Bounds on eigenvalues of the Hadamard product and the Fan product of matrices

We prove an upper bound for the spectral radius of the Hadamard product of nonnegative matrices and a lower bound for the minimum eigenvalue of the Fan product of M-matrices. These improve two existing results.     

矩阵Hadamard积和Fan积的特征值界的一些新估计式

本文研究了非奇异M-矩阵A与B的Fan积的最小特征值下界和非负矩阵A与B的Hadamard积 的谱半径上界的估计问题.利用Brauer定理,得到了一些只依赖于矩阵的元素且易于计算的新估计式,改进 了文献[4]现有的一些结果.     

Some new bounds on the spectral radius of matrices

A new lower bound on the smallest eigenvalue τ(AB) for the Fan product of two nonsingular M-matrices A and B is given. Meanwhile, we also obtain a new upper bound on the spectral radius ρ(A°B) for nonnegative matrices A and B. These bounds improve some results of Huang (2008) [R. Huang, Some inequalities for the Hadamard product and the Fan product of matrices, Linear Algebra Appl. 428 (2008) 1551-1559].     

New lower bounds on eigenvalue of the Hadamard product of an M-matrix and its inverse

Some new lower bounds for the minimum eigenvalue of the Hadamard product of an M-matrix and its inverse are given. These bounds improve the results of [H.B. Li, T.Z. Huang, S.Q. Shen, H. Li, Lower bounds for the minimum eigenvalue of Hadamard product of an M-matrix and its inverse, Linear Algebra Appl. 420 (2007) 235-247].     

New inequalities for the minimum eigenvalue of M-matrices

Some new inequalities for the minimum eigenvalue of M-matrices are established. These inequalities improve the results in [G. Tian and T. Huang, Inequalities for the minimum eigenvalue of M-matrices, Electr. J. Linear Algebra 20 (2010), pp. 291–302].     

The inverse positivity of perturbed tridiagonal M-matrices

A well-known property of an M-matrix M is that the inverse is element-wise non-negative, which we write as M^-1?0. In this paper, we consider element-wise perturbations of non-symmetric tridiagonal M-matrices and obtain sufficient bounds on the perturbations so that the non-negative inverse persists. These bounds improve the bounds recently given by Kennedy and Haynes [Inverse positivity of perturbed tridiagonal M-matrices, Linear Algebra Appl. 430 (2009) 2312-2323]. In particular, when perturbing the second diagonals (elements (l,l+2) and (l,l-2)) of M, these sufficient bounds are shown to be the actual maximum allowable perturbations. Numerical examples are given to demonstrate the effectiveness of our estimates.     

Some inequalities for the Hadamard product and the Fan product of matrices

If A and B are nonsingular M-matrices, a sharp lower bound on the smallest eigenvalue τ(A★B) for the Fan product of A and B is given, and a sharp lower bound on τ(A°B^-1) for the Hadamard product of A and B^-1 is derived. In addition, we also give a sharp upper bound on the spectral radius ρ(A°B) for nonnegative matrices A and B.     

Lower bounds for the minimum eigenvalue of Hadamard product of an M-matrix and its inverse

For the Hadamard product A ° A⁻¹ of an M-matrix A and its inverse A⁻¹, we give new lower bounds for the minimum eigenvalue of A ° A⁻¹. These bounds are strong enough to prove the conjecture of Fiedler and Markham [An inequality for the Hadamard product of an M-matrix and inverse M-matrix, Linear Algebra Appl. 101 (1988) 1-8].     

Bounds for the infinity norm of the inverse for certain M- and H-matrices

The paper presents new two-sided bounds for the infinity norm of the inverse for the so-called PM-matrices, which form a subclass of the class of nonsingular M-matrices and contain the class of strictly diagonally dominant matrices. These bounds are shown to be monotone with respect to the underlying partitioning of the index set, and the equality cases are analyzed. Also an upper bound for the infinity norm of the inverse of a PH-matrix (whose comparison matrix is a PM-matrix) is derived. The known Ostrowski, Ahlberg–Nilson–Varah, and Mora?a bounds are shown to be special cases of the upper bound obtained.     

On a generalized Fan inequality

It is shown that the ω- and τ-matrices, the weakly sign symmetric matrices, the R- and V-matrices, and the matrices c-equivalent to an M-matrix or to a real matrix with nonpositive off-diagonal elements, can all be characterized by the same determinantal inequality, which we call a generalized Fan inequality.     

A trace inequality for M-matrices and the symmetrizability of a real matrix by a positive diagonal matrix

The question of whether a real matrix is symmetrizable via multiplication by a diagonal matrix with positive diagonal entries is reduced to the corresponding question for M-matrices and related to Hadamard products. In the process, for a nonsingular M-matrix A, it is shown that tr(A^-1A^T) ? n, with equality if and only if A is symmetric, and that the minimum eigenvalue of A^-1 ° A is ? 1 with equality in the irreducible case if and only if A is positive diagonally symmetrizable.     

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.     

New error bounds for the linear complementarity problem with an SB-matrix

Error bounds for SB-matrices linear complementarity problems are given in the paper (Dai et al., Numer Algorithms 61:121–139, 2012). In this paper, new error bounds for the linear complementarity problem when the matrix involved is an SB-matrix are presented and some sufficient conditions that new bounds are sharper than those of the previous paper under certain assumptions are provided. New perturbation bounds of SB-matrices linear complementarity problems are also considered.     

Error bounds for linear complementarity problems for SB-matrices

S-strictly dominant B-matrices (SB-matrices) are introduced by Li et al. (Numer Linear Algebra Appl 14:391?C405, 2007). In this paper, we give error bounds for the linear complementarity problem when the matrix involved is an SB-matrix, which generalize those of DB-matrix linear complementarity problem and show advantages with respect to the computational cost. Then the perturbation bounds of SB-matrices linear complementarity problems are also provided. The preliminary numerical results show the sharpness of the bounds.     

Characterizations of inverse M-matrices with special zero patterns

In this paper, we provide some characterizations of inverse M-matrices with special zero patterns. In particular, we give necessary and sufficient conditions for k-diagonal matrices and symmetric k-diagonal matrices to be inverse M-matrices. In addition, results for triadic matrices, tridiagonal matrices and symmetric 5-diagonal matrices are presented as corollaries.     

A generalization of N-matrices

Ky Fan defines an N-matrix to be a matrix of the form A = tI ? B, B ? 0, λ < t < ?(B), where ?(B) is the spectral radius of B and λ is the maximum of the spectral radii of all principal submatrices of B of order n ? 1. In this paper, we define the closure (N₀-matrices) of N-matrices by letting λ ? t. It is shown that if A ∈ Z and A^-1 < 0, then A ∈ N. Certain inequalities of N-matrices are shown to hold for N₀-matrices, and a method for constructing an N-matrix from an M-matrix is given.     

Classes of general H-matrices

Let M(A) denote the comparison matrix of a square H-matrix A, that is, M(A) is an M-matrix. H-matrices such that their comparison matrices are nonsingular are well studied in the literature. In this paper, we study characterizations of H-matrices with either singular or nonsingular comparison matrices. The spectral radius of the Jacobi matrix of M(A) and the generalized diagonal dominance property are used in the characterizations. Finally, a classification of the set of general H-matrices is obtained.     

Lower bounds on the minimum average distance of binary codes

Let β(n,M) denote the minimum average Hamming distance of a binary code of length n and cardinality M. In this paper we consider lower bounds on β(n,M). All the known lower bounds on β(n,M) are useful when M is at least of size about 2ⁿ⁻¹/n. We derive new lower bounds which give good estimations when size of M is about n. These bounds are obtained using a linear programming approach. In particular, it is proved that lim_n→∞β(n,2n)=5/2. We also give a new recursive inequality for β(n,M).     

On the minimum average distance of binary constant weight codes

Let β(n,M,w) denote the minimum average Hamming distance of a binary constant weight code with length n, size M and weight w. In this paper, we study the problem of determining β(n,M,w). Using the methods from coding theory and linear programming, we derive several lower bounds on the average Hamming distance of a binary constant weight code. These lower bounds enable us to determine the exact value for β(n,M,w) in several cases.     

A comparison of error bounds for linear complementarity problems of H-matrices

We give new error bounds for the linear complementarity problem when the involved matrix is an H-matrix with positive diagonals. We find classes of H-matrices for which the new bounds improve considerably other previous bounds. We also show advantages of these new bounds with respect the computational cost. A new perturbation bound of H-matrices linear complementarity problems is also presented.