The {2}-inverse with applications in statistics

An inverse G of a given matrix A which satisfies the property GAG = G is known as a {2}-inverse. This paper presents a three-phase inversion procedure for which the {2}-inverse is a special case. We present the geometry of {2}-inverses and show that, starting from {2}-inverses, various types of generalized inverses can be derived. Two examples of the occurrence of {2}-inverses in statistics are given: one concerning the constrained least-squares estimator, the other concerning a necessary and sufficient condition for a quadratic form of singular multivariate normal variates to follow a chi-square distribution.     

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.     

Rank equalities related to outer inverses of matrices and applications

A matrix X is called an outer inverse for a matrix A if XAX=X. In this paper, we present some basic rank equalities for difference and sum of outer inverses of a matrix, and apply them to characterize various equalities related to outer inverses, Moore-Penrose inverses, group inverses, Drazin inverses and weighted Moore-Penrose inverses of matrices.     

Boolean designs and self-dual matroids

A proper splitting of a rectangular matrix A is one of the form A = M ? N, where A and M have the same range and null spaces. This concept was introduced by R. Plemmons as a means of generalizing to rectangular and singular matrices the concept of a regular splitting of a nonsingular matrix as introduced by R. Varga. In consideration of the linear system Ax=b, A. Berman and R. Plemmons used a proper splitting of A into M ? N and showed that the iteration x⁽ⁱ⁺¹⁾=M⁺Nx⁽ⁱ⁾+M⁺b converges to A⁺b, the best least-squares solution to the system, if and only if the spectral radius of M⁺N is less than one. The purpose of this paper is to further develop the characteristics of proper splittings and to extend these previous results by replacing the Moore-Penrose generalized inverse with a least-squares g-inverse, a minimum-norm g-inverse, or a g-inverse. Also, some criteria are given for comparing convergence rates of M_i^?N_i, where A = M₁?N₁ = M₂?N₂, and a method is developed for constructing proper splittings of special types of matrices.     

Generalized inverses on normed vector spaces

We extend the concepts, introduced by C.R. Rao for Euclidean norms, of minimum g-inverses and least square g-inverses, using arbitrary norms. We give a characterization of such generalized inverses and an application to the case in which the norm is l_∞. As a result of this application we obtain that when $\text{A?}\text{C}{}^{\mathrm{(n+1)\times n}}$ has rank n, there exists a generalized inverse of A, which serves the same purpose as the Moore-Penrose inverse, when the norm is l_∞.     

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.     

On generalized inverses of Boolean Matrices

Generalized inverses of Boolean Matrices are defined and the general form of matrices having generalized inverses is determined. Some structure theorems are proved, from which, some known results are obtained as corollaries. An algorithm to compute a generalized inverse of a matrix, when it exists, is given. The existence of various types of g-inverses is also investigated. All the results are obtained first for the {0,1}-Boolean algebra and then extended to an arbitrary Boolean algebra.     

1–2 Inverses and the invariance of BA+C

The most general 1–2 inverse is found for a matrix over a regular ring. This result is used to give necessary and sufficient conditions for the invariance of the triplet BA'C, where A' is a 1-inverse or a 1–2 inverse of A.     

Existence Criteria and Expressions of the (<Emphasis Type="Italic">b</Emphasis>, <Emphasis Type="Italic">c</Emphasis>)-Inverse in Rings and Their Applications

Let R be a ring. Existence criteria for the (b, c)-inverse are given. We present explicit expressions for the (b, c)-inverse by using inner inverses. We answer the question when the (b, c)-inverse of \(a\in R\) is an inner inverse of a. As applications, we give a unified theory of some well-known results of the \(\{1,3\}\)-inverse, the \(\{1,4\}\)-inverse, the Moore–Penrose inverse, the group inverse and the core inverse.     

Generalized inverses of tensors via a general product of tensors

We define the {i}-inverse (i = 1, 2, 5) and group inverse of tensors based on a general product of tensors. We explore properties of the generalized inverses of tensors on solving tensor equations and computing formulas of block tensors. We use the {1}-inverse of tensors to give the solutions of a multilinear system represented by tensors. The representations for the {1}-inverse and group inverse of some block tensors are established.     

二元Boolean矩阵的加权Moore-Penrose逆

主要研究了二元Boolean矩阵A的加权Moore-Penrose逆的存在性问题,给出了二元Boolean矩阵A的加权Moore-Penrose逆存在的一些充分必要条件,并讨论了加权Moore-Penrose逆存在时的若干等价刻画及惟一性问题.     

The Moore-Penrose inverse of a morphism with factorization

Given an m-by-n matrix A of rank r over a field with an involutory automorphism, it is well known that A has a Moore-Penrose inverse if and only if rank A¹A=r= rank AA¹. By use of the full-rank factorization theorem, this result may be restated in the category of finite matrices as follows: if (A₁, r, A₂) is an (epic, monic) factorization of A:m→n through r, then A has a Moore-Penrose inverse if and only if (A¹A₁, r, A₂) and (A₁, r, A₂A¹) are, respectively, (epic, monic) factorizations of A¹A:n→n and AA¹:m→m through r. This characterization of the existence of Moore-Penrose inverses is extended to arbitrary morphisms with (epic, monic) factorizations.     

Inverses of banded matrices

We establish a correspondence between the vanishing of a certain set of minors of a matrix A and the vanishing of a related set of minors of A^×1. In particular, inverses of banded matrices are characterized. We then use our results to find patterns for Toeplitz matrices with banded inverses. Finally we give an interesting determinant formula for inverses of banded matrices, and show that in general a “banded partial” matrix may be completed in a unique way to give a banded inverse of the same bandwidth.     

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.     

Perturbations and expressions for generalized inverses in Banach spaces and Moore-Penrose inverses in Hilbert spaces of closed linear operators

The problems of perturbation and expression for the generalized inverses of closed linear operators in Banach spaces and for the Moore-Penrose inverses of closed linear operators in Hilbert spaces are studied. We first provide some stability characterizations of generalized inverses of closed linear operators under T-bounded perturbation in Banach spaces, which are exactly equivalent to that the generalized inverse of the perturbed operator has the simplest expression T⁺(I+δTT⁺)^-1. Utilizing these results, we investigate the expression for the Moore-Penrose inverse of the perturbed operator in Hilbert spaces and provide a unified approach to deal with the range preserving or null space preserving perturbation. An explicit representation for the Moore-Penrose inverse of the perturbation is also given. Moreover, we give an equivalent condition for the Moore-Penrose inverse to have the simplest expression T^†(I+δTT^†)^-1. The results obtained in this paper extend and improve many recent results in this area.     

The Moore-Penrose inverse of a retrocirculant

In a recent paper Chao [2] has determined the eigenvalues of a matrix of the form A=PC where P is a permutation matrix which commutes with a certain unitary matrix and C is a circulant. Here we determine the Moore-Penrose inverse of such a “retrocirculant” and show that the nonzero eigenvalues of the Moore-Penrose inverse are the reciprocals of the nonzero eigenvalues of the retrocirculant.     

New results on (b,c)–inverses

In this paper, we present some necessary and sufficient conditions for the existence of the (b, c)-inverse and several representations for the (b, c)-inverse related to the group inverse. Since Mary inverse, core inverse, dual core inverse and Bott–Duffin (e, f)-inverse are all the special cases of the (b, c)-inverse, related results for these inverses are obtained.     

Unicyclic graphs with bicyclic inverses

A graph is nonsingular if its adjacency matrix A(G) is nonsingular. The inverse of a nonsingular graph G is a graph whose adjacency matrix is similar to A(G)^?1 via a particular type of similarity. Let H denote the class of connected bipartite graphs with unique perfect matchings. Tifenbach and Kirkland (2009) characterized the unicyclic graphs in H which possess unicyclic inverses. We present a characterization of unicyclic graphs in H which possess bicyclic inverses.     

The rank of a difference of matrices and associated generalized inverses

Various representations are given to characterize the rank of A-S in terms of rank A+k where A and S are arbitrary complex matrices and k is a function of A and S. It is shown that if S=AMA for some matrix M, and if G is any matrix satisfying A=AGA, then

$\text{rank(A-S) = rankA-nullity (I-SG)}$

. Several alternative forms of this result are established, as are many equivalent conditions to have

$\text{rank(A-S) = rankA-rankS}$

. General forms for the Moore-Penrose inverse of matrices A-S are developed which include as special cases various results by Penrose, Wedin, Hartwig and others.     

About the generalized LM-inverse and the weighted Moore-Penrose inverse

The recursive method for computing the generalized LM-inverse of a constant rectangular matrix augmented by a column vector is proposed in Udwadia and Phohomsiri (2007) [16] and [17]. The corresponding algorithm for the sequential determination of the generalized LM-inverse is established in the present paper. We prove that the introduced algorithm for computing the generalized LM-inverse and the algorithm for the computation of the weighted Moore-Penrose inverse developed by Wang and Chen (1986) in [23] are equivalent algorithms. Both of the algorithms are implemented in the present paper using the package MATHEMATICA. Several rational test matrices and randomly generated constant matrices are tested and the CPU time is compared and discussed.