arXiv subjects

Nestor Thome

Publications and source records attributed to Nestor Thome.

7 recordsLinked to original sources

arXiv2026

Hyper-dual group inverse: existence, characterizations, and applications

Motivated by the recent work of Xiao and Zhong [AIMS Math. 9 (2024), 35125--35150: MR4840882], we propose a generalized inverse for a hyper-dual matrix called hyper-dual group generalized inverse (HDGGI). Firstly, we characterize the existence of the HDGGI of a hyper-dual matrix and we show that it is unique, whenever exists. The HDGGI is then used to solve a linear hyper-dual system. We discuss the minimal $P$-norm least-squares properties of hyper-dual group inverse. We also exploit some sufficient conditions under which the reverse and forward-order laws for a particular form of the HDGGI and hyper-dual Moore-Penrose generalized inverse (HDMPGI) hold. Using the definition of dual matrix of order $n$, we finally establish necessary and sufficient condition for the existence of the group inverse of a dual matrix of order $n$.

math.RA↗

arXiv2026

On the $(b, c)$-inverse of a sum with a radical element in a ring

Let $R$ be a ring with identity and $J(R)$ be its Jacobson radical. Assume that $a\in R$ is $(b,c)$-invertible and $j_a,j_b,j_c\in J(R)$. This paper provides necessary and sufficient conditions for $a+j_a$ to be $(b+j_b,c+j_c)$-invertible. As an application, corresponding results on $(\widehat{B},\widehat{C})$-inverses of a dual matrix $\widehat{A}$ are derived.

math.RA↗

arXiv2025

Additive properties and absorption laws for generalized inverses

Let $a,~f$ be elements in a ring with pseudo core inverses $a^{\scriptsize\textcircled{\tiny D}}$, $f^{\scriptsize\textcircled{\tiny D}}$, and let $b=f-a$. We prove that the absorption law $a^{\scriptsize\textcircled{\tiny D}}(a+f)f^{\scriptsize\textcircled{\tiny D}}=a^{\scriptsize\textcircled{\tiny D}}+f^{\scriptsize\textcircled{\tiny D}}$ holds if and only if $1+a^{\scriptsize\textcircled{\tiny D}}b$ is invertible and the additive property $f^{\scriptsize\textcircled{\tiny D}}=(1+a^{\scriptsize\textcircled{\tiny D}}b)^{-1}a^{\scriptsize\textcircled{\tiny D}}$ is satisfied. We further characterize these properties and establish analogous results for other generalized inverses. Finally, we apply these results to the case of complex matrices.

math.RA↗

arXiv2025

Dual Core-EP Generalized Inverse and Decomposition

In this work, we introduce a new type of generalized inverse called dual core-EP generalized inverse (in short DCEPGI) for dual square matrices. We analyze the existence and uniqueness of the DCEPGI inverse and its compact formula using dual Drazin and dual MP inverse. Moreover, some characterizations using core-EP decomposition are obtained. We present a new dual matrix decomposition named the dual core-EP decomposition for square dual matrices. In addition, some relationships with other dual generalized inverses are established. As an application, solutions to some inconsistent system of linear dual equations are derived.

math.NA↗

arXiv2025

Some results on core EP Drazin matrices and partial isometries

In this paper, by using the core EP inverse and the Drazin inverse which are two well known generalized inverses, a new class of matrices entitled core EP Drazin matrices (shortly, CEPD matrices) is introduced. This class contains the set of all EP matrices and also the set of normal matrices. Some algebraic properties of these matrices are also investigated. Moreover, some results about the Drazin inverse and the core EP inverse of partial isometries are derived, and using them, some conditions for which partial isometries are CEPD, are obtained. To illustrate the main results, some numerical examples are given.

math.NA↗

arXiv2024

The core-EP inverse: A numerical approach for its acute perturbation

This paper studies the concept of stable perturbation $B\in\mathbb{C}^{n\times n}$ for the core-EP inverse of a matrix $A\in\mathbb{C}^{n\times n}$ with index $k$. For a given stable perturbation $B$ of $A$, explicit expressions of its core-EP inverse $B^{\scriptsize\textcircled{\tiny $\dagger$}}$ and its projection at zero $B^π$ are presented. Then, the perturbation bounds of $\parallel B^{\scriptsize\textcircled{\tiny $\dagger$}}-A^{\scriptsize\textcircled{\tiny $\dagger$}}\parallel/\parallel A^{\scriptsize\textcircled{\tiny $\dagger$}}\parallel$ and $\parallel B^π-A^π\parallel$ are given provided that $B$ is a stable perturbation of $A$. In addition, we investigate the concept of acute perturbation of $A$. We give a perturbation analysis with respect to core-EP inverses. We provide a condition under which the acute perturbation coincides with the stable perturbation for core-EP inverses.

math.RA↗

arXiv2024

G-Drazin inverse combined with inner inverse

This paper introduces new classes of generalized inverses for square matrices named GD1, and the dual, called 1GD inverse. In addition, we discuss a few characterizations and representations of these inverses. The explicit expressions of these inverses have been established via core-nilpotent decomposition. Further, we introduce a binary relation for GD1 inverse and 1GD inverse, along with a few derived properties.

math.NA↗