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.     

Re-nnd generalized inverses

In this paper, we give some necessary and sufficient conditions for the existence of Re-nnd and nonnegative definite {1,3}$\{1,3\}$- and {1,4}$\{1,4\}$-inverses of a matrix A∈C^n×n$A\in {\mathbb{C}}^{n\times n}$ and completely described these sets. Moreover, we prove that the existence of nonnegative definite {1,3}$\{1,3\}$-inverse of a matrix A   is equivalent with the existence of its nonnegative definite {1,2,3}$\{1,2,3\}$-inverse and present the necessary and sufficient conditions for the existence of Re-nnd {1,3,4}$\{1,3,4\}$-inverse of A.     

Gauss and generalized inverses

Although Gauss did not formalize the notion of a generalized inverse, he provided the essential ingredients to produce one.     

Algorithms for generalized inverses

Every rectangular matrix has a Moore-Penrose generalized inverse. The purpose of this paper is to present several computational approaches to the determination of this generalized inverse, which plays a role in least-squares problems.     

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.     

Rate of strong consistency of quasi maximum likelihood estimate in generalized linear models

Under the assumption that in the generalized linear model (GLM) the expectation of the response variable has a correct specification and some other smooth conditions, it is shown that with probability one the quasi-likelihood equation for the GLM has a solution when the sample size n is sufficiently large. The rate of this solution tending to the true value is determined. In an important special case, this rate is the same as specified in the LIL for iid partial sums and thus cannot be improved anymore.     

Analytic perturbation of generalized inverses

We investigate the analytic perturbation of generalized inverses. Firstly we analyze the analytic perturbation of the Drazin generalized inverse (also known as reduced resolvent in operator theory). Our approach is based on spectral theory of linear operators as well as on a new notion of group reduced resolvent. It allows us to treat regular and singular perturbations in a unified framework. We provide an algorithm for computing the coefficients of the Laurent series of the perturbed Drazin generalized inverse. In particular, the regular part coefficients can be efficiently calculated by recursive formulae. Finally we apply the obtained results to the perturbation analysis of the Moore–Penrose generalized inverse in the real domain.     

The representation and approximations of outer generalized inverses

We present a unified representation theorem for the class of all outer generalized inverses of a bounded linear operator. Using this representation we develop a few specific expressions and computational procedures for the set of outer generalized inverses. The obtained result is a generalization of the well-known representation theorem of the Moore--Penrose inverse as well as a generalization of the well-known results for the Drazin inverse and the generalized inverse A_T,S ⁽²⁾. Also, as corollaries we get corresponding results for reflexive generalized inverses.     

A note on generalized inverses

Motivated by too restrictive or even incorrect statements about generalized inverses in the literature, properties about these functions are investigated and proven. Examples and counterexamples show the importance of generalized inverses in mathematical theory and its applications.     

Generalized inverses and generalized splines

An abstract framework in Hilbert space is provided for generalized splines and generalized inverses of operators.     

Banach triples with generalized inverses

Penrose showed in [6] that for every complex rectangular matrix aM_nxm() there exists a unique bM_nxm(), the generalized inverse of a, satisfying the following four conditions: i) a a^*b=a, ii) b a^*a=a, iii) b b^*a=b, iv) a b^*b=b.These condition can be expressed in terms of the triple product     

The representations of generalized inverses of lower triangular operators

In this note, the explicit representations of Moore-Penrose inverses of lower triangular operator matrices are established. As an application, the explicit representations of Bott-Duffin inverses of bounded linear operators on a Hilbert space with respect to a closed subspace are obtained.     

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”.     

Existence theorems of a maximum likelihood estimate from a generalized censored data sample

Summary A new type of random sample, called a generalized censored data sample, is defined. An approach to finding criteria for the existence of a maximum likelihood estiamte from a finite generalized censored data sample is presented. This approach, named the probability contents boundary analysis, gives systematically a number of practical criteria, each of which is effective for various kinds of typical distribution families in statistical analysis.     

Computing determinantal representation of generalized inverses

We investigate implementation of the determinantal representation of generalized inverses for complex and rational matrices in the symbolic package MATHEMATICA. We also introduce an implementation which is applicable to sparse matrices.