Moore–Penrose inverse of an invertible infinite matrix

In this article we show that, contrary to finite matrices (with real or complex entries) an invertible infinite matrix V could have a Moore–Penrose inverse that is not a classical inverse of V. This also answers a recent open problem on infinite matrices.     

The Moore–Penrose inverse of block magic rectangles

As is known, a semi-magic square is an n?×?n matrix having the sum of entries in each row and each column equal to a constant. This note generalizes this notion and introduce a special class of block matrices called block magic rectangles. It is proved that the Moore–Penrose inverse of a block magic rectangle is also a block magic rectangle.     

The Moore–Penrose inverse of the normalized graph Laplacian

We prove a formula that relates the Moore–Penrose inverses of two matrices A,B$A,B$ such that A=N⁻¹BM⁻¹$A=N^{-1}B{M}^{-1}$ and discuss some applications, in particular to the representation of the Moore–Penrose inverse of the normalized Laplacian of a graph. The Laplacian matrix of an undirected graph is symmetric and is strictly related to its connectivity properties. However, our formula applies to asymmetric matrices, so that we can generalize our results for asymmetric Laplacians, whose importance for the study of directed graphs is increasing.     

Reverse order law for the weighted Moore–Penrose inverse in C *-algebras

In this paper we give a positive solution to a conjecture on the reverse order law for the weighted Moore–Penrose inverse in C ^*-algebras (Mosi? and Djordjevi? in Electron. J. Linear Algebra 22:92–111, 2011).     

Range projections and the Moore–Penrose inverse in rings with involution

In this article we study the existence of range projections in rings with involution, relating it to the existence of the Moore–Penrose inverse. The results are applied to the solution of the equation xbx?=?x in rings with involution, extending the results of Greville for matrices. Simpler new proofs are given of the Moore–Penrose invertibility of regular elements in rings with involution, and of the Ljance's formula.     

Convergence of Rump’s method for computing the Moore-Penrose inverse

We extend Rump’s verified method (S.Oishi, K.Tanabe, T.Ogita, S.M.Rump (2007)) for computing the inverse of extremely ill-conditioned square matrices to computing the Moore-Penrose inverse of extremely ill-conditioned rectangular matrices with full column (row) rank. We establish the convergence of our numerical verified method for computing the Moore-Penrose inverse. We also discuss the rank-deficient case and test some ill-conditioned examples. We provide our Matlab codes for computing the Moore-Penrose inverse.     

Some identities for Moore–Penrose inverses of matrix products

A group of identities are established for the Moore–Penrose inverses and the weighted Moore–Penrose inverses of matrix products AB and ABC. Some consequences and applications are also presented.     

A new method of solving the coefficient inverse problem

This paper is concerned with the new method for solving the coefficient inverse problem in the reproducing kernel space. It is different from the previous studies. This method gives accurate results and shows that it is valid by the numerical example.     

The Moore–Penrose inverse of the incidence matrix of complete multipartite and bi-block graphs

A bi-block graph is a connected graph all of whose blocks are complete bipartite graphs. We give a combinatorial interpretation of the Moore–Penrose inverse of the incidence matrix of a complete multipartite graph and a bi-block graph.     

On Weighted Reverse Order Laws for the Moore–Penrose Inverse and K-Inverses

The main objective of this article is to study several generalizations of the reverse order law for the Moore–Penrose inverse in ring with involution.     

A new central polynomial for 3 × 3 matrices

Abstract

Orthodox semigroups have been studied by many authors, in particular by Hall, Yamada and Petrich. In this paper, we give the standard representation of orthodox semigroups and investigate various e-varieties of orthodox semigroups which are determined by the standard representations.     

A new predictor–corrector method for solving unconstrained minimization problems

This paper presents a new predictor–corrector method for finding a local minimum of a twice continuously differentiable function. The method successively constructs an approximation to the solution curve and determines a predictor on it using a technique similar to that used in trust region methods for unconstrained optimization. The proposed predictor is expected to be more effective than Euler's predictor in the sense that the former is usually much closer to the solution curve than the latter for the same step size. Results of numerical experiments are reported to demonstrate the effectiveness of the proposed method.     

A Riemann–Hilbert approach to computing the inverse spectral map for measures supported on disjoint intervals

We develop a numerical method for computing with orthogonal polynomials that are orthogonal on multiple, disjoint intervals for which analytical formulae are currently unknown. Our approach exploits the Fokas–Its–Kitaev Riemann–Hilbert representation of the orthogonal polynomials to produce an $O(N)$ method to compute the first N recurrence coefficients. The method can also be used for pointwise evaluation of the polynomials and their Cauchy transforms throughout the complex plane. The method encodes the singularity behavior of weight functions using weighted Cauchy integrals of Chebyshev polynomials. This greatly improves the efficiency of the method, outperforming other available techniques. We demonstrate the fast convergence of our method and present applications to integrable systems and approximation theory.