Mixed-Type Reverse Order Laws for Generalized Inverses over Hilbert Space

Let A, B and AB be closed range operators. The explicit matrix expressions for various generalized inverses are obtained by using block operator matrix methods. Some subtle relationships between the properties of sub-blocks in operator matrices A, B and their range relations are built. New necessary and sufficient conditions for the equivalent relations, inclusion relations and mixed-type generalized inverses relations are presented. Some recent mixed-type reverse-order laws results are covered and many new mixed-type generalized inverses relations are established by using this block-operator matrix technique.

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“…The space decomposition forms in (1) first appeared in the paper [21,Theorem 1]. It was also used in the papers [17,18,29,30]. Then…”

Section: The Matrix Representations Of Two Operators a And Bmentioning

confidence: 99%

On the mixed-type generalized inverses of the products of two operators

Liu

Zhang

Deng

2019

Filomat

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“…In fact, people have put a great deal of time and effort into the investigations of various ROLs for generalized inverses of matrix products, in particular, researches of (1.9) – (1.12) have already been flourished and diversified in contents since that beginning (cf. [1] , [13] , [14] , [21] , [24] , [30] , [33] , [34] , [44] , [51] , [52] , [73] , [74] , [75] , [113] , [114] , [115] , [121] , [122] ). In spite of much effort devoted to the researches to ROLs problems in the past several decades, most parts of (1.9) – (1.12) remain unresolved.…”

Section: Introductionmentioning

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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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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Two groups of mixed reverse order laws for generalized inverses of two and three matrix products

Tian

2020

Comp. Appl. Math.

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Generalized inverses of a matrix product can be written as certain matrix expressions that are composed by the given matrices and their generalized inverses, and a challenging task in this respect is to establish various reasonable reverse order laws for generalized inverses of matrix products. In this paper, we present two groups of known and new mixed reverse order laws for the Moore-Penrose inverses of products of two and three matrices through various conventional matrix operations. We also establish four groups of matrix set inclusions that are composed by {1}and {1, 2}-generalized inverses of A, B, C, and their products AB and ABC. Keywords Matrix product • Generalized inverse • Reverse order law • Set inclusion Mathematics Subject Classification 15A09 • 15A24 • 47A05 A matrix X is called a {i,. .. , j}-generalized inverse of A, denoted by A (i,..., j) , if it satisfies the ith,.. . , jth equations in (1.1). The collection of all {i,. .. , j}-generalized inverses of A is denoted by {A (i,..., j) }. There are all 15 types of {i,. .. , j}-generalized inverses of A,

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Mixed-Type Reverse Order Laws for Generalized Inverses over Hilbert Space

Abstract: In this paper, by using a block-operator matrix technique, we study mixed-type reverse order laws for {1,3}-, {1,2,3}-and {1,3,4}-generalized inverses over Hilbert spaces. It is shown that ( ) B ABB is given.

Cited by 3 publications

References 9 publications

On the mixed-type generalized inverses of the products of two operators

On the mixed-type generalized inverses of the products of two operators

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

Two groups of mixed reverse order laws for generalized inverses of two and three matrix products

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