A Note on Postian Matrix Theory

1997 ◽  
Vol 07 (02) ◽  
pp. 161-179 ◽  
Author(s):  
Michel Serfati

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).

2013 ◽  
Vol 457-458 ◽  
pp. 36-39
Author(s):  
Qing Jiang Chen ◽  
Huan Chen ◽  
Hong Wei Gao

Material science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. In this work, we study construction and properties of orthogonal two-direction vector-valued wavelet with poly-scale. Firstly, the concepts concerning two-direct-ional vector-valued waelets and wavelet wraps with multi-scale are provided. Secondly, we prop ose a construction algorim for compactly supported orthogonal two-directional vector-valued wave lets. Lastly, properties of a sort of orthogonal two-directional vector-valued wavelet wraps are char acterized by virtue of the matrix theory and the time-frequency analysis method.


2021 ◽  
Vol 7 (1) ◽  
pp. 384-397
Author(s):  
Yinlan Chen ◽  
◽  
Lina Liu

<abstract><p>In this paper, we consider the common Re-nonnegative definite (Re-nnd) and Re-positive definite (Re-pd) solutions to a pair of linear matrix equations $ A_1XA_1^\ast = C_1, \ A_2XA_2^\ast = C_2 $ and present some necessary and sufficient conditions for their solvability as well as the explicit expressions for the general common Re-nnd and Re-pd solutions when the consistent conditions are satisfied.</p></abstract>


1974 ◽  
Vol 17 (2) ◽  
pp. 179-183 ◽  
Author(s):  
J. L. Brenner ◽  
M. J. S. Lim

In [15], O. Taussky-Todd posed the problem of title, namely to find X, Y, Z when A, B are given. Clearly if X, Y, Z exist then A, B are either both invertible or both noninvertible.In section 1, the problem is reviewed in case A, B are both invertible. The problem is seen to be fundamentally one of group theory rather than matrix theory. Application of results of Shoda, Thompson, Ree to the general group-theoretical results allows specialization to certain matrix groups.In Section 2, examples and counterexamples are given in case A, B are noninvertible. A general necessary condition for solvability (involving ranks) is obtained. This condition may or may not be sufficient. For dim A=2, 3 the problem is settled: there is always a solution in the noninvertible case.


2016 ◽  
Vol 31 ◽  
pp. 465-475
Author(s):  
Jacob Van der Woude

Conditions for the existence of a common solution X for the linear matrix equations U_iXV_j 􏰁 W_{ij} for 1 \leq 􏰃 i,j \leq 􏰂 k with i\leq 􏰀 j \leq 􏰃 k, where the given matrices U_i,V_j,W_{ij} and the unknown matrix X have suitable dimensions, are derived. Verifiable necessary and sufficient solvability conditions, stated directly in terms of the given matrices and not using Kronecker products, are also presented. As an application, a version of the almost triangular decoupling problem is studied, and conditions for its solvability in transfer matrix and state space terms are presented.


2021 ◽  
Vol 6 (12) ◽  
pp. 13845-13886
Author(s):  
Yongge Tian ◽  

<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>


Author(s):  
N.E. Zubov ◽  
V.N. Ryabchenko

New formulas were obtained to solve the linear non-degenerate matrix equations based on zero divisors of numerical matrices. Two theorems were formulated, and a proof to one of them is provided. It is noted that the proof of the second theorem is similar to the proof of the first one. The proved theorem substantiates new formula in solving the equation equivalent in the sense of the solution uniqueness to the known formulas. Its fundamental difference lies in the following: any explicit matrix inversion or determinant calculation is missing; solution is "based" not on the left, but on the right side of the matrix equation; zero divisor method is used (it was never used in classical formulas for solving a matrix equation); zero divisor calculation is reduced to simple operations of permutating the vector elements on the right-hand side of the matrix equation. Examples are provided of applying the proposed method for solving a nondegenerate matrix equation to the numerical matrix equations. High accuracy of the proposed formulas for solving the matrix equations is demonstrated in comparison with standard solvers used in the MATLAB environment. Similar problems arise in the synthesis of fast and ultrafast iterative solvers of linear matrix equations, as well as in nonparametric identification of abnormal (emergency) modes in complex technical systems, for example, in the power systems


2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Kefeng Li ◽  
Chao Zhang

This paper studies the solutions of second-order linear matrix equations on time scales. Firstly, the necessary and sufficient conditions for the existence of a solution of characteristic equation are introduced; then two diverse solutions of characteristic equation are applied to express general solution of the matrix equations on time scales.


The characterization of matrices which can be optimally scaled with respect to various modes of scaling is studied. Particular attention is given to the following two problems: ( a) The characterization of those square matrices for which inf lub (D -1 MD) D is attainable for some non-singular diagonal matrix D . ( b) The characterization of those square non-singular matrices A for which inf cond 12 (D 1 AD 2 ) D 1 , D 2 is attainable for some non-singular diagonal matrices D 1 and D 2 . For norms having certain properties, various necessary and sufficient conditions for optimal scalability are obtained when, in problem ( a ), the matrix A and, in problem ( b ), both A and A -1 have chequerboard sign distribution. The characterizations so established impose various conditions on the combinatorial and spectral structure of the matrices. These are investigated by using results from the Perron-Frobenius theory of non-negative matrices and combinatorial matrix theory. It is shown that the Holder or l p -norms have the required properties, and that, in general, the only norms having all of the properties needed, for both the necessary and the sufficient conditions to be satisfied, are variants of the l p -norms. For the special cases p = 1 and p = oo, the characterizations obtained hold for all matrices, irrespective of sign distribution.


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