On the sharpness of two-sided bounds for the Perron Root and of the related eigenvalue inclusion sets

The paper considers the sharpness problem for certain two-sided bounds for the Perron root of an irreducible nonnegative matrix. The results obtained are applied to prove the sharpness of the related eigenvalue inclusion sets in classes of matrices with fixed diagonal entries, bounded above deleted absolute row sums, and a partly specified irreducible sparsity pattern.     

New eigenvalue inclusion sets for tensors

Two new eigenvalue inclusion sets for tensors are established. It is proved that the new eigenvalue inclusion sets are tighter than that in Qi's paper “Eigenvalues of a real supersymmetric tensor”. As applications, upper bounds for the spectral radius of a nonnegative tensor are obtained, and it is proved that these upper bounds are sharper than that in Yang's paper “Further results for Perron–Frobenius theorem for nonnegative tensors”. And some sufficient conditions of the positive definiteness for an even‐order real supersymmetric tensor are given. Copyright © 2012 John Wiley & Sons, Ltd.     

Upper and lower bounds for the Perron root of a nonnegative matrix

We derive upper and lower bounds for the Perron root of a nonnegative matrix by using generalized Gershgorin inclusion regions. Our bounds seem particularly effective for certain sparse matrices.     

非负矩阵Perron根上界序列(英文)

给出了非负矩阵Perron根的一系列优化上界,即通过相似对角变换与Gerschgorin定理较好的估计了Perron根的上界,并且通过例子来说明这种方法的有效性.     

非负不可约矩阵Perron根的上界序列

给出了非负不可约矩阵Perron根的新上界序列,并指出该序列是收敛到Perron根的,最后给出两个数值例子加以说明,并与文献[1,3,6]中的结论进行了比较．     

Bounds for the singular values of a matrix involving its sparsity pattern

The paper presents new upper and lower bounds for the singular values of rectangularmatrices explicitly involving the matrix sparsity pattern. These bounds are based on an upper bound for the Perron root of a nonnegative matrix and on the sparsity-dependent version of the Ostrowski-Brauer theorem on eigenvalue inclusion regions. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 57–68.     

非负矩阵Perron根的下界序列

非负矩阵Perron根的估计是非负矩阵理论研究的重要课题之一.如果其上下界能够表示为非负矩阵元素的易于计算的函数,那么这种估计价值更高.本文结合非负矩阵的迹分两种情况给出Perron根的下界序列,并且给出数值例子加以说明.     

非负矩阵Perron根的上界

文章针对特殊的非负矩阵，应月简单的相似变换，使矩阵保持非负性且最大行和减小，从而得到行和为正非负矩阵Perron根的新上界．     

计算非负不可约矩阵Perron根的对角变换(英文)

计算非负矩阵Perron根一般通过矩阵的对角变换,但是有的时候是不可行的.本文为非负不可约矩阵的计算给了一列对角变换.此种变换对所有的非负不可约矩阵实用,并且方便计算,最后给出了数值例子.     

计算非负不可约矩阵Perron根的对角变换

计算非负矩阵Perron根一般通过矩阵的对角变换,但是有的时候是不可行的．本文为非负不可约矩阵的计算给了一列对角变换．此种变换对所有的非负不可约矩阵实用,并且方便计算,最后给出了数值例子．     

A property of eigenvalue bounds for a class of symmetric tridiagonal interval matrices

The eigenvalue bounds of interval matrices are often required in some mechanical and engineering fields. In this paper, we consider an interval eigenvalue problem with symmetric tridiagonal matrices. A theoretical result is obtained that under certain assumptions the upper and lower bounds of interval eigenvalues of the problem must be achieved just at some vertex matrices of the interval matrix. Then a sufficient condition is provided to guarantee the assumption to be satisfied. The conclusion is illustrated also by a numerical example. Copyright © 2010 John Wiley & Sons, Ltd.     

关于非负矩阵Perron特征值的上、下界

本文通过构造一可逆矩阵,对一类非负矩阵A进行若干次简单的相似变换,便可同时得到矩阵A之Perron特征值的较好的上、下界.     

非负不可约矩阵Perron根新的下界序列

Estimate bounds for the Perron root of a nonnegative matrix are important in theory of nonnegative matrices. It is more practical when the bounds are expressed as an easily calculated function in elements of matrices. For the Perron root of nonnegative irreducible matrices, three sequences of lower bounds are presented by means of constructing shifted matrices, whose convergence is studied. The comparisons of the sequences with known ones are supplemented with a numerical example.     

On determinantal diagonal dominance conditions

The paper suggests sufficient nonsingularity conditions for matrices in terms of certain determinantal relations of diagonal dominance type, which improve and generalize some known results. These conditions are used to describe new eigenvalue inclusion sets and to derive new two-sided bounds on the determinants of matrices satisfying them. Bibliography: 8 titles.     

非负矩阵Perron根上界序列

给出了非负矩阵Perron根的一系列优化上界,即通过相似对角变换与Gerschgorin定理较好的估计了Perron根的上界,并且通过例子来说明这种方法的有效性．     

非负矩阵最大特征值的新界值

对最大特征值的上下界进行估计是非负矩阵理论的重要部分,借助两个新的矩阵,从而得到一个判定非负矩阵最大特征值范围的界值定理,其结果比有关结论更加精确.     

Conditioning analysis of block incomplete factorizations and its application to elliptic equations

Summary. The paper deals with eigenvalue estimates for block incomplete factorization methods for symmetric matrices. First, some previous results on upper bounds for the maximum eigenvalue of preconditioned matrices are generalized to each eigenvalue. Second, upper bounds for the maximum eigenvalue of the preconditioned matrix are further estimated, which presents a substantial improvement of earlier results. Finally, the results are used to estimate bounds for every eigenvalue of the preconditioned matrices, in particular, for the maximum eigenvalue, when a modified block incomplete factorization is used to solve an elliptic equation with variable coefficients in two dimensions. The analysis yields a new upper bound of type for the condition number of the preconditioned matrix and shows clearly how the coefficients of the differential equation influence the positive constant . Received March 27, 1996 / Revised version received December 27, 1996     

Joint bounds for the Perron roots of nonnegative matrices with applications

Given a finite set {A_x}_{x ∈ X} of nonnegative matrices, we derive joint upper and lower bounds for the row sums of the matrices D⁻¹ A^(x) D, x ∈ X, where D is a specially chosen nonsingular diagonal matrix. These bounds, depending only on the sparsity patterns of the matrices A^(x) and their row sums, are used to obtain joint two-sided bounds for the Perron roots of given nonnegative matrices, joint upper bounds for the spectral radii of given complex matrices, bounds for the joint and lower spectral radii of a matrix set, and conditions sufficient for all convex combinations of given matrices to be Schur stable. Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 30–56.     

非负矩阵Perron根的上界序列

1引言本文讨论非负矩阵Perron根的上界。设     

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.