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On orderings induced by the Loewner partial ordering

Jan Hauke, Augustyn Markiewicz (1994)

Applicationes Mathematicae

The partial ordering induced by the Loewner partial ordering on the convex cone comprising all matrices which multiplied by a given positive definite matrix become nonnegative definite is considered. Its relation to orderings which are induced by the Loewner partial ordering of the squares of matrices is presented. Some extensions of the latter orderings and their comparison to star orderings are given.

On the convergence theory of double K -weak splittings of type II

Vaibhav Shekhar, Nachiketa Mishra, Debasisha Mishra (2022)

Applications of Mathematics

Recently, Wang (2017) has introduced the K -nonnegative double splitting using the notion of matrices that leave a cone K n invariant and studied its convergence theory by generalizing the corresponding results for the nonnegative double splitting by Song and Song (2011). However, the convergence theory for K -weak regular and K -nonnegative double splittings of type II is not yet studied. In this article, we first introduce this class of splittings and then discuss the convergence theory for these sub-classes...

On the Drazin index of regular elements

Pedro Patrício, António Costa (2009)

Open Mathematics

It is known that the existence of the group inverse a # of a ring element a is equivalent to the invertibility of a 2 a − + 1 − aa −, independently of the choice of the von Neumann inverse a − of a. In this paper, we relate the Drazin index of a to the Drazin index of a 2 a − + 1 − aa −. We give an alternative characterization when considering matrices over an algebraically closed field. We close with some questions and remarks.

On the inverse of the adjacency matrix of a graph

Alexander Farrugia, John Baptist Gauci, Irene Sciriha (2013)

Special Matrices

A real symmetric matrix G with zero diagonal encodes the adjacencies of the vertices of a graph G with weighted edges and no loops. A graph associated with a n × n non–singular matrix with zero entries on the diagonal such that all its (n − 1) × (n − 1) principal submatrices are singular is said to be a NSSD. We show that the class of NSSDs is closed under taking the inverse of G. We present results on the nullities of one– and two–vertex deleted subgraphs of a NSSD. It is shown that a necessary...

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