Upper bound for the non-maximal eigenvalues of irreducible nonnegative matrices

We present a lower and an upper bound for the second smallest eigenvalue of Laplacian matrices in terms of the averaged minimal cut of weighted graphs. This is used to obtain an upper bound for the real parts of the non-maximal eigenvalues of irreducible nonnegative matrices. The result can be applied to Markov chains.     

Minimal zonotopes containing the crosspolytope

Motivated by the problem to improve Minkowski’s lower bound on the successive minima for the class of zonotopes we determine the minimal volume of a zonotope containing the standard crosspolytope. It turns out that this volume can be expressed via the maximal determinant of a ±1-matrix, and that in each dimension the set of minimal zonotopes contains a parallelepiped. Based on that link to ±1- matrices, we characterize all zonotopes attaining the minimal volume in dimension 3 and present related results in higher dimensions.     

Even and odd diagonals in doubly stochastic matrices

The existence of even or odd diagonals in doubly stochastic matrices depends on the number of positive elements in the matrix. The optimal general lower bound in order to guarantee the existence of such diagonals is determined, as well as their minimal number for given number of positive elements. The results are related to the characterization of even doubly stochastic matrices in connection with Birkhoff's algorithm.     

A note on matrices with maximal growth factor for Neville elimination

Neville elimination is a direct method for the solution of linear systems of equations with advantages for some classes of matrices and in the context of pivoting strategies for parallel implementations. The growth factor is an indicator of the numerical stability of an algorithm. In the literature, bounds for the growth factor of Neville elimination with some pivoting strategies have appeared. In this work, we determine all the matrices such that the minimal upper bound of the growth factor of Neville elimination with those pivoting strategies is reached.     

The structure of irreducible matrix groups with submultiplicative spectrum

We describe the structure of irreducible matrix groups with submultiplicative spectrum. Since all such groups are nilpotent, the study is focused on p-groups. We obtain a block-monomial structure of matrices in irreducible p-groups and build polycyclic series arising from that structure. We give an upper bound to the exponent of these groups. We determine all minimal irreducible groups of p× p matrices with submultiplicative spectrum and discuss the case of p²× p² matrices if p is an odd prime.     

Bounds for visual cryptography schemes

In this paper, we investigate the best pixel expansion of various models of visual cryptography schemes. In this regard, we consider visual cryptography schemes introduced by Tzeng and Hu (2002) [13]. In such a model, only minimal qualified sets can recover the secret image and the recovered secret image can be darker or lighter than the background. Blundo et al. (2006) [4] introduced a lower bound for the best pixel expansion of this scheme in terms of minimal qualified sets. We present another lower bound for the best pixel expansion of the scheme. As a corollary, we introduce a lower bound, based on an induced matching of hypergraph of qualified sets, for the best pixel expansion of the aforementioned model and the traditional model of visual cryptography scheme realized by basis matrices. Finally, we study access structures based on graphs and we present an upper bound for the smallest pixel expansion in terms of strong chromatic index.     

On the critical ideals of graphs

We introduce some determinantal ideals of the generalized Laplacian matrix associated to a digraph G, that we call critical ideals of G. Critical ideals generalize the critical group and the characteristic polynomials of the adjacency and Laplacian matrices of a digraph. The main results of this article are the determination of some minimal generator sets and the reduced Gröbner basis for the critical ideals of the complete graphs, the cycles and the paths. Also, we establish a bound between the number of trivial critical ideals and the stability and clique numbers of a graph.     

On the accuracy of the Gerschgorin circle theorem for bounding the spread of a real symmetric matrix

The spread of a matrix with real eigenvalues is the difference between its largest and smallest eigenvalues. The Gerschgorin circle theorem can be used to bound the extreme eigenvalues of the matrix and hence its spread. For nonsymmetric matrices the Gerschgorin bound on the spread may be larger by an arbitrary factor than the actual spread even if the matrix is symmetrizable. This is not true for real symmetric matrices. It is shown that for real symmetric matrices (or complex Hermitian matrices) the ratio between the bound and the spread is bounded by $\text{p+1}$, where p is the maximum number of off diagonal nonzeros in any row of the matrix. For full matrices this is just $\text{n}$. This bound is not quite sharp for n greater than 2, but examples with ratios of $\text{n?1}$ for all n are given. For banded matrices with m nonzero bands the maximum ratio is bounded by $\text{m}$ independent of the size of n. This bound is sharp provided only that n is at least 2m. For sparse matrices, p may be quite small and the Gerschgorin bound may be surprisingly accurate.     

A characterization of the nearly sign central matrices and its minimallity

The sign central matrices were characterized by Ando and Brualdi. And, the nearly sign central matrices were characterized by Lee and Cheon. In this paper, we give another characterization of nearly sign central matrices. Also, we introduce the nearly minimal sign central matrices and study the properties of nearly minimal sign central matrices.     

A conforming decomposition theorem,a piecewise linear theorem of the alternative,and scalings of matrices satisfying lower and upper bounds

A scaling of a nonnegative matrixA is a matrixXAY ^?1, whereX andY are nonsingular, nonnegative diagonal matrices. Some condition may be imposed on the scaling, for example, whenA is square,X=Y or detX=detY. We characterize matrices for which there exists a scaling that satisfies predetermined upper and lower bound. Our principal tools are a piecewise linear theorem of the alternative and a theorem decomposing a solution of a system of equations as a sum of minimal support solutions which conform with the given solutions.     

On the generalized indices of boolean matrices

We characterize completely those Boolean matrices with the largest generalized indices in the class of Boolean matrices and in the class of reducible Boolean matrices and derive a new upper bound for the generalized index in terms of period. We also generalize the upper and lower multiexponents of primitive Boolean matrices to general Boolean matrices.     

On the generalized indices of boolean matrices

We characterize completely those Boolean matrices with the largest generalized indices in the class of Boolean matrices and in the class of reducible Boolean matrices and derive a new upper bound for the generalized index in terms of period. We also generalize the upper and lower multiexponents of primitive Boolean matrices to general Boolean matrices.     

The structure of irreducible matrix groups with submultiplicative spectrum

We describe the structure of irreducible matrix groups with submultiplicative spectrum. Since all such groups are nilpotent, the study is focused on p-groups. We obtain a block-monomial structure of matrices in irreducible p-groups and build polycyclic series arising from that structure. We give an upper bound to the exponent of these groups. We determine all minimal irreducible groups of p× p matrices with submultiplicative spectrum and discuss the case of p ²× p ² matrices if p is an odd prime.     

Stopping criteria for the Ando-Li-Mathias and Bini-Meini-Poloni geometric means

We provide an upper bound for the number of iterations necessary to achieve a desired level of accuracy for the Ando-Li-Mathias [Linear Algebra Appl. 385 (2004) 305-334] and Bini-Meini-Poloni [Math. Comput. 79 (2010) 437-452] symmetrization procedures for computing the geometric mean of n positive definite matrices, where accuracy is measured by the spectral norm and the Thompson metric on the convex cone of positive definite matrices. It is shown that the upper bound for the number of iterations depends only on the diameter of the set of n matrices and the desired convergence tolerance. A striking result is that the upper bound decreases as n increases on any bounded region of positive definite matrices.     

A structure of the Bang-Bang representation for 3×3 embeddable matrices

Summary Necessary and sufficient conditions are given for a 3×3 stochastic matrix to be embeddable by 6 elementary stochastic matrices (Poisson matrices). For a 3×3 embeddable matrix, a structure of the minimal Bang-Bang representation, i.e. the one that contains the smallest number of elementary matrices, is obtained. Based on the minimal Bang-Bang representation an algorithm for determining the embeddability of a 3×3 stochastic matrix is given.I would like to thank Søren Johansen for helpful comments and stimulating discussions on the subject of this paper     

双随机矩阵收敛指数的最好上界

Ｍ．Ｌｅｗｉｎ（１９７４）已得到了ｎ阶本原双随机矩阵收敛（本原）指数的最好上界。本文对不可约非本原和可约双随机矩阵的收敛指数作出估值，从而证明了Ｌｅｗｉｎ的上界是适用于一切双随机矩阵的最好上界。     

The behavior of the QR-factorization algorithm with column pivoting

A bound on the performance of QR-factorization with column pivoting is derived and two classes of matrices are constructed for which the bound is sharp or asymptotically sharp.     

矩阵Fan积特征值的界

设Z_n为非对角元素都为非正实数的n阶方阵的集合,令A_k∈Z_n,k∈{1,…,m},给出矩阵Fan积最小特征值的一个新下界,其中p_k>0且and (sum from k=1 to m)1/p_k≥1,这个下界改进了文献中的相关结果.     

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.     

An improvement of the error bounds for linear complementarity problems of Nekrasov matrices

A new error bound for the linear complementarity problems, which involves a parameter, is given when the involved matrices are Nekrasov matrices. It is shown that there exists an optimal value of the parameter such that the new bound is sharper than that provided by Li et al. (Numer Algor. 2017;74:997–1009). Numerical examples are given to illustrate the corresponding results.