Caracterización y representación de una clase de matrices perturbadas en el contexto de la inversa de Drazin

Se presentan resultados que caracterizan una clase de matrices perturbadas, y una representación matricial por bloques, en el contexto de la inversa de Drazin

Characterizations and perturbation analysis of a class of matrices related to core-EP inverses

[EN] Let A, B is an element of C-nxn with ind(A) = k and ind(B) = s and let L-B = (BB)-B-2(sic). A new condition (C-s,C-*): R(A(k)) boolean AND N((B-s)*) = {0} and R(B-s) boolean AND N((A(k))*) = {0}, is defined. Some new characterizations related to core-EP inverses are obtained when B satisfies condition (C-s,C-*). Explicit expressions of B(sic) and BB(sic) are also given. In addition, equivalent conditions, which guarantee that B satisfies condition (C-s,C-*), are investigated. We proved that B satisfies condition (C-s,C-*) if and only if L-B has a fixed matrix form. As an application, upper bounds for the errors parallel to B(sic) - A(sic)parallel to/parallel to A(sic)parallel to and parallel to BB(sic) - AA(sic)parallel to are studied. (c) 2021 Elsevier B.V. All rights reserved.The authors thank the Editor and Reviewers sincerely for their constructive comments and suggestions which have improved the quality of the paper. This research is supported by the National Natural Science Foundation of China (Nos. 11771076, 11871145), the Postgraduate Research and Practice Innovation Program of Jiangsu Province (No. KYCX18 -0053), the China Scholarship Council (File No. 201906090122). The third author is partially supported by Ministerio de Economia y Competitividad of Spain (grant Red de Excelencia MTM2017-90682-REDT) and partially supported by Universidad de Buenos Aires, Argentina. EXP-UBA: 13.019/2017, 20020170100350BAZhou, M.; Chen, J.; Thome, N. (2021). Characterizations and perturbation analysis of a class of matrices related to core-EP inverses. Journal of Computational and Applied Mathematics. 393:1-11. https://doi.org/10.1016/j.cam.2021.113496S11139

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

An Expression of the Drazin Inverse of a Perturbed Matrix

Given a square matrix A and its perturbation matrix E, a new expression for the Drazin inverse BD of B=A+E is derived if AADB2=(AADB)2 or B2AAD=(BAAD)2. Based on the new expression, a bound of the relative error of BD is developed. Some known results in the literature on the Drazin inverse and the perturbation bound are included by this new formula as special cases. A numerical example is given to compare the upper bounds

An Expression of the Drazin Inverse of a Perturbed Matrix

Given a square matrix A and its perturbation matrix E, a new expression for the Drazin inverse BD of B=A+E is derived if AADB2=(AADB)2 or B2AAD=(BAAD)2. Based on the new expression, a bound of the relative error of BD is developed. Some known results in the literature on the Drazin inverse and the perturbation bound are included by this new formula as special cases. A numerical example is given to compare the upper bounds