Further results on integral generalized inverses of integral matrices

This paper further investigates integral generalized inverses of integral matrices.     

Integral generalized inverses of integral matrices

This paper gives necessary and sufficient conditions for the generalized inverse of an integral matrix to be integral. Also, additional conditions are found for the product of two integral matrices with this property to have that same property.     

On generalized inverses of matrices over integral domains

It is proved that a matrix A over an integral domain admits a 1-inverse if and only if a linear combination of all the r × r minors of A is equal to one, where r is the rank of A. Some results on the existence of Moore-Penrose inverses are also obtained.     

Further results on several types of generalized inverses

Let R be an associative ring with unity 1 and let a,b,c∈R. In this paper, several characterizations for hybrid (b,c)-inverses of a are given. Also, the hybrid (b,c)-inverse of a is characterized by the group inverse of ab, under certain hypothesis. In particular, existence criteria for the the inverse along an element are obtained. Finally, we get the double commutant property and the reverse order law of annihilator (b,c)-inverses.     

Further results on generalized centro-invertible matrices

Numerical Algorithms - This paper deals with generalized centro-invertible matrices introduced by the authors in Lebtahi et al. (Appl. Math. Lett. 38, 106–109, 2014). As a first result, we...     

Further results on iterative methods for computing generalized inverses

In this paper, we reconsider the iterative method X_k=X_k−1+βY(I−AX_k−1), k=1,2,…,β∈C?{0} for computing the generalized inverse over Banach spaces or the generalized Drazin inverse a^d of a Banach algebra element a, reveal the intrinsic relationship between the convergence of such iterations and the existence of or a^d, and present the error bounds of the iterative methods for approximating or a^d. Moreover, we deduce some necessary and sufficient conditions for iterative convergence to or a^d.     

Nonnegative integral generalized inverses

Necessary and sufficient conditions are given for a nonnegative integral matrix to have nonnegative integral generalized inverses of various types, and the possible ranks of these inverses are determined. More generally, conditions are also given for matrices to have generalized inverses over certain subsets of the nonnegative reals forming monoids under addition and multiplication. Many of our results are adapted from results of Berman and Plemmons [3–6] on real matrices.     

Further results on generalized inverses of tensors via the Einstein product

The notion of the Moore–Penrose inverse of tensors with the Einstein product was introduced, very recently. In this paper, we further elaborate on this theory by producing a few characterizations of different generalized inverses of tensors. A new method to compute the Moore–Penrose inverse of tensors is proposed. Reverse order laws for several generalized inverses of tensors are also presented. In addition to these, we discuss general solutions of multilinear systems of tensors using such theory.     

Subclasses of generalized inverses of matrices

Summary Structure of allg-inverses of a matrix in a weak sense is shown. Characterizations of main subclasses ofg-inverses are investigated thoroughly. The dualities among subclasses and the relation betweeng-inverses and projections are stressed. The Gauss-Markov theorem reduces to a duality of two types ofg-inverses A preliminary Japanese version was published by the author in theProc. Inst. Statist. Math., Vol. 17, (1969) with the title “Generalized inverses of matrices—Part 1”.     

Integer generalized inverses of incidence matrices

Graphical procedures are used to characterize the integral {1}- and {1, 2}-inverses of the incidence matrix A of a digraph, and to obtain a basis for the space of matrices X such that AXA = 0. These graphical procedures also produce the Smith canonical form of A and a full rank factorization of A using matrices with entries from {-1, 0, 1}. It is also shown how the results on incidence matrices of oriented graphs can be used to find generalized inverses of matrices of unoriented bipartite graphs.     

On generalized inverses of Boolean matrices—II

Analogous to minimum norm g-inverses and least squares g-inverses for real matrices, we introduce the concepts of minimum weight g-inverses and least distance g-inverses for Boolean matrices. All those Boolean matrices which admit such g-inverses are characterized.This paper is a continuation of [2].     

Generalized inverses of circulant and generalized circulant matrices

An expression for the Moore-Penrose inverse of certain singular circulants by S.R. Searle is generalized to include all circulants. Similar expressions are given for the Moore-Penrose inverse of block circulants with circulant blocks, level-q circulants, k-circulants where |k|=1, and certain other matrices which are the product of a permutation matrix and a circulant. Expressions for other generalized inverses are given.     

On constrained generalized inverses of matrices and their properties

A matrix G is called a generalized inverse (g-invserse) of matrix A if AGA = A and is denoted by G = A ⁻. Constrained g-inverses of A are defined through some matrix expressions like E(AE)⁻, (FA)⁻ F and E(FAE)⁻ F. In this paper, we derive a variety of properties of these constrained g-inverses by making use of the matrix rank method. As applications, we give some results on g-inverses of block matrices, and weighted least-squares estimators for the general linear model.     

On the characterization of generalized inverses by bordered matrices

A method to characterize the class of all generalized inverses of any given matrix A is considered. Given a matrix A and a nonsingular bordered matrix T of A,

$\text{T=}\text{A}\text{P}\text{Q}\text{R}$

the submatrix, corresponding to A, of T^-1 is a generalized inverse of A, and conversely, any generalized inverse of A is obtainable by this method. There are different definitions of a generalized inverse, and the arguments are developed with the least restrictive definition. The characterization of the Moore-Penrose inverse, the most restrictive definition, is also considered.     

Differentiation of generalized inverses for rational and polynomial matrices

Our basic motivation is a direct method for computing the gradient of the pseudo-inverse of well-conditioned system with respect to a scalar, proposed in [13] by Layton. In the present paper we combine the Layton’s method together with the representation of the Moore-Penrose inverse of one-variable polynomial matrix from [24] and developed an algorithm for computing the gradient of the Moore-Penrose inverse for one-variable polynomial matrix. Moreover, using the representation of various types of pseudo-inverses from [26], based on the Grevile’s partitioning method, we derive more general algorithms for computing {1}, {1, 3} and {1, 4} inverses of one-variable rational and polynomial matrices. Introduced algorithms are implemented in the programming language MATHEMATICA. Illustrative examples on analytical matrices are presented.     

Weighted generalized inverses of partitioned matrices in Banachiewicz-Schur form

In this paper the conditions under which the weighted generalized inversesA ^(1,3M), A^(1,4N), A _M,N ^Dg andA ^d,W can be expressed in Banachiewicz-Schur form are considered and some interesting results are established. These results contribute to verify recent results obtained by J. K. Baksalary and G. P. Styan [2] and Y. Wei [15] and these extend their works.