A new method for computing Moore–Penrose inverse matrices

The Moore–Penrose inverse of an arbitrary matrix (including singular and rectangular) has many applications in statistics, prediction theory, control system analysis, curve fitting and numerical analysis. In this paper, an algorithm based on the conjugate Gram–Schmidt process and the Moore–Penrose inverse of partitioned matrices is proposed for computing the pseudoinverse of an m×n$m\times n$ real matrix A$A$ with m≥n$m\ge n$ and rank r≤n$r\le n$. Numerical experiments show that the resulting pseudoinverse matrix is reasonably accurate and its computation time is significantly less than that of pseudoinverses obtained by the other methods for large sparse matrices.     

A simultaneous canonical form of a pair of matrices and applications involving the weighted Moore–Penrose inverse

In this paper, a simultaneous canonical form of a pair of rectangular complex matrices is developed. Using this new tool we give a necessary and sufficient condition to assure that the reverse order law is valid for the weighted Moore–Penrose inverse. Additionally, we characterize matrices ordered by the weighted star partial order and adjacent matrices as applications.     

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.     

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.     

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 a companion matrix

Necessary and sufficient conditions are given for the Moore–Penrose inverse of a companion matrix over an arbitrary ring to exist.     

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.     

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.     

Determinantal representation of the Moore–Penrose inverse matrix over the quaternion skew field

Within the framework of the theory of column and row determinants, we have obtained the determinantal representation of the Moore–Penrose inverse matrix over the quaternion skew field.     

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).     

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.     

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.     

Further results on the Moore–Penrose invertibility of projectors and its applications

The aim of this article is to investigate new results on the Moore–Penrose invertibility of the products and differences of projectors and generalized projectors. The range relations of projectors and the detailed representations for Moore–Penrose inverses are presented.