Reverse order law for generalized inverses of multiple operator product

Radenković, Jovana Nikolov

doi:10.1080/03081087.2015.1082961

Cited by 9 publications

(7 citation statements)

References 14 publications

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“…We first mention that several cases of the set inclusions in (1.10) were considered in the literature, such as,

(cf. [20] , [21] , [23] , [28] , [50] , [52] , [54] , [64] , [77] , [81] , [92] , [114] , [116] , [117] , [119] , [123] , [126] ). Despite the fact that ROLs have been around for a long time, there are still many open problems regarding ROLs under various assumptions.…”

Section: Set Inclusions For {1}- 1 2 mentioning

confidence: 99%

“…This section focuses on the four groups of set inclusions for {( ) (1) }, {( ) (1,2) }, {( ) (1,3) }, {( ) (1,4) } in (1.10). We first mention that several cases of the set inclusions in (1.10) were considered in the literature, such as, [20,21,23,28,50,52,54,64,77,81,92,114,116,117,119,123,126]). Despite the fact that ROLs have been around for a long time, there are still many open problems regarding ROLs under various assumptions.…”

Section: Lemma 83 ([93]mentioning

confidence: 99%

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A family of 512 reverse order laws for generalized inverses of a matrix product: A review

Tian

2020

Heliyon

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Reverse order laws for generalized inverses of matrix products are a classic object of study in the theory of generalized inverses. One of the well-known reverse order laws for a matrix product AB is , where denotes a -generalized inverse of matrix. Because -generalized inverse of a singular matrix is not unique, the relationships between both sides of the reverse order law can be divided into four situations for consideration. The aim of this paper is to give an overview of plenty of results concerning reverse order laws for -generalized inverses of the product AB , from the development of background and preliminary tools to the collection of miscellaneous formulas and facts on the reverse order laws in one place with cogent introduction and references for further study. We begin with the introduction of a linear mixed model and the presentation of two least-squares methodologies for estimating the fixed parameter vector β in the model, and the description of connections between the two types of least-squares estimators and the reverse order laws for generalized inverses of AB . We then prepare various necessary matrix study tools, including a general theory on linear or nonlinear algebraic matrix identities, a group of expansion formulas for calculating ranks of block matrices, two groups of explicit formulas for calculating the maximum and minimum ranks of , as well as necessary and sufficient conditions for to be invariant with respect to the choice of the generalized inverses, etc. Subsequently, we present a unified approach to the 512 matrix set inclusion problems associated with the above reverse order laws for the eight commonly-used types of generalized inverses of A , B , and AB by means of the definitions of generalized inverses, the block matrix methodology (BMM), the matrix equation methodology (MEM), and the matrix rank methodology (MRM).

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“…We first mention that several cases of the set inclusions in (1.10) were considered in the literature, such as,

Section: Set Inclusions For {1}- 1 2 mentioning

confidence: 99%

Section: Lemma 83 ([93]mentioning

confidence: 99%

A family of 512 reverse order laws for generalized inverses of a matrix product: A review

Tian

2020

Heliyon

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“…Further results on ROL for Moore‐Penrsoe inverse can be found in [13, 15, 21]. The ROL for K ‐inverses, where

K\subseteq \lbrace 1,2,3,4\rbrace

, are considered in [3, 4, 6–9, 16–19, 22, 24, 26]. Certain results on weighted generalized inverses are given in [25, 27], while some results regarding the ROL for the group inverse can be found in [2, 10, 20].…”

Section: Introductionmentioning

confidence: 99%

Improvements on the reverse order laws

Cvetković-Ilić

Milenković

2023

Mathematische Nachrichten

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In many papers concerning properties of generalized inverses in different settings, we can find the results with many redundant instances of assuming regularity of certain elements. We have made an effort to widen the range of applicability of concrete results by considering more general cases of the problems without imposing any such additional assumptions. This was the main motivation for this paper, and we present several significant improvements of the reverse order laws on false{1,3false}$\lbrace 1,3\rbrace$ and false{1,2,3false}$\lbrace 1,2,3\rbrace$‐inverses in the ring setting.

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“…The reverse order laws for the generalized inverse of multiple matrix products (1) yield a class of interesting problems that are fundamental in the theory of the generalized inverse of matrices; see [1,[4][5][6]. As a hot topic in current matrix research, the necessary and sufficient conditions for the reverse order laws for the generalized inverse of matrix products are useful in both theoretical study and practical scientific computing; hence, this has attracted considerable attention and several interesting results have been obtained; see [10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Forward Order Law for the Reflexive Inner Inverse of Multiple Matrix Products

Zhou

Xiong

Qin

2022

Axioms

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The generalized inverse has numerous important applications in aspects of the theoretic research of matrices and statistics. One of the core problems of generalized inverse is finding the necessary and sufficient conditions for the reverse (or the forward) order laws for the generalized inverse of matrix products. In this paper, by using the extremal ranks of the generalized Schur complement, some necessary and sufficient conditions are given for the forward order law for A1{1,2}A2{1,2}…An{1,2}⊆(A1A2…An){1,2}.

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Reverse order law for generalized inverses of multiple operator product

Cited by 9 publications

References 14 publications

A family of 512 reverse order laws for generalized inverses of a matrix product: A review

A family of 512 reverse order laws for generalized inverses of a matrix product: A review

Improvements on the reverse order laws

Forward Order Law for the Reflexive Inner Inverse of Multiple Matrix Products

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