Néstor Thome

On a new generalized inverse for matrices of an arbitrary index

The purpose of this paper is to introduce a new generalized inverse, called DMP inverse, associated with a square complex matrix using its Drazin and Moore-Penrose inverses. DMP inverse extends the notion of core inverse, introduced by Baksalary and Trenkler for matrices of index at most 1 in (Baksalary and Trenkler (2010) [1]) to matrices of an arbitrary index. DMP inverses are analyzed from both algebraic as well as geometrical approaches establishing the equivalence between them.

{ k }-Group Periodic Matrices

In this paper we deal with two problems related to {k}-group periodic matrices (i.e.,

The generalized Schur complement in group inverses and (k + 1)-potent matrices

A^{\#} = A^{k-1}

Idempotency of linear combinations of an idempotent matrix and a t -potent matrix that do not commute

, where

An algorithm to check the nonnegativity of singular systems

A^{\#}

A geometrical approach on generalized inverses by Neumann-type series

is the group inverse of a matrix A). First, we give different characterizations of {k}-group periodic matrices. Later, we present characterizations of the {k}-group periodic matrices for linear combinations of projectors. This work extends some well-known results in the literature.

Further properties on the core partial order and other matrix partial orders

In this article, two facts related to the generalized Schur complement are studied. The first one is to find necessary and sufficient conditions to characterize when the group inverse of a partitioned matrix can be expressed in the Schur form. The other one is to develop a formula for any power of the generalized Schur complement of an idempotent partitioned matrix and then to characterize when this generalized Schur complement is a (k+1)-potent matrix. In addition, some spectral theory related to this complement is analyzed.

The diamond partial order in rings

The problem of characterizing all situations in which aA + bB is an idempotent matrix when A 2 = A, B k + 1 = B, AB ≠ BA, and a, b are nonzero complex numbers is studied.

The star partial order and the eigenprojection at 0 on EP matrices

In the literature, an important class of generalized inverse matrices corresponds to the group inverse, that is, matrices of index 1. Recently, the nonnegativity of a singular system has been applied to different fields. In this paper, an algorithm to check the nonnegativity of a singular linear control system of index 1 is presented. To this purpose, the nonnegativity of this kind of systems is characterized using a block partition of the original matrices. In this way, we can work with matrices having smaller sizes and keeping the original information. Finally, numerical examples illustrating the obtained results are shown.

Weighted binary relations involving the Drazin inverse

The convergence of the Neumann-type series to {1,2}-inverses has been shown by K. Tanabe [Linear Algebra Appl. 10 (1975) 163]. In this paper, these results indicating conditions characterizing the convergence of this series to different generalized inverses are extended. In addition, these results for obtaining different generalized inverses from the hyperpower method are applied. Finally, generalized involutory matrices are introduced and characterized using the obtained results.