On the independence of blocks of generalized inverses of bordered matrices

Necessary and sufficient conditions are given for the blocks in the (1)-, (1,3)-, and (1,4)-inverses of a general bordered matrix to be independent of each other.     

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.     

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].     

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.     

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.     

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 matrices over an arbitrary semiring and their generalized inverses

In this paper, we consider matrices with entries from a semiring S. We first discuss some generalized inverses of rectangular and square matrices. We establish necessary and sufficient conditions for the existence of the Moore–Penrose inverse of a regular matrix. For an m×n$m\times n$ matrix A  , an n×m$n\times m$ matrix P and a square matrix Q of order m, we present necessary and sufficient conditions for the existence of the group inverse of QAP   with the additional property that P(QAP)^#Q$P{(QAP)}^{\mathrm{\#}}Q$ is a {1,2}$\{1,2\}$ inverse of A  . The matrix product used here is the usual matrix multiplication. The result provides a method for generating elements in the set of {1,2}$\{1,2\}$ inverses of an m×n$m\times n$ matrix A starting from an initial {1} inverse of A  . We also establish a criterion for the existence of the group inverse of a regular square matrix. We then consider a semiring structure (_Mm×n(S),+,°)$({\mathbf{M}}_{m\times n}(\mathbf{S}),+,°)$ made up of m×n$m\times n$ matrices with the addition defined entry-wise and the multiplication defined as in the case of the Hadamard product of complex matrices. In the semiring (_Mm×n(S),+,°)$({\mathbf{M}}_{m\times n}(\mathbf{S}),+,°)$, we present criteria for the existence of the Drazin inverse and the Moore–Penrose inverse of an m×n$m\times n$ matrix. When S is commutative, we show that the Hadamard product preserves the Hermitian property, and provide a Schur-type product theorem for the product A°(CC^?)$A°(C{C}^{?})$ of a positive semidefinite n×n$n\times n$ matrix A   and an n×n$n\times n$ matrix C.     

On Hermitian generalized inverses and positive semidefinite generalized inverses

The main aim of this paper is to investigate the Hermitian and positive semidefinite generalized inverses of a square matrix. First, we present some conditions for the existence of Hermitian and positive semidefinite generalized inverses. Further, expressions of these generalized inverses are given. Finally, we give two numerical examples to demonstrate our results.     

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 inverses of Hessenberg matrices

The lower half of the inverse of a lower Hessenberg matrix is shown to have a simple structure. The result is applied to find an algorithm for finding the inverse of a tridiagonal matrix. With minor modifications, the technique applies to block Hessenberg matrices.     

Further results on integral generalized inverses of integral matrices

This paper further investigates integral generalized inverses of integral matrices.     

On the existence of generalized inverses

Conditions are given for the existence of a generalized inverse of a ring morphism, and the results are related to the theory of semi-simple Artinian rings. Necessary and sufficient conditions are also given for the existence of a generalized inverse of a morphism on real inner product spaces.     

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.     

Further results on integral generalized inverses of integral matrices

This paper further investigates integral generalized inverses of integral matrices.     

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.     

On the inverses of general tridiagonal matrices

In this work, the sign distribution for all inverse elements of general tridiagonal H-matrices is presented. In addition, some computable upper and lower bounds for the entries of the inverses of diagonally dominant tridiagonal matrices are obtained. Based on the sign distribution, these bounds greatly improve some well-known results due to Ostrowski (1952) 23, Shivakumar and Ji (1996) 26, Nabben (1999) [21] and [22] and recently given by Peluso and Politi (2001) 24, Peluso and Popolizio (2008) 25 and so forth. It is also stated that the inverse of a general tridiagonal matrix may be described by 2n-2 parameters ( and ) instead of 2n+2 ones as given by El-Mikkawy (2004) 3, El-Mikkawy and Karawia (2006) 4 and Huang and McColl (1997) 10. According to these results, a new symbolic algorithm for finding the inverse of a tridiagonal matrix without imposing any restrictive conditions is presented, which improves some recent results. Finally, several applications to the preconditioning technology, the numerical solution of differential equations and the birth-death processes together with numerical tests are given.