Characterizations of a class of matrices and perturbation of the Drazin inverse

Este trabajo supone un avance en la caracterización y representación de una clase de matrices perturbadas, para el estudio de la perturbación de la inversa de Drazin. Se obtienen diversas caracterizaciones de las matrices perturbadas: geométrica, algebraica, en función de los rangos, y respecto una representación matricial por bloques. Con estas caracterizaciones se alcanzan expresiones explícitas de la inversa de Drazin de la matriz perturbada, y cotas del error relativo de la perturbación de la inversa de Drazin. Se presentan ejemplos numéricos en los que se comparan las cotas dadas con otras publicadas recientemente en la literatura. Como aplicación, se presentan resultados relativos a la continuidad de la inversa de Drazin. Given a singular square matrix $A$ with index $r$, $\operatorname{ind}(A)=r$, we establish several characterizations in the Drazin inverse framework of the class of matrices $B$, which satisfy the conditions $\mathcal{N}(B^s)\cap\mathcal{R}(A^r)=\{0\}$ and $\mathcal{R}(B^s)\cap\mathcal{N}(A^r)=\{0\}$ with $\operatorname{ind}(B)=s$, where $\mathcal{N}(A)$ and $\mathcal{R}(A)$ denote the null space and the range space of a matrix $A$, respectively. We give explicit representations for $B^{\rm D}$ and $BB^{\rm D}$ and upper bounds for the errors $\|B^{\rm D}-A^{\rm D}\|/\|A^{\rm D}\|$ and $\|BB^{\rm D}-AA^{\rm D}\|$. In a numerical example we show that our bounds are better than others given in the literature

On a partial order defined by the weighted Moore Penrose inverse

The weighted Moore-Penrose inverse of a matrix can be used to define a partial order on the set of m x n complex matrices and to introduce the concept of weighted-EP matrices. In this paper we study the weighted star partial order on the set of weighted-EP matrices. In addition, some properties that relate the eigenprojection at zero with the weighted star partial order are obtained. (C) 2013 Elsevier Inc. All rights reserved.This author was partially supported by Ministry of Education of Spain (Grant DGI MTM2010-18228).Hernández, AE.; Lattanzi, MB.; Thome, N. (2013). On a partial order defined by the weighted Moore Penrose inverse. Applied Mathematics and Computation. 219(14):7310-7318. https://doi.org/10.1016/j.amc.2013.02.010S731073182191

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

[EN] The space of n x n complex matrices with the star partial order is considered in the first part of this paper. The class of EP matrices is analyzed and several properties related to this order are given. In addition, some information about predecessors and successors of a given EP matrix is obtained. The second part is dedicated to the study of some properties that relate the eigenprojection at 0 with the star and sharp partial orders. 2012 Elsevier Inc. All rights reserved.This paper was partially supported by Ministry of Education of Argentina (PPUA, Grant Resol. 228, SPU, 14-15-222) and by Universidad Nacional de La Pampa, Facultad de Ingenieria (Grant Resol. No 049/11).Hernández, AE.; Lattanzi, MB.; Thome, N.; Urquiza, F. (2012). The star partial order and the eigenprojection at 0 on EP matrices. Applied Mathematics and Computation. 218(21):10669-10678. https://doi.org/10.1016/J.AMC.2012.04.034S10669106782182

Nonnegative singular control systems using the Drazin projector

In this work we study conditions for guaranteeing the nonnegativity of a discrete-time singular control system. A first approach can be found in the literature for general systems, using the whole coefficient matrices. Also, the particular case of matrices of index 1 has been treated by using a block decomposition and the group-projector of the matrix that gives the singularity to the system. In order to complete this study, an analysis of the nonnegativity of a singular control system for matrices having arbitrary index is done by means of the core nilpotent decomposition. This technique allows us to reduce the size of the original matrices, improving the results where the whole coefficients are involved.This paper was partially supported by the Ministry of Education of Spain (grant DGI MTM2010-18228).Herrero Debón, A.; Thome Coppo, NJ. (2013). Nonnegative singular control systems using the Drazin projector. Applied Mathematics Letters. 26:799-803. doi:10.1016/j.aml.2013.03.0047998032

Moore-Penrose invertibility in involutory rings: the case aa+=bb+

In this article, we consider Moore-Penrose invertibility in rings with a general involution. Given two von Neumann regular elements a, b in a general ring with an arbitrary involution, we aim to give necessary and sufficient conditions to aa† = bb†. As a special case, EP elements are considered.Fundação para a Ciência e a Tecnologia (FCT)POCT

The class of m-EP and m-normal matrices

The well-known classes of EP matrices and normal matrices are de- fined by the matrices that commute with their Moore-Penrose inverse and with their conjugate transpose, respectively. This paper investigates the class of m-EP matrices and m-normal matrices that provide a generalization of EP matrices and normal matrices, respectively, and analyzes both of them for their properties and characterizations.Third author was partially supported by Ministerio de Economia y Competitividad of Spain [grant number DGI MTM2013-43678-P], [Red de Excelencia MTM2015-68805-REDT].Malik, SB.; Rueda, L.; Thome, N. (2016). The class of m-EP and m-normal matrices. Linear and Multilinear Algebra. 64(11):2119-2132. https://doi.org/10.1080/03081087.2016.1139037S21192132641

Spectral stability of shock profiles for hyperbolically regularized systems of conservation laws

We report a proof that under natural assumptions shock profiles viewed as heteroclinic travelling wave solutions to a hyperbolically regularized system of conservation laws are spectrally stable, if the shock amplitude is sufficiently small. This means that an associated Evans function $\mathcal{E}:\Lambda\rightarrow\mathbb{C}$ with $\Lambda\subset\mathbb{C}$ an open superset of the closed right half plane $\mathbb{H}^+\equiv\{\kappa\in\mathbb{C}:\text{Re}\,\kappa \geq 0\}$, has only one zero, namely a simple zero at $0$. The result is analogous to the one obtained in [FS02] and [PZ04] for parabolically regularized systems of conservation laws, and also distinctly extends findings on hyperbolic relaxation systems in [PZ04], [MZ09], [Ued09]