Sharp bounds for spectral radius of nonnegative weakly irreducible tensors

We obtain the sharp upper and lower bounds for the spectral radius of a nonnegative weakly irreducible tensor. By using the technique of the representation associate matrix of a tensor and the associate directed graph of the matrix, the equality cases of the bounds are completely characterized by graph theory methods. Applying these bounds to a nonnegative irreducible matrix or a connected graph (digraph), we can improve the results of L. H. You, Y. J. Shu, and P. Z. Yuan [Linear Multilinear Algebra, 2017, 65(1): 113–128], and obtain some new or known results. Applying these bounds to a uniform hypergraph, we obtain some new results and improve some known results of X. Y. Yuan, M. Zhang, and M. Lu [Linear Algebra Appl., 2015, 484: 540–549]. Finally, we give a characterization of a strongly connected k-uniform directed hypergraph, and obtain some new results by applying these bounds to a uniform directed hypergraph.     

矩阵奇异值的下界估计

本文中总记mxn复（实）矩阵空间以C"""（R"""），q二min｛。，n）．设A一（a；。）e＊-"－，A的q个奇异值按递减次序排列为。1川三。2（AZ...Z内科三0．对A的奇异值，特别是最小奇异值的下界估计，是矩阵分析的重要课题，在目前已有重要估计【回叫，C．R．Johnson给出的下述最小奇异值下界估计是最好的结果11］：矩阵Cassini型谱包含域得到了矩阵奇异值的一个下界估计式．进而给出了达到下界估计式时的矩阵表征，所得结果改进了山一［4］之相应结果．我们首先讨论方阵的情况．引理1．设A二（。ti）EC"""，人（）＝｛Al（A），...，A…     

Bounds and perturbation bounds for the matrix exponential

Some new types of bounds and perturbation bounds, based on the Jordan normal form, for the matrix exponential are derived. These bounds are compared to known bounds, both theoretically and by numerical examples. Some recent results on the matrix exponential and the logarithmic norm are also included.     

Perturbation Theory for Simultaneous Bases of Singular Subspaces

New perturbation theorems are proved for simultaneous bases of singular subspaces of real matrices. These results improve the absolute bounds previously obtained in [6] for general (complex) matrices. Unlike previous results, which are valid only for the Frobenius norm, the new bounds, as well as those in [6] for complex matrices, are extended to any unitarily invariant matrix norm. The bounds are complemented with numerical experiments which show their relevance for the algorithms computing the singular value decomposition. Additionally, the differential calculus approach employed allows to easily prove new sin perturbation theorems for singular subspaces which deal independently with left and right singular subspaces.     

Estimates for the inverse elements of tridiagonal matrices

In this work, new upper and lower bounds for the inverse entries of the tridiagonal matrices are presented. The bounds improve the bounds in D. Kershaw [Inequalities on the elements of the inverse of a certain tridiagonal matrix, Math. Comput. 24 (1970) 155–158], P.N. Shivakumar, C.X. Ji [Upper and lower bounds for inverse elements of finite and infinite tridiagonal matrices, Linear Algebr. Appl. 247 (1996) 297–316], R. Nabben [Two-sided bounds on the inverse of diagonally dominant tridiagonal matrices, Linear Algebr. Appl. 287 (1999) 289–305] and R. Peluso, T. Politi [Some improvements for two-sided bounds on the inverse of diagonally dominant tridiagonal matrices, Linear. Algebr. Appl. 330 (2001) 1–14].     

关于矩阵多项式特征值界的注记

本文讨论矩阵多项式特征值定域问题.首先对Higham和Tisseur[Linear Algebra Appl.,358(2003),5-22]得到的结果给出较详细的比较.然后利用分块矩阵谱半径的估计给出了获取特征值界的一种新办法.利用这种新办法,不但可以简明地得出很多已有的界,且对椭圆及双曲矩阵多项式得出了特征值的新的界.     

New upper solution bounds of the discrete algebraic Riccati matrix equation

In this note, we present upper matrix bounds for the solution of the discrete algebraic Riccati equation (DARE). Using the matrix bound of Theorem 2.2, we then give several eigenvalue upper bounds for the solution of the DARE and make comparisons with existing results. The advantage of our results over existing upper bounds is that the new upper bounds of Theorem 2.2 and Corollary 2.1 are always calculated if the stabilizing solution of the DARE exists, whilst all existing upper matrix bounds might not be calculated because they have been derived under stronger conditions. Finally, we give numerical examples to demonstrate the effectiveness of the derived results.     

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.     

New perturbation bounds for the W-weighted Drazin inverse

The constructive perturbation bounds for the W-weighted Drazin inverse are derived by two approaches in this paper. One uses the matrixG = [(A+E)W]^l?(AW)^l, whereA, E ∈ C ^mxn ,W ∈ C ^nxm ,l = max Ind(AW), Ind[(A + E)W]. The other uses a technique proposed by G. Stewart and based on perturbation theory for invariant subspaces of a matrix. The new approaches to develop perturbation bounds for W-weighted Drazin inverse of a matrix extend the previous results in [19, 29, 31, 36, 42, 44]. Several examples which indicate the sharpness of the new perturbation bounds are presented.     

Bounds for relative errors of complex matrix factorizations

The goal of this paper is to obtain optimal first order bounds for absolute and relative errors of unitary and Hermitian factors of some commonly used matrix factorizations. We have chosen the strong derivative calculus approach and we have expressed the factors as a differentiable function of the data but since these expressions define the functions implicitly, the inverse function theorem plays a central role in finding the Jacobian matrix. Then, first order bounds are deduced by means of the mean value theorem for the derivatives. We either improve or generalize some of the bounds proposed by Bhatia [1], Stewart [2], and Sun [3].     

G -函数与迭代阵谱半径的估计

该文应用G -函数概念, 获得了迭代矩阵谱半径新的上、下界, 所得结果推广和改进了文献[1--6]中的相应结果.这些结果适合于更广泛的矩阵类, 数值结果也表明在相同的条件下这些新界优于文献[1--6]中的界.     

严格对角占优M-矩阵的逆矩阵的无穷大范数的新上界

给出了严格对角占优M-矩阵的逆矩阵的无穷大范数上界新的估计式,进而给出严格对角占优M-矩阵的最小特征值下界的估计式.新估计式改进了已有文献的结果.     

Error bounds for linear complementarity problems of DB-matrices

Doubly B-matrices (DB-matrices), which properly contain B-matrices, are introduced by Peña (2003) [2]. In this paper we present error bounds for the linear complementarity problem when the matrix involved is a DB-matrix and a new bound for linear complementarity problem of a B-matrix. The numerical examples show that the bounds are sharp.     

Bounds and Majorization Relations for the Zeros of a Class of Fibonacci-Type Polynomials

We apply certain matrix inequalities involving eigenvalues, the numerical radius, and the spectral radius to obtain new bounds and majorization relations for the zeros of a class of Fibonacci-type polynomials. Our results improve upon some earlier bounds for the zeros of these polynomials.     

Perturbation d'une matrice hermitienne ou normale

Summary Given a hermitian (normal) matrixA with known eigenelements, we study the behavior of these elements under a hermitian perturbationH. With a symmetric 2×2 matrix, the problem is explicit (algebraic equation of 2nd degree), and we try, in the case of ann×n hermitian matrix, to obtain upper bounds which are as close as possible of exact results forn=2. The results are collected in § I. They state in a unified manner some results of Davis [2], Gavurin [4], Golub [5], Kato-Temple [7], Ortega [3], Wilkinson [8] and others.In § II, we apply this theory to produce error bounds for eigenelements computed by theQR and Jacobi methods. The given error bounds are realistic and easy to compute during the algorithms. When the separation distance between eigenvalues ofA approaches zero, the problem of computing eigenelements ofA+H is ill-conditionned with respect to the eigenvalues, the eigenvectors being orthogonal. The precision then obtained is given.

---

Perturbation d'une matrice hermitienne ou normale

     

Computation of Error Bounds for P-matrix Linear Complementarity Problems

We give new error bounds for the linear complementarity problem where the involved matrix is a P-matrix. Computation of rigorous error bounds can be turned into a P-matrix linear interval system. Moreover, for the involved matrix being an H-matrix with positive diagonals, an error bound can be found by solving a linear system of equations, which is sharper than the Mathias-Pang error bound. Preliminary numerical results show that the proposed error bound is efficient for verifying accuracy of approximate solutions. This work is partly supported by a Grant-in-Aid from Japan Society for the Promotion of Science.     

Perturbation analysis of singular subspaces and deflating subspaces

Summary. Perturbation expansions for singular subspaces of a matrix and for deflating subspaces of a regular matrix pair are derived by using a technique previously described by the author. The perturbation expansions are then used to derive Fr\'echet derivatives, condition numbers, and th-order perturbation bounds for the subspaces. Vaccaro's result on second-order perturbation expansions for a special class of singular subspaces can be obtained from a general result of this paper. Besides, new perturbation bounds for singular subspaces and deflating subspaces are derived by applying a general theorem on solution of a system of nonlinear equations. The results of this paper reveal an important fact: Each singular subspace and each deflating subspace have individual perturbation bounds and individual condition numbers. Received July 26, 1994     

B-矩阵线性互补问题的误差界估计

利用严格对角占优M-矩阵的逆矩阵的无穷大范数的范围,给出了B-矩阵线性互补问题误差界新的估计式.相应数值算例表明了结果的有效性.     

一类中立型泛函微分系统周期解存在性问题

首先对线性差分算子M:[Mx](t)=z(t)-Cx(t-r)的性质进行了研究,在此基础上利用Mawhin重合度拓展定理研究了一类具偏差变元的中立型泛函微分系统的周期解问题,得到了周期解存在性的新结论.本文的矩阵C仅为一般的实方阵,不必为实对称阵,因而结果改进和推广了已有工作.此外,本文周期解先验界估计方法与已有工作也不相同.     

Large deviation bounds for matrix Brownian motion

We prove large deviation bounds for the convergence of Hermitian matrix valued Brownian motion towards free Brownian motion. As a consequence, we obtain upper and lower bounds on the microstates entropy introduced by Voiculescu [24]. Oblatum 5-VIII-2002 & 18-XI-2002?Published online: 24 February 2003