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In this paper, the quaternion differential equation, A x ′ (t) + B x (t) = f (t) , is studied in the case when the quaternion matrix A is singular and f (t) is a quaternion-valued vector function of a real variable t. Its solvability conditions are derived, and the general solution is represented by an explicit formula using generalized inverses of coefficient matrices. The dual (left) quaternion differential equation is explored as well. Also, we consider their partial cases when there is a constant quaternion matrix instead of a quaternion-valued vector function. At that, their general solutions are represented by closed formulas using the Moore-Penrose and Drazin inverses. To illustrate the effectiveness of these results, algorithms, and examples for finding solutions to considered differential quaternion matrix equations are provided. To compute generalized inverse matrices, we use a direct method, namely, their determinantal representations based on the theory of row-column noncommutative determinants previously introduced by the author. [ABSTRACT FROM AUTHOR]

Algebraic expressions and equalities can be constructed arbitrarily in a given algebraic framework according to the operational rules provided, and thus it is a prominent and necessary task in mathematics and applications to construct, classify, and characterize various simple general algebraic expressions and equalities. As an update to this prominent topic in matrix algebra, this article reviews and improves upon the well-known block matrix methodology and matrix rank methodology in the construction and characterization of matrix equalities. We present a collection of fundamental and useful formulas for calculating the ranks of a wide range of block matrices and then derive from these rank formulas various valuable consequences. In particular, we present several groups of equivalent conditions in the characterizations of the Hermitian matrix, the skew-Hermitian matrix, the normal matrix, etc. [ABSTRACT FROM AUTHOR]

This article includes the core inverses of operators on Hilbert spaces. In addition, the group inverses and Moore-Penrose inverses of these operators are characterised. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the general errors-in-variables (EIV) model, allowing both the uncertain coefficient matrix and the dispersion matrix to be rank-deficient. We derive the weighted total least-squares (WTLS) solution in the general case and find that with the model consistency condition: (1) If the coefficient matrix is of full column rank, the parameter vector and the residual vector can be uniquely determined independently of the singularity of the dispersion matrix, which naturally extends the Neitzel/Schaffrin rank condition (NSC) in previous work. (2) In the rank-deficient case, the estimable functions and the residual vector can be uniquely determined. As a result, a unified approach for WTLS is provided by using generalized inverse matrices (g-inverses) as a principal tool. This method is unified because it fully considers the generality of the model setup, such as singularity of the dispersion matrix and multicollinearity of the coefficient matrix. It is flexible because it does not require to distinguish different cases before the adjustment. We analyze two examples, including the adjustment of the translation elimination model, where the centralized coordinates for the symmetric transformation are applied, and the unified adjustment, where the higher-dimensional transformation model is explicitly compatible with the lower-dimensional transformation problem. [ABSTRACT FROM AUTHOR]

Inversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most notable is the Moore–Penrose inverse, widely used in physics, statistics, and various fields of engineering. This work investigates generalized inversion of nonlinear operators. We first address broadly the desired properties of generalized inverses, guided by the Moore–Penrose axioms. We define the notion for general sets and then a refinement, termed pseudo-inverse, for normed spaces. We present conditions for existence and uniqueness of a pseudo-inverse and establish theoretical results investigating its properties, such as continuity, its value for operator compositions and projection operators, and others. Analytic expressions are given for the pseudo-inverse of some well-known, non-invertible, nonlinear operators, such as hard- or soft-thresholding and ReLU. We analyze a neural layer and discuss relations to wavelet thresholding. Next, the Drazin inverse, and a relaxation, are investigated for operators with equal domain and range. We present scenarios where inversion is expressible as a linear combination of forward applications of the operator. Such scenarios arise for classes of nonlinear operators with vanishing polynomials, similar to the minimal or characteristic polynomials for matrices. Inversion using forward applications may facilitate the development of new efficient algorithms for approximating generalized inversion of complex nonlinear operators. [ABSTRACT FROM AUTHOR]

This paper is concerned with different properties of the numerical range for a bounded linear operator from the point of view of its similarities to the numerical range of certain generalized inverses such as the Moore–Penrose inverse, Drazin inverse, DMP-inverse, MDP-inverse and CMP-inverse. We will consider the question whether the numerical ranges of an operator A and a certain generalized inverse of A simultaneously contain the origin as well as whether this is true in the case of the closure, boundary, extreme points and sharp points of the corresponding numerical ranges. To illustrate our results we will exhibit some numerical examples. [ABSTRACT FROM AUTHOR]

We develop a new Kantorovich-like convergence analysis of Newton-type methods to solve nonsingular and singular nonlinear equations in Banach spaces. The outer or generalized inverses are exchanged by a finite sum of linear operators making the implementation of these methods easier than in earlier studies. The analysis uses relaxed generalized continuity of the derivatives of operators involved required to control the derivative and on real majorizing sequences. The same approach can also be implemented on other iterative methods with inverses. The examples complement the theory by verifying the convergence conditions and demonstrating the performance of the methods. [ABSTRACT FROM AUTHOR]

Let X be a Banach space and let P: X → X be a bounded linear operator. Using an algebraic inequality on the spectrum of P, we give a new sufficient condition that guarantees the existence of (I–P)−1 as a bounded linear operator on X, and a bound on its spectral radius is also obtained. This generalizes the classic Banach lemma. We apply the result to the perturbation analysis of general bounded linear operators on X with commutative perturbations. [ABSTRACT FROM AUTHOR]

New-typed matrix inverses based on the Hartwig-Spindelböck decomposition were investigated, which is called the strong B-T inverse, as a generalization of the B-T inverse. The relationships between the above inverse and other matrix inverses were established. Several sufficient and necessary conditions of the strong B-T inverse are obtained via the column and null spaces.

Large-scale linear systems, Ax = b, frequently arise in practice and demand effective iterative solvers. Often, these systems are noisy due to operational errors or faulty data-collection processes. In the past decade, the randomized Kaczmarz algorithm (RK) has been studied extensively as an efficient iterative solver for such systems. However, the convergence study of RK in the noisy regime is limited and considers measurement noise in the right-hand side vector, b. Unfortunately, in practice, that is not always the case; the coefficient matrix A can also be noisy. In this paper, we analyze the convergence of RK for doubly noisy linear systems, i.e., when the coefficient matrix, A, has additive or multiplicative noise, and b is also noisy. In our analyses, the quantity R = ||Æ ||²||Ã||²F influences the convergence of RK, where A represents a noisy version of A. We claim that our analysis is robust and realistically applicable, as we do not require information about the noiseless coefficient matrix, A, and by considering different conditions on noise, we can control the convergence of RK. We perform numerical experiments to substantiate our theoretical findings. [ABSTRACT FROM AUTHOR]

Algebraic expressions and equalities can be constructed arbitrarily in a given algebraic framework according to the operational rules provided, and thus it is a prominent and necessary task in mathematics and applications to construct, classify, and characterize various simple general algebraic expressions and equalities. As an update to this prominent topic in matrix algebra, this article reviews and improves upon the well-known block matrix methodology and matrix rank methodology in the construction and characterization of matrix equalities. We present a collection of fundamental and useful formulas for calculating the ranks of a wide range of block matrices and then derive from these rank formulas various valuable consequences. In particular, we present several groups of equivalent conditions in the characterizations of the Hermitian matrix, the skew-Hermitian matrix, the normal matrix, etc.

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) π -regular property. In fact, the class of such decomposition has been characterized. In case of regular ring, observed that an element is right (left) π -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. [ABSTRACT FROM AUTHOR]

Let A be a relatively regular linear relation defined on a Banach space X. In this paper, characterizations of some classes of relatively regular linear relations via the set D (A) : = { B 1 - B 2 ; B 1 and B 2 are pseudo inverses of A } are given. Further, if B is a generalized inverse of A, some spectral properties of B are investigated. More precisely, conditions under which the nonzero points in the spectrum of B are the reciprocals of the nonzero points in the spectrum of A are provided. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a binary relation in the set of all (b, c)-invertible elements of a ring, defined in the manner like minus, star, sharp, core, and dual core partial orders. We prove that this binary relation is actually an equivalence relation and we investigate some of its properties. Furthermore, we define another equivalence relation on the set of all (b, c)-invertible elements of a ring, as an improvement of the previous mentioned equivalence relation. In addition, the equivalence criteria are given for the equality of (b, c)-related idempotents of two elements and their (b, c)-invertibility is studied. [ABSTRACT FROM AUTHOR]

A square matrix A over the field of complex numbers is said to be Hermitian if A = A ∗ , the conjugate transpose of A, while Hermitian matrices are known to be an important class of matrices. In addition to the definition, a Hermitian matrix can be characterized by some other matrix equalities. This fact can be described in the implication form f (A , A ∗) = 0 ⇔ A = A ∗ , where f (·) denotes certain ordinary algebraic operation of A and A ∗ . In this note, we show two special cases of the equivalent facts: A A ∗ A = A ∗ A A ∗ ⇔ A 3 = A A ∗ A ⇔ A = A ∗ without assuming the invertibility of A through the skillful use of decompositions and determinants of matrices. Several consequences and extensions are presented to a selection of matrix equalities composed of multiple products of A and A ∗ . [ABSTRACT FROM AUTHOR]

This paper introduces a new class of generalized inverses for square matrices: core-EP G-Drazin (CEPGD) inverse. The CEPGD inverse is not unique and defined as a proper composition of the core-EP and the G-Drazin inverse. Representations of CEPGD inverses related to the core-nilpotent decomposition and the Hartwig–Spindelböck decomposition are established. The existence of CEPGD inverses as well as a few characterizations and representations of this inverse are discussed. In addition, we consider some additional properties of the CEPGD inverses through an induced binary relation. [ABSTRACT FROM AUTHOR]

Motivated by the works of Wang and Liu (Linear Algebra Appl 488:235–248, 2016) and Mosić (Results Math 75(2):1–21, 2020), we provide further results on GD inverses, and then introduce two new classes for square matrices called GD-star (generalized-Drazin-star) and GD-star-one (generalized-Drazin-star-one) using a GD inverse of a matrix. We exploit their various properties and characterize them in terms of various generalized inverses. We present a representation of a GD-star matrix by using the core-EP decomposition and Hartwig–Spindelböck decomposition. We also define a binary relation called GD-star order using this class of matrices. Further, we obtain some analogous results for the class of star-GD matrices. Moreover, the reverse-order law and forward-order law for GD inverse along with its monotonicity criteria are obtained. [ABSTRACT FROM AUTHOR]

The paper deals with the third-order quaternion tensor equation. Based on the Qt multiplication operation, we derive solvability conditions and also get the general solution, the least-squares solution, the minimum-norm solution and the minimum-norm least-squares solution of the tensor equation $ \mathcal{A} \ast_{\mathbb{Q}} \mathcal{X} = \mathcal{B} $. Finally, two numerical examples are presented.

It is common knowledge that matrix equalities involving ordinary algebraic operations of inverses or generalized inverses of given matrices can be constructed arbitrarily from theoretical and applied points of view because of the noncommutativity of the matrix algebra and singularity of given matrices. Two of such matrix equality examples are given by $ A_1B_1^{g_1}C_1 + A_2B_2^{g_2}C_2 + \cdots + A_kB_k^{g_k}C_{k} = D $ and $ A_1B_1^{g_1}A_2B_2^{g_2} \cdots A_kB_k^{g_k}A_{k+1} = A $, where $ A_1 $, $ A_2 $, $ \ldots $, $ A_{k+1} $, $ C_1 $, $ C_2 $, $ \ldots $, $ C_{k} $ and $ A $ and $ D $ are given, and $ B_1^{g_1} $, $ B_2^{g_2} $, $ \ldots $, $ B_k^{g_k} $ are generalized inverses of matrices $ B_1 $, $ B_2 $, $ \ldots $, $ B_k $. These two matrix equalities include many concrete cases for different choices of the generalized inverses, and they have been attractive research topics in the area of generalized inverse theory. As an ongoing investigation of this subject, the present author presents in this article several groups of new results and facts on constructing and characterizing the above matrix equalities for the mixed combinations of $ \{1\} $- and $ \{1, 2\} $-generalized inverses of matrices with $ k = 2, 3 $ by using some elementary methods, including a series of explicit rank equalities for block matrices.

Dual quaternions can represent rigid body motion in 3D spaces, and have found wide applications in robotics, 3D motion modelling and control, and computer graphics. In this paper, we introduce three different right linear independency concepts for a set of dual quaternion vectors, and study some related basic properties for dual quaternion vectors and dual quaternion matrices. We present a minimax principle for eigenvalues of dual quaternion Hermitian matrices. Based upon a newly established Cauchy-Schwarz inequality for dual quaternion vectors and singular value decomposition of dual quaternion matrices, we propose an inequality for singular values of dual quaternion matrices. Finally, we introduce the concept of generalized inverses of dual quaternion matrices, and present necessary and sufficient conditions for a dual quaternion matrix to be one of four types of generalized inverses of another dual quaternion matrix. [ABSTRACT FROM AUTHOR]

It is common knowledge that matrix equalities involving ordinary algebraic operations of inverses or generalized inverses of given matrices can be constructed arbitrarily from theoretical and applied points of view because of the noncommutativity of the matrix algebra and singularity of given matrices. Two of such matrix equality examples are given by A 1 B 1 g 1 C 1 + A 2 B 2 g 2 C 2 + ⋯ + A k B k g k C k = D and A 1 B 1 g 1 A 2 B 2 g 2 ⋯ A k B k g k A k + 1 = A , where A 1 , A 2 , ... , A k + 1 , C 1 , C 2 , ... , C k and A and D are given, and B 1 g 1 , B 2 g 2 , ... , B k g k are generalized inverses of matrices B 1 , B 2 , ... , B k . These two matrix equalities include many concrete cases for different choices of the generalized inverses, and they have been attractive research topics in the area of generalized inverse theory. As an ongoing investigation of this subject, the present author presents in this article several groups of new results and facts on constructing and characterizing the above matrix equalities for the mixed combinations of { 1 } - and { 1 , 2 } -generalized inverses of matrices with k = 2 , 3 by using some elementary methods, including a series of explicit rank equalities for block matrices. [ABSTRACT FROM AUTHOR]

The inverse square root of a covariance matrix is often desirable for performing data whitening in the process of applying many common multivariate data analysis methods. Direct calculation of the inverse square root is not available when the covariance matrix is either singular or nearly singular, as often occurs in high dimensions. We develop new methods, which we broadly call polynomial whitening, to construct a low-degree polynomial in the empirical covariance matrix which has similar properties to the true inverse square root of the covariance matrix (should it exist). Our method does not suffer in singular or near-singular settings, and is computationally tractable in high dimensions. We demonstrate that our construction of low-degree polynomials provides a good substitute for high-dimensional inverse square root covariance matrices, in both d < N and d ≥ N cases. We offer examples on data whitening, outlier detection and principal component analysis to demonstrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper point spectrum, compactness, reducibility, generalized inverse and some weak normal classes of substitution conditional type operators acting on L 2 (Σ) will be investigated. [ABSTRACT FROM AUTHOR]

Abstract–In this article, some interesting applications of generalized inverses in the graph theory are revisited. Interesting properties of generalized inverses are employed to make the proof of several known results simpler, and several techniques such as bordering method and inverse complemented matrix methods are used to obtain simple expressions for the Moore-Penrose inverse of incidence matrix and Laplacian matrix. Some interesting and simpler expressions are obtained in some special cases such as tree graph, complete graph and complete bipartite graph.

Utilizing the stability characterizations of generalized inverses, we investigate the generalized resolvent of linear operators in Banach spaces. We first prove that the local analyticity of the generalized resolvent is equivalent to the continuity and the local boundedness of generalized inverse functions. We also prove that several properties of the classical spectrum remain true in the case of the generalized one. Finally, we elaborate on the reason why we use the generalized inverse rather than the Moore-Penrose inverse or the group inverse to define the generalized resolvent.

We develop a new Kantorovich-like convergence analysis of Newton-type methods to solve nonsingular and singular nonlinear equations in Banach spaces. The outer or generalized inverses are exchanged by a finite sum of linear operators making the implementation of these methods easier than in earlier studies. The analysis uses relaxed generalized continuity of the derivatives of operators involved required to control the derivative and on real majorizing sequences. The same approach can also be implemented on other iterative methods with inverses. The examples complement the theory by verifying the convergence conditions and demonstrating the performance of the methods.

This work deals with some generalizations of the Drazin invertibility, involving the pseudo-generalized invertibility that has been studied recently by the authors in Lahmar and Skhiri (Filomat 36:2551-2572, 2022) and Lahmar and Skhiri (Filomat 36:4575-4590, 2022). We also introduce a new notion of DPG inverse and the relationships between the notions of DPG inverse, Drazin inverse and EP operator, are studied in this paper. An application of DPG inverses to some operator equations is also provided. [ABSTRACT FROM AUTHOR]

For any semigroup S and any a , b , c ∈ S , the author (Drazin (2012) [3]) defined a as being " (b , c) -invertible", with " (b , c) -inverse" y , if there exists y ∈ S such that y ∈ b S y ∩ y S c (say with b h y = y = y g c), y a b = b and c a y = c. This article is concerned with pairwise implications among y a = a y and other "EP-like" properties of a , b , c , g , h , y such as y c = b y , a h = g a , h y = y g , c a = a b and b h = g c. Each implication is either proved to be always true, or shown by an example to be false. The important special case where b = c and g = h is considered separately, again with a counter-example for every false implication. [ABSTRACT FROM AUTHOR]

In many papers concerning properties of generalized inverses in different settings, we can find the results with many redundant instances of assuming regularity of certain elements. We have made an effort to widen the range of applicability of concrete results by considering more general cases of the problems without imposing any such additional assumptions. This was the main motivation for this paper, and we present several significant improvements of the reverse order laws on {1,3}$\lbrace 1,3\rbrace$ and {1,2,3}$\lbrace 1,2,3\rbrace$‐inverses in the ring setting. [ABSTRACT FROM AUTHOR]

This article considers the iterative approach for finding the Moore–Penrose inverse of a matrix. A convergence analysis is presented under certain conditions, demonstrating that the scheme attains third-order convergence. Moreover, theoretical discussions suggest that selecting a particular parameter could further improve the convergence order. The proposed scheme defines the special cases of third-order methods for β = 0 , 1 / 2 , and 1 / 4 . Various large sparse, ill-conditioned, and rectangular matrices obtained from real-life problems were included from the Matrix-Market Library to test the presented scheme. The scheme's performance was measured on randomly generated complex and real matrices, to verify the theoretical results and demonstrate its superiority over the existing methods. Furthermore, a large number of distinct approaches derived using the proposed family were tested numerically, to determine the optimal parametric value, leading to a successful conclusion. [ABSTRACT FROM AUTHOR]

Let H be a separable infinite dimensional complex Hilbert space and B (H) be the set of all bounded linear operators on H. In this paper, we present several conditions under which the distributive law of the parallel sum is valid. It is proved that the parallel sum for positive operators with closed ranges is continued at 0. For A , B ∈ B (H) with closed ranges, it is proved that A ≤ ¯ B if and only if A and B − A are parallel summable with the parallel sum A : (B − A) = 0 , where the symbol " ≤ ¯ " denotes the minus partial order. [ABSTRACT FROM AUTHOR]

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One of the fundamental matrix equalities that involve multiple products of matrices and their generalized inverses is given by A 1 B 1 - A 2 B 2 - ... A k B k - A k + 1 = A , where A 1 , B 1 , A 2 , B 2 , ... , A k , B k , A k + 1 , and A are matrices of appropriate sizes, and B 1 - , B 2 - , ... , B k - are generalized inverses of matrices. Many matrix equality problems related to generalized inverses, including various reverse-order laws for generalized inverses of matrix products, can be reduced to the special situations of this general equality. Generally speaking, this equality does not necessarily hold, and therefore, it is a primary task to establish necessary and sufficient conditions for it to hold for some generalized inverses of matrices, or to always hold for all generalized inverses of matrices. The two basic cases of the equality problem for k = 1 , 2 and their special forms were well known and approached in the discipline of generalized inverses of matrices. As a further exploration on this kind of matrix, this note explores performances of the equality for k = 3 . We shall present an algebraic procedure to derive explicit necessary and sufficient conditions for the equality to always hold through the skilful use of rank equalities for some block matrices that are constructed from the given matrices, and then mentions the ideas and techniques of block matrix representations in the characterization of the above general matrix equality. [ABSTRACT FROM AUTHOR]

For a closed linear relation everywhere defined on a Hilbert space the concepts of isometry, co-isometry, partial isometry, and generalized inverse are introduced and studied. Part of the results proved in this paper improve and generalize some results known for these concepts. In particular, we extend those of [Acta Sci. Math. (Szeged), 70 (2004), 767–781] and [Studia Math. 205 (2011), no. 1, 71–82]. [ABSTRACT FROM AUTHOR]

Utilizing the stability characterizations of generalized inverses, we investigate the generalized resolvent of linear operators in Banach spaces. We first prove that the local analyticity of the generalized resolvent is equivalent to the continuity and the local boundedness of generalized inverse functions. We also prove that several properties of the classical spectrum remain true in the case of the generalized one. Finally, we elaborate on the reason why we use the generalized inverse rather than the Moore-Penrose inverse or the group inverse to define the generalized resolvent. [ABSTRACT FROM AUTHOR]

In this paper, we study (b, c)-invertibility of some arbitrary (b, c)-inverse and we give an equivalent condition for its (b, c)-invertibility. Moreover, we investigate when some well known properties of generalized inverses hold for (b, c)-inverses. Further, we study when the (b, c)-inverse of the unity is equal to the unity itself and we obtain equivalent conditions for this to hold. In addition, we investigate the reverse order law for the (b, c)-inverse. Finally, we prove that in the case when the unity is (b, c)-invertible, the set of all (b, c)-inverses, with respect to multiplication, is a group. [ABSTRACT FROM AUTHOR]

Skew-symmetric matrices of order 7 defined through the 2-fold vector cross product in ℂ7, and other related matrices, are presented. More concretely, matrix properties, namely invertibility, nullspace, powers and index, are studied. As a consequence, results on vector cross product equations, vector cross product differential equations and vector cross product difference equations in ℂ7 are established.

We give a constructive sufficient condition for a matrix over a commutative ring to be von Neumann regular, and we show that it is also necessary over local rings. Specifically, we prove that a matrix A over a local commutative ring is von Neumann regular if and only if A has an invertible ρ (A) × ρ (A) -submatrix if and only if the determinantal rank ρ (A) and the McCoy rank of A coincide. We deduce consequences to (products of local) commutative rings, and we determine the number of von Neumann regular matrices over some finite rings of residue classes and group algebras. [ABSTRACT FROM AUTHOR]

The common characterizations and various individual properties of different generalized inverses are established. Several equivalent conditions for the core-EP, weak group inverse, m-weak group inverse and weak core inverse are presented. The brand new explicit expressions for the operator binary relations defined by various generalized inverses are obtained. We derive properties and study the relationship between two different elements A , B ∈ B (H) and their (T , S) -outer generalized inverses A T , S (2) and B T , S (2) through which the reverse order law results about (A B) ◯ w = B ◯ w A ◯ w and (A B) ◯ d = B ◯ d A ◯ d are obtained. [ABSTRACT FROM AUTHOR]

Summary: This article introduces a robust and affordable method to compute nullspace and generalized inverse of finite element operators involved in dual domain decomposition methods. The methodology relies on the operator partial factorization and on the analysis of a well chosen Schur complement. The sparse linear operator is interpreted as a network and graph centrality measures are used to select the condensation variables. Eigenvector, Katz and Page Rank centralities are evaluated. An extension to deal with symmetric indefinite systems arising from mixed finite elements is also presented. The approach is assessed on highly heterogeneous problems and one industrial application is presented: the numerical homogenization of solid propellant. [ABSTRACT FROM AUTHOR]

We characterize some variations of pseudoskeleton (also called CUR) decompositions for matrices and tensors over arbitrary fields. These characterizations extend previous results to arbitrary fields and to decompositions which use generalized inverses of the constituent matrices, in contrast to Moore–Penrose pseudoinverses in prior works which are specific to real or complex valued matrices, and are significantly more structured. [ABSTRACT FROM AUTHOR]

This article presents the "Generalized Modified Matrix Bessel Distribution (GMMaB)", which is a generalization of the "Generalized Multivariate Modified Bessel Distribution" (GMMB). The features of the suggested distribution, including characteristic functions (C.f.), moments, marginal and conditional distributions, and the linear combination of Matrix variate in several forms have been investigated. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate solvability of systems of constrained matrix equations in the form of constrained minimization problems. The main novelty of this paper is the unification of solutions of considered matrix equations with corresponding minimization problems. For a particular case we extend some well-known results and give several new results for the weak Drazin inverse. The main characterizations of the Drazin inverse, group inverse and Moore–Penrose inverse are obtained as consequences. [ABSTRACT FROM AUTHOR]

Given graphs G1 and G2, let E(G1) = {e1, e2,... em1} be the edge set of G1. A new join of two graph G1 and G2, denoted by G1 ⊙ G2, can be obtained from one copy of G1 and m1 copied of G2 by adding a new vertex corresponding to each edge of G1, letting the resulting new vertex set be U = {u1, u2,...,um1}, and joinning ui with each vertex of i-th copy of G2 and with endpoints of ei for i = 1, 2,...m1. In this paper, the explicit closed formulas of resistance distance and Kirchhoff index of G1⊙G2 whenever G1 is an r1-regular graph and G2 is arbitrary graphs are obtianed. [ABSTRACT FROM AUTHOR]

Let f (X 1 , X 2 , ... , X k) be a matrix function over the field of complex numbers, where X 1 , X 2 , ... , X k are a family of matrices with variable entries. The purpose of this paper is to propose and investigate the relationships between certain linear matrix functions that regularly appear in matrix theory and its applications. We shall derive a series of meaningful, necessary, and sufficient conditions for the collections of values of two given matrix functions to be equal through the cogent use of some highly selective formulas and facts regarding ranks, ranges, and generalized inverses of block matrix operations. As applications, we discuss some concrete topics concerning the algebraic connections between general solutions of a given linear matrix equation and its reduced equations. [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]

Skew-symmetric matrices of order 7 defined through the 2-fold vector cross product in C7, and other related matrices, are presented. More concretely, matrix properties, namely invertibility, nullspace, powers and index, are studied. As a consequence, results on vector cross product equations, vector cross product differential equations and vector cross product difference equations in C7 are established. [ABSTRACT FROM AUTHOR]

Skew-symmetric matrices of order 7 defined through the 2-fold vector cross product in C7, and other related matrices, are presented. More concretely, matrix properties, namely invertibility, nullspace, powers and index, are studied. As a consequence, results on vector cross product equations, vector cross product differential equations and vector cross product difference equations in C7 are established. [ABSTRACT FROM AUTHOR]