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Simultaneous extension of generalized BT-inverses and core-EP inverses

Abdessalam Kara, Néstor Thome, Dragan S. Djordjevi'c

TL;DR

The paper introduces the generalized inverse of a square matrix with respect to another same-sized matrix, denoted $A^{(B)}$, and shows how it generalizes both the BT-inverse and core-EP inverse via natural special cases. It develops two representations of $A^{(B)}$ through the $B$-decomposition and a core-EP–based decomposition, providing explicit canonical forms. The work then establishes key properties and characterizations of $A^{(B)}$ and connects $A^{(B)}$ to weighted inverses, including $A^{circ,W}$ and $A^{\diamond,W}$, with concrete block representations and a discussion of when these notions coincide. Overall, the results extend the framework of generalized inverses with prescribed range and null-space, offering new tools for analysis and potential numerical applications.

Abstract

In this paper we introduce the generalized inverse of complex square matrix with respect to other matrix having same size. Some of its representations, properties and characterizations are obtained. Also some new representation matrices of W-weighted BT-inverse and W-weighted core-EP inverse are determined as well as characterizations of generalized inverses A A^\odagger, A^{odagger,W}, A^\diamond, A^{\diamond,W}.

Simultaneous extension of generalized BT-inverses and core-EP inverses

TL;DR

The paper introduces the generalized inverse of a square matrix with respect to another same-sized matrix, denoted , and shows how it generalizes both the BT-inverse and core-EP inverse via natural special cases. It develops two representations of through the -decomposition and a core-EP–based decomposition, providing explicit canonical forms. The work then establishes key properties and characterizations of and connects to weighted inverses, including and , with concrete block representations and a discussion of when these notions coincide. Overall, the results extend the framework of generalized inverses with prescribed range and null-space, offering new tools for analysis and potential numerical applications.

Abstract

In this paper we introduce the generalized inverse of complex square matrix with respect to other matrix having same size. Some of its representations, properties and characterizations are obtained. Also some new representation matrices of W-weighted BT-inverse and W-weighted core-EP inverse are determined as well as characterizations of generalized inverses A A^\odagger, A^{odagger,W}, A^\diamond, A^{\diamond,W}.
Paper Structure (7 sections, 20 theorems, 41 equations)

This paper contains 7 sections, 20 theorems, 41 equations.

Table of Contents

  1. Introduction and preliminary
  2. Definition of $A^{(B)}$ and some representations
  3. Some properties of $A^{(B)}$
  4. Some characterizations of $A^{(B)}$
  5. Some characterizations of $A^{\mathrel{\text{\textcircled{$\dagger$}}}},A^{\mathrel{\text{\textcircled{$\dagger$}}},W}, A^{\diamond}, A^{\diamond,W}$
  6. Acknowledgements
  7. Funding

Key Result

Lemma 1.1

FeLeTh1 Let $B\in {\mathbb C}^{n \times m}$ be a nonzero matrix, $A\in {\mathbb C}^{m \times n}$ and $k= \max\{{\rm Ind}(AB), {\rm Ind}(BA)\}$. Then there exist two unitary matrices $U \in {\mathbb C}^{m \times m}$, $V \in {\mathbb C}^{n \times n}$, two nonsingular matrices $A_{1}, B_{1}\in {\mathbb

Theorems & Definitions (38)

  • Lemma 1.1
  • Lemma 1.2
  • Theorem 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 3.1
  • ...and 28 more