverse is given.

Suppose that A₁X=C₁, XB₂=C₂, A₃XB₃=C₃ is a consistent system of matrix equations and partition its solution X into a 2 × 2 block form. In this paper, we give formulas for the maximal and minimal ranks of the submatrices in a solution X to the system. We also investigate the uniqueness and the independence of submatrices in a solution X. As applications, we give some properties of submatrices in generalized inverses of matrices. These extend some known results in the literature.

In this paper we investigate the system of linear matrix equations A₁X=C₁, YB₂=C₂, A₃XB₃=C₃, A₄YB₄=C₄, BX+YC=A. We present some necessary and sufficient conditions for the existence of a solution to this system and give an expression of the general solution to the system when the solvability conditions are satisfied.

In this paper, a new necessary and sufficient condition for the existence of a Hermitian solution as well as a new expression of the general Hermitian solution to the system of matrix equations A₁X=C₁ and A₃XB₃=C₃ are derived. The max-min ranks and inertias of these Hermitian solutions with some interesting applications are shown. In particular, the max-min ranks and inertias of the Hermitian part of the general solution to this system are presented.

We in this paper derive necessary and sufficient conditions for the system of the periodic discrete-time coupled Sylvester matrix equations $A_k{X}_k+Y_k{B}_k=M_k,C_k{X}_{k+1}+Y_k{D}_k=N_k(k=1,2)$ over the quaternion algebra to be consistent in terms of ranks and generalized inverses of the coefficient matrices. We also give an expression of the general solution to the system when it is solvable. The findings of this paper generalize some known results in the literature.

In this paper,we give a necessary and sufficient condition for the solvability to a system of linear matrix equations A₁X₁ = C₁, X₁B₁ = C₂, A₂X₂ = C₃. X₂B₂ = C₄, A₃X₃ = C₅, X₃B₃ = C₆, A₄X₁B₄+A₅X₂B₅+A₆X₃B₆ = C₇ over an arbitrary division ring. The findings of this paper extend some known results in the literature.