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Core EP, Dual Core EP and Composite Generalized Inverses for a Class of Structured Matrices

Faustino Maciala, C. Mendes Araújo, Pedro Patrício

Abstract

We study generalized inverses for matrices associated with double star digraphs. Explicit block formulas and existence criteria are obtained for core, dual core, core EP, and dual core EP inverses, expressed in terms of explicit algebraic criteria derived from the underlying block structure. Other combined outer pseudoinverses, combining Moore--Penrose and core-type inverses, are derived with existence criteria.

Core EP, Dual Core EP and Composite Generalized Inverses for a Class of Structured Matrices

Abstract

We study generalized inverses for matrices associated with double star digraphs. Explicit block formulas and existence criteria are obtained for core, dual core, core EP, and dual core EP inverses, expressed in terms of explicit algebraic criteria derived from the underlying block structure. Other combined outer pseudoinverses, combining Moore--Penrose and core-type inverses, are derived with existence criteria.
Paper Structure (10 sections, 14 theorems, 88 equations, 1 figure, 1 table)

This paper contains 10 sections, 14 theorems, 88 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Case I: $x^T y=0=z^T w.$
  4. Case II: $x^T y \neq 0=z^T w$ and $\zeta=x^T y + ab \neq 0.$
  5. Case III: $x^T y \neq 0=z^T w$ and $\zeta=x^T y + ab =0.$
  6. Core and dual core invertibility
  7. Core EP and dual core EP invertibility
  8. MPCEP and *CEPMP Inverses
  9. GDC and GC Inverses
  10. Summary of existence conditions

Key Result

Proposition 2.1

Given a full rank factorization of a matrix $A=FG$,

Figures (1)

  • Figure 1: Double star digraph $S_{(m+1),(n+1)}$ with $m=3$ leaves on the first star (center $u$) and $n=4$ leaves on the second star (center $v$).

Theorems & Definitions (14)

  • Proposition 2.1
  • Lemma 2.1
  • Lemma 2.2: gao2018pseudo,Theorem 2.3
  • Proposition 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Lemma 5.1
  • Lemma 5.2
  • Lemma 5.3
  • Lemma 5.4
  • ...and 4 more