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In this paper, further characterizations of selfadjoint operators on Hilbert spaces are established. Thus the recent results concerning two-sided removal and cancellation laws associated with a Hermitian matrix (Central University of Finance and Economics, DOI: https://doi.org/10.21203/rs.3.rs-1104901/v1) are extended to the infinite dimensional setting with proofs based on operator matrices. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in a special class of integer matrices, namely the power-product matrix, defined with two positive integers n and d. Each matrix element is computed by a power-product of two weak compositions of d into n parts. The power-product matrix has several interesting applications such as the power-sum representation of polynomials and the difference-of-convex-sums-of-squares decomposition of polynomials. We investigate some properties of this matrix including: nonsingularity, sparsity and determinant. Based on techniques in enumerative combinatorics, we prove that the power-product matrix is nonsingular and the number of nonzero entries can be computed exactly. This matrix shows sparse structure which is a good feature in numerical computation of its inverse required in some applications. Special attention is devoted to the computation of the determinant for n = 2 whose explicit formulation is obtained. [ABSTRACT FROM AUTHOR]

The notion of the weighted core inverse in a ring with involution was introduced recently [Mosić et al. Comm. Algebra, 2018; 46(6); 2332–2345]. In this paper, we explore new characterizations of the weighted core inverse of sum and the difference between two weighted core invertible elements in a ring with involution under different conditions. Further, we discuss reverse order and mixed-type reverse order laws for the weighted core invertible elements in a ring. [ABSTRACT FROM AUTHOR]

Theory of numerical range and numerical radius for tensors is not studied much in the literature. Ke et al. [Linear Algebra Appl. 508 (2016), 100-132: MR3542984] introduced first the notion of the numerical range of a tensor via the k-mode product. However, the convexity of the numerical range via the k-mode product was not proved by them. In this paper, the notion of numerical range and numerical radius for even-order square tensors via the Einstein product are introduced first. Using the notion of the numerical radius of a tensor, we provide some sufficient conditions for a tensor to be unitary. The convexity of the numerical range is also proved. We also provide an algorithm to plot the numerical range of a tensor. Furthermore, some properties of the numerical range for the Moore–Penrose inverse of a tensor are discussed. [ABSTRACT FROM AUTHOR]

Since the introduction of majorization in R n by Hardy, Littlewood and Polya, several extensions of this concept have been studied in the literature. Recently, Kosuru and Saha [Cone type majorization and its strong linear preservers. Electron J Linear Algebra. 2020;36(36):511–518.] defined the concept of cone type majorization. In this paper, we focus on the study of the behavior of the linear preservers of majorization and cone type majorization under generalized inversion, namely, Drazin inversion and Moore-Penrose inversion. A characterization of these linear preservers, given by Ando [Majorization, doubly stochastic matrices and comparisons of eigenvalues. Linear Algebra Appl. 1989;118:163–248.] for majorization, and by Kosuru and Saha [Cone type majorization and its strong linear preservers. Electron J Linear Algebra. 2020;36(36):511–518.] for cone type majorization, prove to be crucial in our proofs. [ABSTRACT FROM AUTHOR]

A square complex matrix A is EP if it satisfies R (A) = R (A ∗) (or, equivalently, R (A) ⊆ R (A ∗)). This paper aims to introduce two natural generalizations of EP matrices: a square matrix A is defined to be left or right power-EP if it satisfies R (A m) ⊆ R (A ∗) or N (A ∗) ⊆ N (A m) for some integer m ≥ 0 , respectively. Some properties, characterizations and applications of these two classes of matrices are given. [ABSTRACT FROM AUTHOR]

Recently, Ferreyra and Malik (Core and strongly core orthogonal matrices. Linear Multilinear Algebra. 2021;1–16. doi:) 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 and get some new characterizations of strong core orthogonality. [ABSTRACT FROM AUTHOR]

In this paper, we establish a different approach for solving the system of three coupled two-sided Sylvester-type quaternion matrix equations A i X i B i + C i X i + 1 D i = E i , i = 1 , 3 ¯ . We give some new necessary and sufficient conditions for the existence of a solution to this system in terms of Moore-Penrose inverses of the matrices involved. We show that these new solvability conditions are equivalent with the solvability conditions which were presented in a recent paper [Linear Algebra Appl. 2016;496:549–593]. The general solution to the system is given when the solvability conditions are satisfied. Applications that are discussed include the solvability conditions and general η-Hermitian solution to a system of quaternion matrix equations. [ABSTRACT FROM AUTHOR]

This paper contributes to define one-sided versions of 'w-core inverse' introduced by the writer. Given any ∗ -ring R and a , w ∈ R , a is called right w-core invertible if there exists some x ∈ R satisfying awxa = a, a w x 2 = x and a w x = (a w x) ∗ . Several characterizations for this type of generalized inverses are given, and it is shown that a is right w-core invertible if and only if a is right w (a w) n − 1 -core invertible if and only if there exists a Hermitian element p such that pa = 0 and p + (a w) n is right invertible for any integer n ≥ 1 , in which case, the expression of right w-core inverses is given. Finally, it is proved that right w-core inverses are instances of right inverses along an element, right (b , c) -inverses and right annihilator (b , c) -inverses. As an application, the characterization for the Moore–Penrose inverse is given. [ABSTRACT FROM AUTHOR]

Necessary and sufficient conditions for the absorption laws of the weighted core inverse, { 1 } , { 1 , 2 } and { 1 , 3 e } -inverse are investigated in this paper. In addition, mixed absorption laws for { 1 } , { 1 , 2 } , { 1 , 3 e } , { 1 , 2 , 3 e } and { 1 , 6 } -inverse are considered. [ABSTRACT FROM AUTHOR]

The paper considers various problems concerned with the Moore–Penrose inverse of a sum of two matrices. By establishing several original results and by combining various facts known in the literature, the article reveals a number of emerging features of the inverse. The investigations shed also a spotlight on different proving approaches useful to cope with the problems in question. Rather than focusing exclusively on the main topic, the considerations endeavour to place it in a wider context, linking it with different matrix notions, facts, and tools, as well as indicating areas of its applications originating from, e.g. computational methods, physics, statistics, discriminant analysis, or neuroscience. [ABSTRACT FROM AUTHOR]

In this paper, we present several representations of the W-weighted WG inverse. These representations are expressed in terms of matrix powers as well as in terms of matrix products involving only the Moore–Penrose inverse. In addition, a new characterization of the W-weighted WG inverse is presented by using a rank equation. [ABSTRACT FROM AUTHOR]

In this paper, we give simple characterizations of the CMP inverse to be an EP matrix and simplify Theorem 2.10 in [New characterizations of the CMP inverse of matrices. Linear Multilinear Algebra. 2020;68(4):790–804]. Also, we characterize the CMP inverse of a matrix based on its range space and null space. Several different representations of the CMP inverse are given. In addition, some new characterizations of k-commutative equalities for some outer generalized inverse are investigated in terms of the CMP inverse. [ABSTRACT FROM AUTHOR]

In this paper, a novel method is presented to compute an explicit formula for the inverse of the confluent Vandermonde matrices. The proposed results may have many interesting perspectives in diverse areas of mathematics and natural sciences, notably on the situations where the Vandermonde matrices have acquired much usefulness. The method presented here is direct and straightforward, it gives explicit and compact formulas. Several examples are presented to highlight the results. The main tools are some elementary basic linear algebra. [ABSTRACT FROM AUTHOR]

As is well known, the generalized inverse, Moore–Penrose inverse and group inverse are not continuous, i.e. for θ = {1, 2}, {1, 2, 3, 4} and {1, 2, 5}, a linear bounded operator T has a θ-inverse T θ , the perturbed operator T ¯ = T + δ T is not necessary θ-invertible and even if it is θ-invertible, lim δ T → 0 T ¯ θ = T θ may not be true. In this paper, we prove that T + T T θ δ T T θ T is θ-invertible and its θ-inverse (T + T T θ δ T T θ T) θ has the simplest possible expression, which satisfies lim δ T → 0 (T + T T θ δ T T θ T) θ = T θ . Thus, we have found a continuous orbit for the generalized inverse, Moore–Penrose inverse and group inverse. [ABSTRACT FROM AUTHOR]

This paper introduces and investigates a new class of generalized inverses, called GDMP-inverses (and their duals), as a generalization of DMP-inverses. GDMP-inverses are defined from G-Drazin inverses and the Moore-Penrose inverse of a complex square matrix. In contrast to most other generalized inverses, GDMP-inverses are not only outer inverses but also inner inverses. Characterizations and representations of GDMP-inverses are obtained by means of the core-nilpotent and the Hartwig-Spindelböck decompositions. [ABSTRACT FROM AUTHOR]

In this paper, we analyse the matrix structures of various generalized inverses, suggest their applicable scopes and build their relationships with particular projections. As applications, several equivalent conditions for core–EP (Range-Hermitian) relation, star order, sharp order and MP–Core–EP (MPCEP) relation are presented. The relationships among these orders or relations are built. In addition, we introduce some conditions for which the core–EP invertible operators satisfy the reverse order law. [ABSTRACT FROM AUTHOR]

In this paper, we extend the operations of parallel addition A:B and parallel subtraction A ÷ B from the cone of bounded nonnegative self-adjoint operators to the linear bounded operators on a Hilbert space. Some conditions for the relations A † : B † = (A + B) † , B = (A : B) ÷ A , (A C) ÷ (B C) = (A ÷ B) C , A ÷ B = (P (A † − B †) P) † , B = A : (B ÷ A) , (C A) : (C B) = C (A : B) to be true are studied and the solution of the equation A:X = B is investigated. Moreover, some relationships between the parallel addition and subtraction of projections are obtained. [ABSTRACT FROM AUTHOR]

The main contribution of this paper is to present new generalized inverses as weaker versions of a G-outer inverse. In particular, we define and characterize left and right G-outer inverses of rectangular matrices. Solvability of matrix equation systems as AXA = AEA and BAEAX = B; or AXA = AEA and XAEAD = D, where A ∈ C m × n , B ∈ C p × m , D ∈ C n × q and E ∈ C n × m , is studied by means of left and right G-outer inverses. The general solution forms of these systems give descriptions of the sets of all left and right G-outer inverses. Using left and right G-outer inverses, we introduce new partial orders and establish their relations with minus partial order and space pre-order. We apply these results to present and investigate left and right G-Drazin inverses of square matrices and corresponding partial orders. [ABSTRACT FROM AUTHOR]

In this paper, two formulas, which were studied by Wang and Liu (2015) and Ma and Li (2019), respectively, for the core inverse, are simplified. Then two methods for computing the core inverse A ◯ # and dual core inverse A ◯ # are investigated through Gauss–Jordan elimination on the two appropriate block partitioned matrices. The corresponding algorithms are also summarized. The computational complexities of the these two algorithms are analysed in detail. In the end, some numerical examples are presented to demonstrate the efficiency of the two algorithms. [ABSTRACT FROM AUTHOR]

Iterative methods based on matrix splittings are useful tools in solving real large sparse linear systems. In this aspect, the type I double splitting approaches are straight forward from the formulation of the iteration scheme and its convergence theory is well established in the literature. However, if a double splitting is of type II, then the convergence of the iteration scheme seems not to be straight forward. In this paper, we develop convergence theory for type II double splittings to make the implementation quite simple. In this direction, we first introduce two new subclasses of double splittings and establish their convergence theory. Using this theory, we prove a new characterization of a monotone matrix. Finally, we apply our theoretical findings to the double splitting of an M-matrix in the Gauss–Seidel double SOR method to obtain a comparison result. [ABSTRACT FROM AUTHOR]

The main topic of this paper is the group invertibility and expressions of group inverses of linear operators in Banach spaces. We first give some properties of regular factorizations and then derive some new characterizations of the group invertibility of linear operators. Based on these results, we use any one of generalized inverses to give some concise expressions of the group inverse. [ABSTRACT FROM AUTHOR]

In this paper, we obtain new additive perturbation bounds of the Moore-Penrose inverse under the unitarily invariant norm and the Q-norm by using the singular value decomposition, respectively. These bounds always improve the corresponding ones in [Cai L et al. Additive and multiplicative perturbation bounds for the Moore-Penrose inverse. Linear Algebra Appl. 2011;434:480–489]. [ABSTRACT FROM AUTHOR]

In this paper, we give algebraic conditions for the existence of a solution and the expression for the general solution of the system of the equations a 1 x b 1 = c 1 , a 2 x b 2 = c 2 , a 3 x b 3 = c 3 , when each element belongs to a ring with a unit. As an application, we get necessary and sufficient conditions for the existence of common inner inverse of three regular elements. [ABSTRACT FROM AUTHOR]

Applications of the theory and computations of Boolean matrices are of fundamental importance to study a variety of discrete structural models. But the increasing ability of data collection systems to store huge volumes of multidimensional data, the Boolean matrix representation of data analysis is not enough to represent all the information content of the multiway data in different fields. From this perspective, it is appropriate to develop an infrastructure that supports reasoning about the theory and computations. In this paper, we discuss the generalized inverses of the Boolean tensors with the Einstein product. Further, we elaborate on this theory by producing a few characterizations of different generalized inverses and several equivalence results on Boolean tensors. We explore the space decomposition of the Boolean tensors and present reflexive generalized inverses through it. In addition to this, we address rank and the weight for the Boolean tensor. [ABSTRACT FROM AUTHOR]

In this paper we discuss different properties of DMP-inverse of a square matrix introduced by Malik and Thome [On a new generalized inverse for matrices of an arbitrary index. Applied Mathematics and Computation. 2014;226: 575–580]. We characterize DMP-inverse of a square matrix as an outer inverse with prescribed range and null space, as a (B , C) -inverse and as a Drazin inverse. Also we present several different characterizations based on its range and null space as well some algebraic ones. [ABSTRACT FROM AUTHOR]

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 the Minkowski space. [ABSTRACT FROM AUTHOR]

Let R be a ring and a , b , c ∈ R . In this paper, it is shown that if a is left (b , c) -invertible and cab is regular, then a has at least one regular left (b , c) -inverse. And the expression of the (b , c) -inverse of a is given in terms of the inner inverse of cab. Moreover, the strongly left (or right) (b , c) -inverse is introduced in terms of the regularity of cab. It is shown when (a b) ∗ = a b , a is left (b , b) -invertible implies that a is strongly left (b , b) -invertible, and left (b , b) -invertibility coincides with right (b , b) -invertibility. As applications, we consider when the one-sided core inverse is core invertible. It is shown that left core invertibility coincides with right core invertibility in every strongly π-regular ring. [ABSTRACT FROM AUTHOR]

In this paper, characterizations of right core inverse are given by one-sided invertibility. The necessary and sufficient conditions, which guarantee that paq has right core inverses, are investigated. Furthermore, characterizations of right core inverses of triangular matrices, 2 × 2 matrices and a companion matrix are considered. [ABSTRACT FROM AUTHOR]

Let E,F,G,D be infinite complex Banach spaces and B (F , E) the Banach space of all bounded linear operators from F into E. Consider A 1 , A 2 ∈ B (F , E) , B 1 , B 2 ∈ B (D , G). Let M A 1 , B 1 : X → A 1 X B 1 be the multiplication operator on B (G , F) induced by A 1 , B 1 . In particular, L A 1 = M A 1 , I and R B 1 = M I , B 1 , where I is the identity operator are the left and the right multiplication operators, respectively. The elementary operator Ψ defined on B (G , F) is the sum of two multiplication operators Ψ = M A 1 , B 1 + M A 2 , B 2 . This paper gives necessary and sufficient conditions for the existence of a common solution of the operator equations M A 1 , B 1 (X) = C 1 and M A 2 , B 2 (X) = C 2 and derive a new representation of the general common solution via the inner inverse of the elementary operator Ψ; we apply this result to determine new necessary and sufficient conditions for the existence of a Hermitian solution and a representation of the general Hermitian solution to the operator equation M A , B (X) = C. As a consequence, we obtain well-known results of Daji c ´ and Koliha. [ABSTRACT FROM AUTHOR]

Although solvability conditions for a system of two linear equations are well-known even in the case of rings and, for three linear equations, in the case of matrices, in the case of four linear equations there are no results. In this paper, we consider systems of four linear matrix equations A i X B i = C i , i = 1 , 4 ¯ and present some necessary and sufficient conditions for their solvability as well as an expression for the general solution. There are two advantages to our results: the presented solvability conditions in many cases can be presented in a purely algebraic form and the method used in the proof allows for a generalization of the obtained results to some more general structures such as algebras of bounded linear operators or rings, under some additional assumptions concerning regularity only. We present several applications of our results. [ABSTRACT FROM AUTHOR]

One of the fundamental research problems in the theory of generalized inverses has certainly been establishments of various matrix equalities that involve generalized inverses. A simplest form of such matrix equalities is given by A (i , ... , j) = B (k , ... , l) , where A and B are matrices of the same size, and (⋅) (i , ... , j) denotes the { i , ... , j } -generalized inverse of a matrix. Because generalized inverses of a matrix are not necessarily unique, the equality A (i , ... , j) = B (k , ... , l) does not imply A=B for singular matrices. In this paper, we derive necessary and sufficient conditions for this kind of equalities to hold for the eight commonly-used types of generalized inverse of A and B using the matrix equation method and the matrix rank method. [ABSTRACT FROM AUTHOR]

In this paper, we present equivalent characterizations for the minimal ∞-norm of generalized inverses of the incidence matrix of a tree. We also compute the minimal ∞-norm for a class of trees. [ABSTRACT FROM AUTHOR]