关于正定矩阵一不等式的简单证明

设A=(aij)是一n阶正定实对称矩阵,本文用代数方法证明了|A|≤a11a22…ann,当且仅当A是对角矩阵时等号成立.且证法简单.     

α-对角占优与广义严格对角占优矩阵的判定

1 引言与记号 广义严格对角占优矩阵在数学、物理、控制论及经济学等许多领域有着重要的研究价值和实用价值.广义严格对角占优矩阵就是非奇异日一矩阵,它是一类范围很广的特殊矩阵,熟知的严格对角占优矩阵,不可约对角占优矩阵,非奇异M-矩阵等都是其特殊情形.如何在实际应用中简便地判别一个矩阵是否是日一矩阵,一直是人们关注的问题.     

More refined enumerations of alternating sign matrices

We study a further refinement of the standard refined enumeration of alternating sign matrices (ASMs) according to their first two rows instead of just the first row, and more general “d-refined” enumerations of ASMs according to the first d rows. For the doubly-refined case of d=2, we derive a system of linear equations satisfied by the doubly-refined enumeration numbers A_n,i,j that enumerate such matrices. We give a conjectural explicit formula for A_n,i,j and formulate several other conjectures about the sufficiency of the linear equations to determine the A_n,i,j's and about an extension of the linear equations to the general d-refined enumerations.     

Constructing integral matrices with given line sums

D. Gale, in 1957 and H.J. Ryser, in 1963, independently proved the famous Gale–Ryser theorem on the existence of (0, 1)–matrices with prescribed row and column sums. Around the same time, in 1968, Mirsky solved the more general problem of finding conditions for the existence of a nonnegative integral matrix with entries less than or equal to p and prescribed row and column sums. Using the results of Mirsky, Brualdi shows that a modified version of the domination condition of Gale–Ryser is still necessary and sufficient for the existence of a matrix under the same constraints. In this article we prove another extension of Gale–Ryser’s domination condition. Furthermore we present a method to build nonnegative integral matrices with entries less than or equal to p and prescribed row and column sums.     

Sums of commuting square-zero transformations

We show that a linear transformation on a vector space is a sum of two commuting square-zero transformations if and only if it is a nilpotent transformation with index of nilpotency at most 3 and the codimension of in kerT is greater than or equal to the dimension of the space . We also characterize products of two commuting unipotent transformations with index 2.     

A review of infinite matrices and their applications

Infinite matrices, the forerunner and a main constituent of many branches of classical mathematics (infinite quadratic forms, integral equations, differential equations, etc.) and of the modern operator theory, is revisited to demonstrate its deep influence on the development of many branches of mathematics, classical and modern, replete with applications. This review does not claim to be exhaustive, but attempts to present research by the authors in a variety of applications. These include the theory of infinite and related finite matrices, such as sections or truncations and their relationship to the linear operator theory on separable and sequence spaces. Matrices considered here have special structures like diagonal dominance, tridiagonal, sign distributions, etc. and are frequently nonsingular. Moreover, diagonally dominant finite and infinite matrices occur largely in numerical solutions of elliptic partial differential equations.The main focus is the theoretical and computational aspects concerning infinite linear algebraic and differential systems, using techniques like conformal mapping, iterations, truncations etc. to derive estimates based solutions. Particular attention is paid to computable precise error estimates, and explicit lower and upper bounds. Topics include Bessel’s, Mathieu equations, viscous fluid flow, simply and doubly connected regions, digital dynamics, eigenvalues of the Laplacian, etc. Also presented are results in generalized inverses and semi-infinite linear programming.     

Spectrum of certain tridiagonal matrices when their dimension goes to infinity

This paper deals with the spectra of matrices similar to infinite tridiagonal Toeplitz matrices with perturbations and with positive off-diagonal elements. We will discuss the asymptotic behavior of the spectrum of such matrices and we use them to determine the values of a matrix function, for an entire function. In particular we determine the matrix powers and matrix exponentials.     

On the parallel GSAOR method for block diagonally dominant matrices

In this paper we consider the parallel generalized SAOR iterative method based on the generalized AOR iterative method presented by James for solving large nonsingular system. We obtain some convergence theorems for the case when coefficient matrix is a block diagonally dominant matrix or a generalized block diagonal dominant matrix. A numerical example is given to illustrate to our results.