scholarly journals Miscellaneous reverse order laws and their equivalent facts for generalized inverses of a triple matrix product

2021 ◽  
Vol 6 (12) ◽  
pp. 13845-13886
Author(s):  
Yongge Tian ◽  
Keyword(s):  
Generalized Inverses ◽  
Matrix Analysis ◽  
Reverse Order ◽  
Matrix Product ◽  
Commutative Algebras ◽  
Matrix Rank Method ◽  
The Matrix ◽  
Research Problems ◽  
Generalized Inverses Of Matrices

<abstract><p>Reverse order laws for generalized inverses of products of matrices are a class of algebraic matrix equalities that are composed of matrices and their generalized inverses, which can be used to describe the links between products of matrix and their generalized inverses and have been widely used to deal with various computational and applied problems in matrix analysis and applications. ROLs have been proposed and studied since 1950s and have thrown up many interesting but challenging problems concerning the establishment and characterization of various algebraic equalities in the theory of generalized inverses of matrices and the setting of non-commutative algebras. The aim of this paper is to provide a family of carefully thought-out research problems regarding reverse order laws for generalized inverses of a triple matrix product $ ABC $ of appropriate sizes, including the preparation of lots of useful formulas and facts on generalized inverses of matrices, presentation of known groups of results concerning nested reverse order laws for generalized inverses of the product $ AB $, and the derivation of several groups of equivalent facts regarding various nested reverse order laws and matrix equalities. The main results of the paper and their proofs are established by means of the matrix rank method, the matrix range method, and the block matrix method, so that they are easy to understand within the scope of traditional matrix algebra and can be taken as prototypes of various complicated reverse order laws for generalized inverses of products of multiple matrices.</p></abstract>

scholarly journals A Family of 512 Reverse Order Laws for Generalized Inverses of Two Matrix Product: A Review

2020 ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Least Squares ◽  
Mixed Model ◽  
Linear Mixed Model ◽  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Generalized Inverses ◽  
Matrix Analysis ◽  
Reverse Order ◽  
Matrix Product ◽  
The Matrix

Reverse order laws for generalized inverses of matrix products is 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 $(AB)^{(i,\ldots,j)} = B^{(s_2,\ldots,t_2)}A^{(s_1,\ldots,t_1)}$, where $(\cdot)^{(i,\ldots,j)}$ denotes an $\{i,\ldots, j\}$-generalized inverse of matrix. Because $\{i,\ldots, j\}$-generalized inverse of a singular matrix is unique, the relationships between both sides of the reverse order law can be divided into four situations for consideration. This paper provides a thorough coverage of the reverse order laws for $\{i,\ldots, j\}$-generalized inverses of $AB$, 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 $y = AB\beta + A\gamma + \epsilon$ and the presentation of two least-squares methodologies to estimate the fixed parameter vector $\beta$ 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 some valued matrix analysis tools, including a general theory on linear or nonlinear 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 $B^{(s_2,\ldots,t_2)}A^{(s_1,\ldots,t_1)}$, as well as necessary and sufficient conditions for $B^{(s_2,\ldots,t_2)}A^{(s_1,\ldots,t_1)}$ to be invariant with respect to the choice of $B^{(s_2,\ldots,t_2)}A^{(s_1,\ldots,t_1)}$. We then 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 $AB$ through use of the definitions of generalized inverses, the block matrix method (BMM), the matrix rank method (MRM), the matrix equation method (MEM), and various algebraic calculations of matrices.

scholarly journals On mixed-type reverse-order laws for the Moore-Penrose inverse of a matrix product

2004 ◽  
Vol 2004 (58) ◽  
pp. 3103-3116 ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Mixed Type ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Reverse Order ◽  
Matrix Product ◽  
Matrix Rank ◽  
Matrix Rank Method ◽  
The Matrix ◽  
Necessary And Sufficient

Some mixed-type reverse-order laws for the Moore-Penrose inverse of a matrix product are established. Necessary and sufficient conditions for these laws to hold are found by the matrix rank method. Some applications and extensions of these reverse-order laws to the weighted Moore-Penrose inverse are also given.

scholarly journals Linear Statistical Models, Least-Squares Estimators, and Classication Analysis to Reverse-Order Laws for Generalized Inverses of Matrix Products

2018 ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Least Squares ◽  
Linear Mixed Model ◽  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Generalized Inverses ◽  
Matrix Analysis ◽  
Reverse Order ◽  
Matrix Product ◽  
Least Squares Estimators ◽  
Matrix Products

Reverse-order laws for generalized inverses of matrix products is a classic object of study in the theory of&nbsp;generalized inverses. One of the well-known reverse-order laws for a matrix product AB is (AB)(i,...,j) = B(i,...,j)A(i,...,j),&nbsp;where (&middot;)i,...,j denotes an {i,...,j}-generalized inverse of matrix. Because {i,...,j}-generalized inverse of a general&nbsp;matrix is not necessarily unique, the relationships between both sides of the reverse-order law can be divided into&nbsp;four situations for consideration. In this article, we first introduce a linear mixed model y = AB&beta; + A&gamma; + &epsilon;, present&nbsp;two least-squares&nbsp; methodologies to estimate the fixed parameter vector in the model, and describe the connections&nbsp;between the two least-squares estimators and the reverse-order laws for generalized inverses of the matrix product&nbsp;AB. We then prepare some valued matrix analysis tools, including a general theory on linear or nonlinear matrix&nbsp;identities, a group of expansion formulas for calculating ranks of block matrices, two groups of explicit formulas for&nbsp;calculating the maximum and minimum ranks of B(i,...,j)A(i,...,j), as well as necessary and sufficient conditions for&nbsp;B(i,...,j)A(i,...,j) to be invariant with respect to the choice of A(i,...,j) and B(i,...,j). We then present a unied approach&nbsp;to the 512 set inclusion problems {(AB)(i,...,j)&nbsp;&supe;&nbsp;{B(i,...,j)A(i,...,j)}for the eight commonly-used types of generalized&nbsp;inverses of A, B, and AB using the block matrix representation method (BMRM), matrix equation method (MEM),&nbsp;and matrix rank method (MRM), where {(&middot;)(i,...,j)}&nbsp;denotes the collection of all {i,...,j}-generalized inverse of a matrix.

scholarly journals A CANCELLATION PROPERTY OF THE MOORE–PENROSE INVERSE OF TRIPLE PRODUCTS

2009 ◽  
Vol 86 (1) ◽  
pp. 33-44 ◽  
Author(s):  
TOBIAS DAMM ◽  
HARALD K. WIMMER
Keyword(s):  
Molecular Dynamics ◽  
Matrix Equation ◽  
Generalized Inverses ◽  
Reverse Order ◽  
Cancellation Property ◽  
Singular Vectors ◽  
The Matrix ◽  
Compliance Matrices

AbstractWe study the matrix equation C(BXC)†B=X†, where X† denotes the Moore–Penrose inverse. We derive conditions for the consistency of the equation and express all its solutions using singular vectors of B and C. Applications to compliance matrices in molecular dynamics, to mixed reverse-order laws of generalized inverses and to weighted Moore–Penrose inverses are given.

A Note on Postian Matrix Theory

1997 ◽  
Vol 07 (02) ◽  
pp. 161-179 ◽  
Author(s):  
Michel Serfati
Keyword(s):  
Matrix Theory ◽  
Consistency Condition ◽  
Heyting Algebra ◽  
Linear Matrix ◽  
Matrix Product ◽  
Large Set ◽  
Matrix Equations ◽  
Post Algebra ◽  
The Matrix

The present paper is devoted to some aspects of postian matrix theory over an arbitrary Post algebra, through results in Postian relative–pseudocomplementation. It is well known (for instance Rousseau [10] or Dwinger [6]) that any r-Post algebra is a Brouwerian lattice (or a Heyting algebra), that is to say, for every (a, b) in P2, the set of x in P such as a. x ≤ b admits a greatest element (a.x is inf {a, x}), called the relative inf–pseudocomplement of a in b, and denoted (b|a). In the first part of our paper, we compute the explicit expression of the disjunctive components of the relative inf–pseudocomplement (Theorem 2), which allows us to state some specific new properties of Postian pseudocomplementation, among which the cases where (b|a) is a Boolean element, and the equation (x|a) = c (Theorem 4), for which we give a consistency condition. Relative pseudo complementation in fact plays a major role in Postian structures, since we show (Theorem 5), that every element a in P is completely determined by the sequence of the (ek|a): in fact, being given (r-1) elements of P submitted to the ascending chain condition: α1 ≤ α2 ≤ … ≤ αr-1, there exists exactly one a in P such as (ek|a) = αk for every k, where the ek are the elements of the underlying chain in P. Study of pseudocomplementation properties in some Post algebra P actually helps us to enlighten relations between P and, on one hand, its center B (the set of its complemented elements, which is a Boolean algebra), on the other hand, the underlying chain. The second part is devoted to Postian linear matrix equations and inequations: in fact just like in Boolean algebras, the existence of the inf–pseudocomplement in the underlying lattice implies the residuation property is valid over the ordered semi-group of Postian matrices, equipped with the matrix product ⊗ (this was a general theorem from Blyth [3]): the Postian matrix inequation A ⊗ X ≤ B thus admits a greatest solution (Theorem 6), which is explicitly computed. This provides a consistency condition for the matrix equation A ⊗ X = B (Theorem 7). Another result states a characterization of inversible square Postian matrices. On this point of inversibility, as it is well known, the Boolean results were built in three successive steps by Wedderbrun [17], Luce [9] and Rutherford [13]. As to the Postian case, we prove in turn that a Postian matrix is inversible if and only if it is Boolean and orthogonal (Theorem 11). To prove this result, as well as Theorem 2 in the first part, we make a systematic use of the representation theorem for Post algebras by the Boolean way, as enunciated in the Preliminaries (Theorem 1). Repeated applications of the method provide a large set of conditions equivalent to the inversibility of a Postian matrix (Theorem 12). Afterwards, we examine various other Postian matrix equations and inequations, among which t A ⊗ A ≤ I, leading to the characterization of right–distributive over conjunction–Postian matrices (Theorem 9), and also the equation A ⊗ X = Ek, for which it is given a complete consistency condition (Theorem 13).

scholarly journals Invariance property of a five matrix product involving two generalized inverses

2021 ◽  
Vol 29 (1) ◽  
pp. 83-92
Author(s):  
Bo Jiang ◽  
Yongge Tian
Keyword(s):  
Sufficient Conditions ◽  
Generalized Inverses ◽  
Invariance Property ◽  
Matrix Product ◽  
Special Cases ◽  
The Matrix ◽  
Matrix Equality ◽  
Representation Method ◽  
Necessary And Sufficient ◽  
General Method

Abstract Matrix expressions composed by generalized inverses can generally be written as f(A − 1, A − 2, . . ., A − k ), where A 1, A 2, . . ., A k are a family of given matrices of appropriate sizes, and (·)− denotes a generalized inverse of matrix. Once such an expression is given, people are primarily interested in its uniqueness (invariance property) with respect to the choice of the generalized inverses. As such an example, this article describes a general method for deriving necessary and sufficient conditions for the matrix equality A 1 A − 2 A 3 A − 4 A 5 = A to always hold for all generalized inverses A − 2 and A − 4 of A 2 and A 4 through use of the block matrix representation method and the matrix rank method, and discusses some special cases of the equality for different choices of the five matrices.

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