An inverse eigenvalue problem for totally nonnegative matrices

In a previous paper we proved that the diagonal elements of a totally nonnegative matrix are majorized by its eigenvalues. In this note we show that the majorization of a vector of nonnegative real numbers by another vector of nonnegative real numbers is not sufficient for the existence of a totally nonnegative matrix with diagonal elements taken from the entries of the majorized vector and eigenvalues taken from the entries of the majorizing vector.     

On the Square Root of an H-matrix with Positive Diagonal Elements

This paper proves that for an H-matrix with positive diagonal elements there exists one and only one square root which is also an H-matrix with positive diagonal elements. An algorithm approaching the square root is proposed.     

Some improvements for two-sided bounds on the inverse of diagonally dominant tridiagonal matrices

In this paper, we obtain lower and upper bounds for the entries of the inverses of diagonally dominant tridiagonal matrices. First of all we derive the bounds for off-diagonal elements of the inverse as a function of the diagonal ones, then we improve the two-sided bounds for the diagonal entries obtaining sharper lower and upper bounds for all the elements of the inverse.     

对角方阵的变换群

在不改变对角方阵各行、各列、主对角线、次对角线的元素之集的条件下,其变换群是n次对称群S_n的直积S_n×S_n的子群,因对角拉丁方、对角拉丁方正交侣、幻方、高次幻方、加乘幻方均属此类方阵,本文对构作这类对象及研究它们的计数有重要意义.     

On the product of the diagonal elements of hermitian matrices

In [5] M. Marcus stated a conjecture concerning the product of the diagonal elements of normal matrices which is false [6]. In this paper we prove that such a conjecture is true in the Hermitian case.     

Regularization techniques in interior point methods

Regularization techniques, i.e., modifications on the diagonal elements of the scaling matrix, are considered to be important methods in interior point implementations. So far, regularization in interior point methods has been described for linear programming problems, in which case the scaling matrix is diagonal. It was shown that by regularization, free variables can be handled in a numerically stable way by avoiding column splitting that makes the set of optimal solutions unbounded. Regularization also proved to be efficient for increasing the numerical stability of the computations during the solutions of ill-posed linear programming problems. In this paper, we study the factorization of the augmented system arising in interior point methods. In our investigation, we generalize the methods developed and used in linear programming to the case when the scaling matrix is positive semidefinite, but not diagonal. We show that regularization techniques may be applied beyond the linear programming case.     

A two-step modulus-based multisplitting iteration method for the nonlinear complementarity problem

In this paper, we construct a two-step modulus-based multisplitting iteration method based on multiple splittings of the system matrix for the nonlinear complementarity problem. And we prove its convergence when the system matrix is an $H$-matrix with positive diagonal elements. Numerical experiments show that the proposed method is efficient.     

Estimates for the inverse of a matrix and bounds for eigenvalues

This paper gives new bounds for the relationship between the diagonal elements of a square matrix and the corresponding diagonal elements of the matrix inverse, as well as bounds for the eigenvalues of the matrix. The results given here generalize those of Ostrowski and Ky Fan, and have their origin in engineering application.     

A new formulation of regularized meshless method applied to interior and exterior anisotropic potential problems

This paper proposes a new formulation of regularized meshless method (RMM), which differs from the traditional RMM in that the traditional formulation generates the diagonal elements of influence matrix via null-field integral equations, while our new one directly employs the boundary integral equations at the domain point to evaluate the diagonal elements. We test the present RMM formulation to two-dimensional anisotropic potential problems in finite and infinite domains in comparison with the traditional RMM. Numerical results show that the present RMM sharply outperforms the traditional RMM in the solution of interior problems, while the latter is clearly superior for exterior problems. A rigorous theoretical analysis of circular domain case also corroborates such numerical experiment observations and is provided in the appendix of this paper.     

相容次序矩阵PSD迭代法的收敛性

本文是在求解大型线性方程组Ax=b的系数矩阵A为(1,1)相容次序矩阵且其Jacobi迭代矩阵的特征值均为纯虚数或零的条件下,得到PSD迭代法收敛的充分必要性定理,并在特殊情况下得到了相应的最优参数.     

非负不可约矩阵Perron根的新下界

给出了非负不可约矩阵Perron根的一些新下界.特别的,若矩阵对角元素均相同,设为a,则(?)该结果易于计算且优于相关文献的下界.     

Extreme determinants on the convex hull of matrices with prescribed singular values

In a recent paper[1] the author carried through a comprehesive analysis of the diagonal elements of matrices having prescribed singular values, and as an application of this analysis a characterization was obtained of matrices with prescribed singular values. In the present note we obtain the greatest and least values for the determinants of the matrices in such convex hulls.     

Upper bounds for nearly optimal diagonal scaling of matrices

Optimal diagonal scaling of an n×n matrix A consists in finding a diagonal matrix D that minimizes a condition number of AD. Often a nearly optimal scaling of A is achieved by taking a diagonal matrix D₁ such that all diagonal elements of D₁A^TAD₁ are equal to one. It is shown in this paper that the condition number of AD₁ can be at least (n/2)^1/2 times the minimal one. Some questions for a further research are posed.     

Extreme determinants on the convex hull of matrices with prescribed singular values

In a recent paper[1] the author carried through a comprehesive analysis of the diagonal elements of matrices having prescribed singular values, and as an application of this analysis a characterization was obtained of matrices with prescribed singular values. In the present note we obtain the greatest and least values for the determinants of the matrices in such convex hulls.     

Bounds on the diagonal elements of a unitary matrix

It was shown by A. Horn that the diagonal elements of a unitary n×n matrix satisfy a set of linear inequalities (Theorem I). We give a simple proof of this result, and we show that the diagonal elements satisfy additional linear inequalities (Theorem II) if the matrix is also symmetric.     

A Functional CLT for Partial Traces of Random Matrices

Journal of Theoretical Probability - In this paper, we show a functional central limit theorem for the sum of the first $$\lfloor t n \rfloor $$ diagonal elements of f(Z) as a function in t, for Z...     

The community problem for Stieltjes matrices

The communality problem in factor analysis is that of reducing the diagonal elements of a correlation matrix so that the resulting matrix will be positive semidefinite and of minimum rank. The problem is well studied, but no effective solution procedures have been devised. In this paper, we propose a variant problem and give an algorithm for its solution. We prove that a solution to this problem also solves the communality problem if the correlation matrix is Stieltjes.     

预条件同时置换(PSD)迭代法的收敛性分析

1引言求解线性方程组Ax=6,(1.1)其中A∈R~(n×n)非奇异阵且对角元非零,x,b∈R~n,x未知,b已知.不失一般性,我们假设A=I-L-U,(1.2)其中L,U分别为A的严格下和上三角矩阵,相应的Jacobi迭代矩阵为B=L U.(1.3)若Q是非奇异阵且Q~(-1)易计算,于是(1.1)可以变成     

A theorem on inverse of tridiagonal matrices

Tridiagonal or Jacobi matrices arise in many diverse branches of mathematics and have been studied extensively. However, there is little written about the inverses of such matrices. In this paper we characterize those matrices with nonzero diagonal elements whose inverses are tridiagonal. The arguments given are elementaryand show that matrices with tridiagonal inverses have an interesting structure.