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Square matrix
About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.
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TL;DR: It is shown that a subclass of almost fully copositive matrices intorduced in (Linear Algebra Appl 400:243–252 2005) with $$Q_{0}$$Q0-property is captured by sufficient matrices introduced by Cottle et al.
Abstract: In this article, we study the properties of some matrix classes using principal pivot transform (PPT). These matrices with some additional conditions have nonnegative principal minors. We show that a subclass of almost fully copositive matrices intorduced in (Linear Algebra Appl 400:243–252 2005) with $$Q_{0}$$
-property is captured by sufficient matrices introduced by Cottle et al. in (Linear Algebra Appl 114/115:231–249 1989) and the solution set of a linear complementarity problem is the same as the set of Karush–Kuhn–Tucker stationary points of the corresponding quadratic programming problem. We introduce some more PPT based matrix classes in continuation of (Linear Algebra Appl 400:243–252 2005) and study the properties of these classes.
27 citations
01 Jan 2016
TL;DR: In this article, a graph-theoretic approach to linear models and to Minkowski-Leontief matrices is presented, which has significant realizations in mathematical economics and in the theory of stochastic processes.
Abstract: THIS PAPER presents a graph-theoretic approach to linear models and to Minkowski-Leontief matrices which have significant realizations in mathematical economics and in the theory of stochastic processes [1, 2, 3, 5, 6, 13, 14]. We designate any finite nonnegative square matrix A with no row sum greater than unity as a matrix of Minkowsski-Leontief type. In the body of this paper a simple graph-theoretic formulation is set forth of necessary and sufficient conditions for the existence of the inverse (I -A)-', where I denotes the identity matrix and A is a matrix of Minkowski-Leontief type. We consider linear systems of the form x(I - A) = w, where w is a nonnegative vector, and give a complete graphtheoretic characterization of all nontrivial, finite and nonnegative solutions x of such systems for (I - A) singular or not. In the present approach, any matrix A of Minkowski-Leontief type is viewed as a principal submatrix of a stochastic matrix A* and it is shown that the properties of essential interest in the development rest on fundamental results in the theory of finite-dimensional Markov chains (discrete time parameter). We cite an algorithm for computing the unique stationary stochastic vector of an indecomposable stochastic matrix and show its relevance in the treatment of linear models. In the course of exposition, an economic interpretation is consistently set forth in the context of certain input-output models. The investigation of the character of solutions is of interest for descriptive as well as for normative economics. A static open input-output model of the form x(I - A) = w may, for example, be treated from an essentially phenomenological standpoint or it may be regarded as a representation compatible with the assumptions of the competitive market [10]. Delineation of the economically meaningful solutions ("equilibria") affords an intrinsic insight into the structure of the linear model.2 In the following development, we employ the graph-theoretic concepts and formulations introduced in [7] for the treatment of finite nonnegative matrices, in particular the concept of "cyclic net." 1 This paper was prepared as a part of the project "Symbolic Methods in the Study of Organizations" under Contract Nonr-1180(00) with the Office of Naval Research. An earlier version of this paper appeared as Technical Report B, July 1956. 2 The relation of the present linear systems to multi-sector trade models and to issues of dynamic coupling and of macroeconomic stability has been well stated in [11] and we accordingly adhere to the economy of a single interpretation. 325
27 citations
TL;DR: In this paper, the spectrum of an asymmetric random matrix with block structured variances was studied and the authors showed that these matrices have a circularly symmetric spectrum, and they gave an explicit formula for their spectral radius and a set of implicit equations for the full density function.
Abstract: We study the spectrum of an asymmetric random matrix with block structured variances. The rows and columns of the random square matrix are divided into $D$ partitions with arbitrary size (linear in $N$). The parameters of the model are the variances of elements in each block, summarized in $g\in\mathbb{R}^{D\times D}_+$. Using the Hermitization approach and by studying the matrix-valued Stieltjes transform we show that these matrices have a circularly symmetric spectrum, we give an explicit formula for their spectral radius and a set of implicit equations for the full density function. We discuss applications of this model to neural networks.
27 citations
28 Sep 2009
TL;DR: MadWeight, a new phase space generator aimed at efficiently computing the weights needed in the matrix element method, creates an optimized phase-space mapping to integrate the product of the square matrix element and the transfer function.
Abstract: We present MadWeight, a new phase space generator aimed at efficiently computing the weights needed in the matrix element method. Given an arbitrary decay chain and a user-defined transfer function, the algorithm creates an optimized phase-space mapping to integrate the product of the square matrix element and the transfer function. We illu strate the code with some applications, such as the use of these weights to discriminate signa l from background in charged Higgs production analyses.
27 citations
TL;DR: In this article, the average Schmidt strength of a bipartite diagonal quantum gate is shown to scale as Θ(log N) in contrast to the Θ((log N −1/n) ) behavior characteristic to random unitary gates.
Abstract: Nonlocal properties of an ensemble of diagonal random unitary matrices of order $N^2$ are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as $\log N$, in contrast to the $\log N^2$ behavior characteristic to random unitary gates. Entangling power of a diagonal gate $U$ is related to the von Neumann entropy of an auxiliary quantum state $\rho=AA^{\dagger}/N^2$, where the square matrix $A$ is obtained by reshaping the vector of diagonal elements of $U$ of length $N^2$ into a square matrix of order $N$. This fact provides a motivation to study the ensemble of non-hermitian unimodular matrices $A$, with all entries of the same modulus and random phases and the ensemble of quantum states $\rho$, such that all their diagonal entries are equal to $1/N$. Such a state is contradiagonal with respect to the computational basis, in sense that among all unitary equivalent states it maximizes the entropy copied to the environment due to the coarse graining process. The first four moments of the squared singular values of the unimodular ensemble are derived, based on which we conjecture a connection to a recently studied combinatorial object called the "Borel triangle". This allows us to find exactly the mean von Neumann entropy for random phase density matrices and the average entanglement for the corresponding ensemble of bipartite pure states.
27 citations