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The $W$-weighted $m$-weak core inverse

D. E. Ferreyra, D. Mosic

Abstract

Recently, Malik and Ferreyra introduced the $m$-weak core inverse for complex square matrices which generalizes the core-EP inverse, the WC inverse, and therefore the core inverse. The main aim of this paper is to extend the concept of $m$-weak core inverse for complex rectangular matrices. This extension is called the $W$-weighted $m$-weak core inverse. We analyse its existence and uniqueness as solution of a system of matrix equations. We present various properties, representations and characterizations of the $W$-weighted $m$-weak core inverse, as well as its applications in solving certain matrix systems. A canonical form of the $W$-weighted $m$-weak core inverse is also provided by using a simultaneous unitary block upper triangularization of a pair of rectangular matrices.

The $W$-weighted $m$-weak core inverse

Abstract

Recently, Malik and Ferreyra introduced the -weak core inverse for complex square matrices which generalizes the core-EP inverse, the WC inverse, and therefore the core inverse. The main aim of this paper is to extend the concept of -weak core inverse for complex rectangular matrices. This extension is called the -weighted -weak core inverse. We analyse its existence and uniqueness as solution of a system of matrix equations. We present various properties, representations and characterizations of the -weighted -weak core inverse, as well as its applications in solving certain matrix systems. A canonical form of the -weighted -weak core inverse is also provided by using a simultaneous unitary block upper triangularization of a pair of rectangular matrices.
Paper Structure (5 sections, 25 theorems, 63 equations)

This paper contains 5 sections, 25 theorems, 63 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Existence, Uniqueness and Representations
  3. Algebraic characterizations of the $W$-weighted $m$-core inverse
  4. Canonical form of the $W$-weighted $m$-weak core inverse
  5. Applications

Key Result

Proposition 2.3

Let $A \in {{\mathbb{C}}^{p\times n}}$, $0\neq W\in {{\mathbb{C}}^{n\times p}}$, $k=\max\{{\text{\rm Ind}}(AW), {\text{\rm Ind}}(WA)\}$, and $m\in \mathbb{N}$. Then

Theorems & Definitions (51)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 41 more