Optimally scalable matrices

1977 ◽  
Vol 287 (1345) ◽  
pp. 307-349 ◽  
Keyword(s):  
Matrix Theory ◽  
Sufficient Conditions ◽  
Spectral Structure ◽  
Combinatorial Matrix Theory ◽  
Sign Distribution ◽  
Special Cases ◽  
The Matrix ◽  
Singular Matrices ◽  
Necessary And Sufficient

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.

Multiconsensus of first-order multiagent systems with directed topologies

2020 ◽  
Vol 34 (23) ◽  
pp. 2050240
Author(s):  
Xiao-Wen Zhao ◽  
Guangsong Han ◽  
Qiang Lai ◽  
Dandan Yue
Keyword(s):  
Multiagent Systems ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Consensus Problem ◽  
Multiple Agents ◽  
First Order ◽  
The Matrix ◽  
Directed Topologies ◽  
Necessary And Sufficient ◽  
Theoretical Results

The multiconsensus problem of first-order multiagent systems with directed topologies is studied. A novel consensus problem is introduced in multiagent systems — multiconsensus. The states of multiple agents in each subnetwork asymptotically converge to an individual consistent value in the presence of information exchanges among subnetworks. Linear multiconsensus protocols are proposed to solve the multiconsensus problem, and the matrix corresponding to the protocol is designed. Necessary and sufficient conditions are derived based on matrix theory, under which the stationary multiconsensus and dynamic multiconsensus can be reached. Simulations are provided to demonstrate the effectiveness of the theoretical results.

scholarly journals Real Representation Approach to Quaternion Matrix Equation Involving ϕ-Hermicity

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Xin Liu ◽  
Huajun Huang ◽  
Zhuo-Heng He
Keyword(s):  
Matrix Equation ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Quaternion Matrix ◽  
Real Representation ◽  
Quaternion Matrix Equation ◽  
The Matrix ◽  
Hermitian Solution ◽  
Necessary And Sufficient

For a quaternion matrix A, we denote by Aϕ the matrix obtained by applying ϕ entrywise to the transposed matrix AT, where ϕ is a nonstandard involution of quaternions. A is said to be ϕ-Hermitian or ϕ-skew-Hermitian if A=Aϕ or A=−Aϕ, respectively. In this paper, we give a complete characterization of the nonstandard involutions ϕ of quaternions and their conjugacy properties; then we establish a new real representation of a quaternion matrix. Based on this, we derive some necessary and sufficient conditions for the existence of a ϕ-Hermitian solution or ϕ-skew-Hermitian solution to the quaternion matrix equation AX=B. Moreover, we give solutions of the quaternion equation when it is solvable.

scholarly journals Matrix Mappings on the Domains of Invertible Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Muhammed Altun
Keyword(s):  
Sufficient Conditions ◽  
Sequence Spaces ◽  
Banach Sequence Spaces ◽  
The Matrix ◽  
Matrix Domains ◽  
Necessary And Sufficient ◽  
Invertible Matrices ◽  
Matrix Operators ◽  
Banach Sequence

We focus on sequence spaces which are matrix domains of Banach sequence spaces. We show that the characterization of a random matrix operator , where and are matrix domains with invertible matrices and , can be reduced to the characterization of the operator . As an application, the necessary and sufficient conditions for the matrix operators between invertible matrix domains of the classical sequence spaces and norms of these operators are given.

scholarly journals Markov chains with transition delta-matrix: ergodicity conditions, invariant probability measures and applications

1991 ◽  
Vol 4 (4) ◽  
pp. 333-355 ◽  
Author(s):  
Lev Abolnikov ◽  
Alexander Dukhovny
Keyword(s):  
Markov Chains ◽  
Generating Functions ◽  
Sufficient Conditions ◽  
Probability Measures ◽  
Transition Matrices ◽  
Special Cases ◽  
Associated Function ◽  
Necessary And Sufficient ◽  
General Method

A large class of Markov chains with so-called Δm,n-and Δ′m,n-transition matrices (“delta-matrices”) which frequently occur in applications (queues, inventories, dams) is analyzed.The authors find some structural properties of both types of Markov chains and develop a simple test for their irreducibility and aperiodicity. Necessary and sufficient conditions for the ergodicity of both chains are found in the article in two equivalent versions. According to one of them, these conditions are expressed in terms of certain restrictions imposed on the generating functions Ai(z) of the elements of the ith row of the transition matrix, i=0,1,2,…; in the other version they are connected with the characterization of the roots of a certain associated function in the unit disc of the complex plane. The invariant probability measures of Markov chains of both kinds are found in terms of generating functions. It is shown that the general method in some important special cases can be simplified and yields convenient and, sometimes, explicit results.As examples, several queueing and inventory (dam) models, each of independent interest, are analyzed with the help of the general methods developed in the article.

scholarly journals The application domain of difference type matrix D(r,0,s,0,t) on some sequence spaces

2020 ◽  
pp. 32-32
Author(s):  
Avinoy Paul ◽  
Binod Tripathy
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sequence Spaces ◽  
Application Domain ◽  
Topological Properties ◽  
Difference Type ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Matrix Mappings

In this paper we introduce new sequence spaces with the help of domain of matrix D(r,0,s,0,t), and study some of their topological properties. Further, we determine ? and ? duals of the new sequence spaces and finally, we establish the necessary and sufficient conditions for characterization of the matrix mappings.

scholarly journals The outer generalized inverse of an even-order tensor

2020 ◽  
Vol 36 (36) ◽  
pp. 599-615
Author(s):  
Jun Ji ◽  
Yimin Wei
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Full Rank ◽  
Outer Inverse ◽  
The Matrix ◽  
Even Order ◽  
Rank Factorization ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

Necessary and sufficient conditions for the existence of the outer inverse of a tensor with the Einstein product are studied. This generalized inverse of a tensor unifies several generalized inverses of tensors introduced recently in the literature, including the weighted Moore-Penrose, the Moore-Penrose, and the Drazin inverses. The outer inverse of a tensor is expressed through the matrix unfolding of a tensor and the tensor folding. This expression is used to find a characterization of the outer inverse through group inverses, establish the behavior of outer inverse under a small perturbation, and show the existence of a full rank factorization of a tensor and obtain the expression of the outer inverse using full rank factorization. The tensor reverse rule of the weighted Moore-Penrose and Moore-Penrose inverses is examined and equivalent conditions are also developed.

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.

scholarly journals On some new paranormed sequence spaces defined by the matrix (D )(r,0,0,s)

2021 ◽  
Vol 40 (3) ◽  
pp. 779-796
Author(s):  
Avinoy Paul
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sequence Spaces ◽  
Topological Properties ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Matrix Mappings

In this paper, we introduce some new paranormed sequence spaces and study some topological properties. Further, we determine α, β and γ-duals of the new sequence spaces and finally, we establish the necessary and sufficient conditions for characterization of matrix mappings.

scholarly journals Matrix characterization of multidimensional subshifts of finite type

2019 ◽  
Vol 20 (2) ◽  
pp. 407
Author(s):  
Puneet Sharma ◽  
Dileep Kumar
Keyword(s):  
Finite Type ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Alternative View ◽  
Subshift Of Finite Type ◽  
Shift Space ◽  
The Matrix ◽  
Subshifts Of Finite Type ◽  
Necessary And Sufficient

<p>Let X ⊂ A<sup>Zd </sup>be a 2-dimensional subshift of finite type. We prove that any 2-dimensional subshift of finite type can be characterized by a square matrix of infinite dimension. We extend our result to a general d-dimensional case. We prove that the multidimensional shift space is non-empty if and only if the matrix obtained is of positive dimension. In the process, we give an alternative view of the necessary and sufficient conditions obtained for the non-emptiness of the multidimensional shift space. We also give sufficient conditions for the shift space X to exhibit periodic points.</p>

scholarly journals Necessary and sufficient conditions for the ergodicity of Markov chains with transition Δm,n(Δ′m,n)-matrix

1987 ◽  
Vol 1 (1) ◽  
pp. 13-24 ◽  
Author(s):  
L. Abolnikov ◽  
A. Dukhovny
Keyword(s):  
Markov Chains ◽  
Transition Matrix ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Transition Matrices ◽  
Triangular Form ◽  
Special Cases ◽  
Associated Function ◽  
Necessary And Sufficient

This paper isolates and studies a class of Markov chains with a special quasi-triangular form of the transition matrix [so-called Δm,n(Δ′m,n)-matrix]. Many discrete stochastic processes encountered in applications (queues, inventories and dams) have transition matrices which are special cases of a Δm,n(Δ′m,n)-matrix. Necessary and sufficient conditions for the ergodicity of a Markov chain with transition Δm,n(Δ′m,n)-matrix are determined in the article in two equivalent versions. According to the first version, these conditions are expressed in terms of certain restrictions imposed on the generating functions Ai(x) of the elements of the i-th row of the transition matrix, i=0,1,2,…; in the other version they are connected with the characterization of the roots of a certain associated function in the unit circle of the complex plane. Results obtained in the article generalize, complement, and refine similar results existing in the literature.

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