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A new generalized inverse for rectangular matrices. A general approach

D. E. Ferreyra, F. E. Levis, R. P. Moas, H. H. Zhu

Abstract

Rao and Mitra in 1972 introduced two different types of constraints to extend the concept of Bott-Duffin inverse and defined a new constrained inverse. Mary in 2011 defined the inverse along an element that generalizes the Moore-Penrose and Drazin inverses in a semigroup. Drazin in 2012 introduced the $(b,c)$-inverse generalizing the Mary inverse. In 2017, Rakić noted that the Rao-Mitra inverse is a direct precursor of the $(b,c)$-inverse. In this paper, we introduce the notion of $EF$-inverse as a unified approach to the aforementioned generalized inverses. Moreover, we show that the recently introduced generalized bilateral inverses that in turn contain the OMP, MPO, and MPOMP inverses can also be considered as special cases of the $EF$-inverse.

A new generalized inverse for rectangular matrices. A general approach

Abstract

Rao and Mitra in 1972 introduced two different types of constraints to extend the concept of Bott-Duffin inverse and defined a new constrained inverse. Mary in 2011 defined the inverse along an element that generalizes the Moore-Penrose and Drazin inverses in a semigroup. Drazin in 2012 introduced the -inverse generalizing the Mary inverse. In 2017, Rakić noted that the Rao-Mitra inverse is a direct precursor of the -inverse. In this paper, we introduce the notion of -inverse as a unified approach to the aforementioned generalized inverses. Moreover, we show that the recently introduced generalized bilateral inverses that in turn contain the OMP, MPO, and MPOMP inverses can also be considered as special cases of the -inverse.
Paper Structure (4 sections, 13 theorems, 60 equations, 2 tables)

This paper contains 4 sections, 13 theorems, 60 equations, 2 tables.

Table of Contents

  1. Introduction
  2. The $EF$-inverse. An extension of the Rao-Mitra inverse
  3. Canonical form of the $EF$-inverse
  4. The $EF$-inverse as an extension of generalized bilateral and non-bilateral inverses

Key Result

Lemma 2.1

PeRaMi Let $A \in \mathbb{C}^{m \times n}$, $B \in \mathbb{C}^{p \times q}$, $C \in \mathbb{C}^{m \times q}$, $A^{(1)}\in A\{1\}$, and $B^{(1)}\in B\{1\}$. Then, the equation $AXB=C$ is consistent (in $X$) if and only if $AA^{(1)}CB^{(1)}B=C$, in which case the general solution is where $Z$ is an arbitrary matrix.

Theorems & Definitions (32)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • proof
  • ...and 22 more