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

Castro González, Nieves, Robles Santamarta, Juan ORCID: https://orcid.org/0000-0002-4509-7446 and Vélez Cerrada, José Ygnacio (2008). Characterizations of a class of matrices and perturbation of the Drazin inverse. "Siam Journal on Matrix Analysis and Applications", v. 30 (n. 2); pp. 882-897. ISSN 0895-4798.

Description

Title: Characterizations of a class of matrices and perturbation of the Drazin inverse
Author/s:
Item Type: Article
Título de Revista/Publicación: Siam Journal on Matrix Analysis and Applications
Date: September 2008
ISSN: 0895-4798
Volume: 30
Subjects:
Faculty: Facultad de Informática (UPM)
Department: Matemática Aplicada
UPM's Research Group: singular matrix, Drazin inverse, eigenprojectors, perturbation
Creative Commons Licenses: Recognition - No derivative works - Non commercial

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Abstract

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.

More information

Item ID: 2159
DC Identifier: https://oa.upm.es/2159/
OAI Identifier: oai:oa.upm.es:2159
Official URL: http://scitation.aip.org/dbt/dbt.jsp?KEY=SJMAEL&Vo...
Deposited by: Memoria Investigacion
Deposited on: 01 Feb 2010 10:56
Last Modified: 20 Apr 2016 11:55
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