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Volume 4 Issue 1

January 2016

Issue of Special Matrices

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Contents
Journal Overview

Regular articles

Open Access December 16, 2015

Pentadiagonal Companion Matrices

Brydon Eastman, Kevin N. Vander Meulen

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Abstract

The class of sparse companion matrices was recently characterized in terms of unit Hessenberg matrices. We determine which sparse companion matrices have the lowest bandwidth, that is, we characterize which sparse companion matrices are permutationally similar to a pentadiagonal matrix and describe how to find the permutation involved. In the process, we determine which of the Fiedler companion matrices are permutationally similar to a pentadiagonal matrix. We also describe how to find a Fiedler factorization, up to transpose, given only its corner entries.

Open Access December 16, 2015

Matrix and discrepancy view of generalized random and quasirandom graphs

Marianna Bolla, Ahmed Elbanna

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Abstract

We will discuss how graph based matrices are capable to find classification of the graph vertices with small within- and between-cluster discrepancies. The structural eigenvalues together with the corresponding spectral subspaces of the normalized modularity matrix are used to find a block-structure in the graph. The notions are extended to rectangular arrays of nonnegative entries and to directed graphs. We also investigate relations between spectral properties, multiway discrepancies, and degree distribution of generalized random graphs. These properties are regarded as generalized quasirandom properties, and we conjecture and partly prove that they are also equivalent for certain deterministic graph sequences, irrespective of stochastic models.

Open Access December 16, 2015

The Smith normal form of product distance matrices

R. B. Bapat, Sivaramakrishnan Sivasubramanian

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Abstract

Let G = ( V , E ) be a connected graph with 2-connected blocks H 1 , H 2 , . . . , H r . Motivated by the exponential distance matrix, Bapat and Sivasubramanian in [4] defined its product distance matrix D G and showed that det D G only depends on det D H i for 1 ≤ i ≤ r and not on the manner in which its blocks are connected. In this work, when distances are symmetric, we generalize this result to the Smith Normal Form of D G and give an explicit formula for the invariant factors of D G .

Open Access December 16, 2015

Nonlinear maps preserving Lie products on triangular algebras

Weiyan Yu

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Abstract

In this paper we prove that every bijection preserving Lie products from a triangular algebra onto a normal triangular algebra is additive modulo centre. As an application, we described the form of bijections preserving Lie products on nest algebras and block upper triangular matrix algebras.

Open Access December 16, 2015

Sufficient conditions to be exceptional

Charles R. Johnson, Robert B. Reams

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Abstract

A copositive matrix A is said to be exceptional if it is not the sum of a positive semidefinite matrix and a nonnegative matrix. We show that with certain assumptions on A −1 , especially on the diagonal entries, we can guarantee that a copositive matrix A is exceptional. We also show that the only 5-by-5 exceptional matrix with a hollow nonnegative inverse is the Horn matrix (up to positive diagonal congruence and permutation similarity).

Open Access December 16, 2015

Orthogonal diagonalization for complex skew-persymmetric anti-tridiagonal Hankel matrices

Jesús Gutiérrez-Gutiérrez, Marta Zárraga-Rodríguez

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Abstract

In this paper, we obtain an eigenvalue decomposition for any complex skew-persymmetric anti-tridiagonal Hankel matrix where the eigenvector matrix is orthogonal.

Open Access January 20, 2016

Preserving zeros of Lie product on alternate matrices

Ajda Fošner, Bojan Kuzma

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Abstract

We study continuous maps on alternate matrices over complex field which preserve zeros of Lie product.

Open Access February 5, 2016

On some characterizations of strong power graphs of finite groups

A. K. Bhuniya, Sudip Bera

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Abstract

Let G be a finite group of order n. The strong power graph P s (G) of G is the undirected graph whose vertices are the elements of G such that two distinct vertices a and b are adjacent if a m1 =b m2 for some positive integers m 1 , m 2 < n. In this article we classify all groups G for which P s (G) is a line graph. Spectrum and permanent of the Laplacian matrix of the strong power graph P s (G) are found for any finite group G.

Open Access February 22, 2016

On the cardinality of complex matrix scalings

George Hutchinson

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Abstract

We disprove a conjecture made by Rajesh Pereira and Joanna Boneng regarding the upper bound on the number of doubly quasi-stochastic scalings of an n × n positive definite matrix. In doing so, we arrive at the true upper bound for 3 × 3 real matrices, and demonstrate that there is no such bound when n ≥ 4.

Open Access June 3, 2016

Ranks of permutative matrices

Xiaonan Hu, Charles R. Johnson, Caroline E. Davis, Yimeng Zhang

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Abstract

A new type of matrix, termed permutative, is defined and motivated herein. The focus is upon identifying circumstances under which square permutative matrices are rank deficient. Two distinct ways, along with variants upon them are given. These are a special kind of grouping of rows and a type of partition in which the blocks are again permutative. Other, results are given, along with some questions and conjectures.

Open Access July 18, 2016

Zero-one completely positive matrices and the A(R, S) classes

G. Dahl, T. A. Haufmann

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Abstract

A matrix of the form A = BBT where B is nonnegative is called completely positive (CP). Berman and Xu (2005) investigated a subclass of CP-matrices, called f0, 1g-completely positive matrices. We introduce a related concept and show connections between the two notions. An important relation to the so-called cut cone is established. Some results are shown for f0, 1g-completely positive matrices with given graphs, and for {0,1}-completely positive matrices constructed from the classes of (0, 1)-matrices with fixed row and column sums.

Open Access October 7, 2016

A class of cyclic (v; k₁, k₂, k₃; λ) difference families with v ≡ 3 (mod 4) a prime

Dragomir Ž. Ðokovic, Ilias S. Kotsireas

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Abstract

We construct several new cyclic (v; k1, k2, k3; λ) difference families, with v ≡ 3 (mod 4) a prime and λ = k1 + k2 + k3 − (3v − 1)/4. Such families can be used in conjunction with the well-known Paley-Todd difference sets to construct skew-Hadamard matrices of order 4v. Our main result is that we have constructed for the first time the examples of skew Hadamard matrices of orders 4 · 239 = 956 and 4 · 331 = 1324.

Special Issue: Proceedings of the 24th International Workshop on Matrices and Statistics

Open Access December 16, 2015

Antieigenvalue analysis for continuum mechanics, economics, and number theory

Karl Gustafson

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Abstract

My recent book Antieigenvalue Analysis , World-Scientific, 2012, presented the theory of antieigenvalues from its inception in 1966 up to 2010, and its applications within those forty-five years to Numerical Analysis, Wavelets, Statistics, Quantum Mechanics, Finance, and Optimization. Here I am able to offer three further areas of application: Continuum Mechanics, Economics, and Number Theory. In particular, the critical angle of repose in a continuum model of granular materials is shown to be exactly my matrix maximum turning angle of the stress tensor of the material. The important Sharpe ratio of the Capital Asset Pricing Model is now seen in terms of my antieigenvalue theory. Euclid’s Formula for Pythagorean triples becomes a special case of my operator trigonometry.

Open Access December 16, 2015

Unified speed estimation of various stabilities

Mu-Fa Chen

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Abstract

The main topic of this talk is the speed estimation of stability/instability. The word “various” comes with no surprising since there are a lot of different types of stability/instability and each of them has its own natural distance to measure. However, the adjective “unified” is very much unexpected. The talk surveys our recent progress on the topic, made in the past five years or so.

Open Access January 21, 2016

Studying the various properties of MIN and MAX matrices - elementary vs. more advanced methods

Mika Mattila, Pentti Haukkanen

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Abstract

Let T = {z1, z2, . . . , zn} be a finite multiset of real numbers, where z1 ≤ z2 ≤ · · · ≤ zn. The purpose of this article is to study the different properties of MIN and MAX matrices of the set T with min(zi , zj) and max(zi , zj) as their ij entries, respectively.We are going to do this by interpreting these matrices as so-called meet and join matrices and by applying some known results for meet and join matrices. Once the theorems are found with the aid of advanced methods, we also consider whether it would be possible to prove these same results by using elementary matrix methods only. In many cases the answer is positive.

Open Access February 2, 2016

Generation of all magic squares of order 5 and interesting patterns finding

Ziqi Lin, Sijie Liu, Kai-Tai Fang, Yuhui Deng

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Abstract

This paper presents an enumeration algorithm to generate all magic squares of order 5 based on the ideas of basic form (Schroeppel [7]) and generating vector which is extension of Frénicle Quads (Ollerenshaw and Bondi [6]). The results lead us to extend Frénicle-Amela patterns from the case of order 4 to the case of order 5, which we refer to Frénicle-Amela-Like patterns. We show that these interesting Frénicle-Amela-Like patterns appear simultaneously. The number of these patterns is also calculated.

Open Access February 12, 2016

Matrix rank/inertia formulas for least-squares solutions with statistical applications

Yongge Tian, Bo Jiang

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Abstract

Least-Squares Solution (LSS) of a linear matrix equation and Ordinary Least-Squares Estimator (OLSE) of unknown parameters in a general linear model are two standard algebraical methods in computational mathematics and regression analysis. Assume that a symmetric quadratic matrix-valued function Φ(Z) = Q − ZPZ0 is given, where Z is taken as the LSS of the linear matrix equation AZ = B. In this paper, we establish a group of formulas for calculating maximum and minimum ranks and inertias of Φ(Z) subject to the LSS of AZ = B, and derive many quadratic matrix equalities and inequalities for LSSs from the rank and inertia formulas. This work is motivated by some inference problems on OLSEs under general linear models, while the results obtained can be applied to characterize many algebraical and statistical properties of the OLSEs.

Open Access March 18, 2016

Accurate calculations of Stationary Distributions and Mean First Passage Times in Markov Renewal Processes and Markov Chains

Jeffrey J. Hunter

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Abstract

This article describes an accurate procedure for computing the mean first passage times of a finite irreducible Markov chain and a Markov renewal process. The method is a refinement to the Kohlas, Zeit fur Oper Res, 30, 197–207, (1986) procedure. The technique is numerically stable in that it doesn’t involve subtractions. Algebraic expressions for the special cases of one, two, three and four states are derived.Aconsequence of the procedure is that the stationary distribution of the embedded Markov chain does not need to be derived in advance but can be found accurately from the derived mean first passage times. MatLab is utilized to carry out the computations, using some test problems from the literature.

Open Access March 18, 2016

Why the Kemeny Time is a constant

Karl Gustafson, Jeffrey J. Hunter

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Abstract

We present a new fundamental intuition forwhy the Kemeny feature of a Markov chain is a constant. This new perspective has interesting further implications.

Open Access April 11, 2016

On a criterion of D-stability for P-matrices

Olga Y. Kushel

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Abstract

In this paper, we study positive stability and D-stability of P-matrices.We introduce the property of Dθ-stability, i.e., the stability with respect to a given order θ. For an n × n P-matrix A, we prove a new criterion of D-stability and Dθ-stability, based on the properties of matrix scalings.

Open Access April 12, 2016

Regularization for high-dimensional covariance matrix

Xiangzhao Cui, Chun Li, Jine Zhao, Li Zeng, Defei Zhang, Jianxin Pan

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Abstract

In many applications, high-dimensional problem may occur often for various reasons, for example, when the number of variables under consideration is much bigger than the sample size, i.e., p >> n. For highdimensional data, the underlying structures of certain covariance matrix estimates are usually blurred due to substantial random noises, which is an obstacle to draw statistical inferences. In this paper, we propose a method to identify the underlying covariance structure by regularizing a given/estimated covariance matrix so that the noises can be filtered. By choosing an optimal structure from a class of candidate structures for the covariance matrix, the regularization is made in terms of minimizing Frobenius-norm discrepancy. The candidate class considered here includes the structures of order-1 moving average, compound symmetry, order-1 autoregressive and order-1 autoregressive moving average. Very intensive simulation studies are conducted to assess the performance of the proposed regularization method for very high-dimensional covariance problem. The simulation studies also show that the sample covariance matrix, although performs very badly in covariance estimation for high-dimensional data, can be used to correctly identify the underlying structure of the covariance matrix. The approach is also applied to real data analysis, which shows that the proposed regularization method works well in practice.

Open Access April 26, 2016

Nonlinear Markov processes in big networks

Quan-Lin Li

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Abstract

Big networks express multiple classes of large-scale networks in many practical areas such as computer networks, internet of things, cloud computation, manufacturing systems, transportation networks, and healthcare systems. This paper analyzes such big networks, and applies the mean-field theory and the nonlinear Markov processes to constructing a broad class of nonlinear continuous-time block-structured Markov processes, which can be used to deal with many practical stochastic systems. Firstly, a nonlinear Markov process is derived from a large number of big networks with weak interactions, where each big network is described as a continuous-time block-structured Markov process. Secondly, some effective algorithms are given for computing the fixed points of the nonlinear Markov process by means of the UL-type RG-factorization. Finally, the Birkhoff center, the locally stable fixed points, the Lyapunov functions and the relative entropy are developed to analyze stability or metastability of the system of weakly interacting big networks, and several interesting open problems are proposed with detailed interpretation. We believe that the methodology and results given in this paper can be useful and effective in the study of big networks.

Open Access April 29, 2016

Positive semidefiniteness of estimated covariance matrices in linear models for sample survey data

Stephen Haslett

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Abstract

Descriptive analysis of sample survey data estimates means, totals and their variances in a design framework. When analysis is extended to linear models, the standard design-based method for regression parameters includes inverse selection probabilities as weights, ignoring the joint selection probabilities. When joint selection probabilities are included to improve estimation, and the error covariance is not a diagonal matrix, the unbiased sample based estimator of the covariance is the Hadamard product of the population covariance, the elementwise inverse of selection probabilities and joint selection probabilities, and a sample selection matrix of rank equal to the sample size. This Hadamard product is however not always positive definite, which has implications for best linear unbiased estimation. Conditions under which a change in covariance structure leaves BLUEs and/or BLUPs are known. Interestingly, this class of “equivalent” matrices for linear models includes non-positive semi-definite matrices. The paper uses these results to explore how the estimated covariance from the sample can be modified so that it meets necessary conditions to be positive semidefinite, while retaining the property that fitting a linear model to the sampled data yields the same BLUEs and/or BLUPs as when the original Hadamard product is used.

Open Access May 2, 2016

Sensitivity analysis in linear models

Shuangzhe Liu, Tiefeng Ma, Yonghui Liu

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Abstract

In this work, we consider the general linear model or its variants with the ordinary least squares, generalised least squares or restricted least squares estimators of the regression coefficients and variance. We propose a newly unified set of definitions for local sensitivity for both situations, one for the estimators of the regression coefficients, and the other for the estimators of the variance. Based on these definitions, we present the estimators’ sensitivity results.We include brief remarks on possible links of these definitions and sensitivity results to local influence and other existing results.

Open Access July 5, 2016

Professor Haruo Yanai and multivariate analysis

Yoshio Takane

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Abstract

The late Professor Yanai has contributed to many fields ranging from aptitude diagnostics, epidemiology, and nursing to psychometrics and statistics. This paper reviews some of his accomplishments in multivariate analysis through his collaborative work with the present author, along with some untold episodes for the inception of key ideas underlying the work. The various topics covered include constrained principal component analysis, extensions of Khatri’s lemma, theWedderburn-Guttman theorem, ridge operators, generalized constrained canonical correlation analysis, and causal inference. A common thread running through all of them is projectors and singular value decomposition, which are the main subject matters of a recent monograph by Yanai, Takeuchi, and Takane [60].

Open Access July 7, 2016

Moore-Penrose inverse of a hollow symmetric matrix and a predistance matrix

Hiroshi Kurata, Ravindra B. Bapat

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Abstract

By a hollow symmetric matrix we mean a symmetric matrix with zero diagonal elements. The notion contains those of predistance matrix and Euclidean distance matrix as its special cases. By a centered symmetric matrix we mean a symmetric matrix with zero row (and hence column) sums. There is a one-toone correspondence between the classes of hollow symmetric matrices and centered symmetric matrices, and thus with any hollow symmetric matrix D we may associate a centered symmetric matrix B, and vice versa. This correspondence extends a similar correspondence between Euclidean distance matrices and positive semidefinite matrices with zero row and column sums.We show that if B has rank r, then the corresponding D must have rank r, r + 1 or r + 2. We give a complete characterization of the three cases.We obtain formulas for the Moore-Penrose inverse D + in terms of B+, extending formulas obtained in Kurata and Bapat (Linear Algebra and Its Applications, 2015). If D is the distance matrix of a weighted tree with the sum of the weights being zero, then B + turns out to be the Laplacian of the tree, and the formula for D + extends a well-known formula due to Graham and Lovász for the inverse of the distance matrix of a tree.

Open Access September 12, 2016

Editors’ Summary for the Special Issue: Proceedings of the 24th International Workshop on Matrices and Statistics

Jeffrey J. Hunter

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Special Issue: 5th International Conference on Matrix Analysis and Applications

Open Access June 16, 2016

Essential sign change numbers of full sign pattern matrices

Xiaofeng Chen, Wei Fang, Wei Gao, Yubin Gao, Guangming Jing, Zhongshan Li, Yanling Shao, Lihua Zhang

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Abstract

A sign pattern (matrix) is a matrix whose entries are from the set {+, −, 0} and a sign vector is a vector whose entries are from the set {+, −, 0}. A sign pattern or sign vector is full if it does not contain any zero entries. The minimum rank of a sign pattern matrix A is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of A. The notions of essential row sign change number and essential column sign change number are introduced for full sign patterns and condensed sign patterns. By inspecting the sign vectors realized by a list of real polynomials in one variable, a lower bound on the essential row and column sign change numbers is obtained. Using point-line confiurations on the plane, it is shown that even for full sign patterns with minimum rank 3, the essential row and column sign change numbers can differ greatly and can be much bigger than the minimum rank. Some open problems concerning square full sign patterns with large minimum ranks are discussed.

Open Access June 16, 2016

A Lie product type formula in Euclidean Jordan algebras

Jiyuan Tao

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Abstract

In this paper,we state and prove an analog of Lie product formula in the setting of Euclidean Jordan algebras.

Open Access July 4, 2016

Some norm inequalities for special Gram matrices

Ramazan Türkmen, Osman Kan, Hasan Gökbas

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Abstract

In this paper we firstly give majorization relations between the vectors Fn = {f 0 , f 1 , . . . , f n−1 },L n = {l 0 , l 1 , . . . , l n−1 } and Pn = {p 0 , p 1 , . . . , p n−1 } which constructed with fibonacci, lucas and pell numbers. Then we give upper and lower bounds for determinants, Euclidean norms and Spectral norms of Gram matrices G F =〈F n ,F ni 〉, G L =〈L n ,L ni 〉, G P =〈P n ,P ni 〉, G FL =〈F n ,L ni 〉, G FP =〈F n ,P ni 〉.

Open Access August 26, 2016

An update on a few permanent conjectures

Fuzhen Zhang

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Abstract

We review and update on a few conjectures concerning matrix permanent that are easily stated, understood, and accessible to general math audience. They are: Soules permanent-on-top conjecture†, Lieb permanent dominance conjecture, Bapat and Sunder conjecture† on Hadamard product and diagonal entries, Chollet conjecture on Hadamard product, Marcus conjecture on permanent of permanents, and several other conjectures. Some of these conjectures are recently settled; some are still open.We also raise a few new questions for future study. (†conjectures have been recently settled negatively.)

Open Access December 5, 2016

On numerical range of sp(2n, C)

Wen Yan, Jicheng Tao, Zhao Lu

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Abstract

In this paper we studied the classical numerical range of matrices in sp(2n, C). We obtained some result on the relationship between the numerical range of a matrix in and that of its diagonal block, the singular values of its off-diagonal block A 2 .

Open Access December 14, 2016

Derivatives of orbital function and an extension of Berezin-Gel’fand’s theorem

Tin-Yau Tam, William C. Hill

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Abstract

A generalization of a result of Berezin and Gel’fand in the context of Eaton triples is given. The generalization and its proof are Lie-theoretic free and requires some basic knowledge of nonsmooth analysis. The result is then applied to determine the distance between a point and a G-orbit or its convex hull.We also discuss the derivatives of some orbital functions.

Open Access December 15, 2016

Characterizations of the distribution of the Demmel condition number of real Wishart matrices

M. Shakil, M. Ahsanullah

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Abstract

The Demmel condition number is an indicator of the matrix condition, and its properties have recently found applications in many practical problems, such as in MIMO communication systems, in the analytical prediction of level-crossing and fade duration statistics of Rayleigh channels, and in spectrum sensing for cognitive radio systems, among others. As the characterizations of a probability distribution play an important role in probability and statistics, in this paper we study the characterizations of the distribution of the Demmel condition number of real Wishart matrices by truncated first moment. Since the truncated distributions arise in practical statistics where the ability of record observations is limited to a given threshold or within a specified range, we hope that these characterizations will be quite useful for practitioners and researchers in the fields of probability, statistics, and other applied sciences, such as actuarial science, linear algebra, multivariate statistics, physics, wireless communications, among others.

Open Access December 19, 2016

Editors’ Summary of the Special Issue dedicated to the 5th International Conference on Matrix Analysis and Applications

Shaun Fallat, Shahla Nasserasr, Qing-Wen Wang, Fuzhen Zhang

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About this journal

Special Matrices publishes original articles of wide significance and originality in all areas of research involving structured matrices present in various branches of pure and applied mathematics and their noteworthy applications in physics, engineering, and other sciences

Aims and Scope

Matrix theory

Spectral graph theory

Combinatorial matrix theory

Nonnegative matrices

Inequalities in matrices

Numerical matrix theory

Operator theory and matrices

Volume 4 Issue 1

Issue of Special Matrices

Pentadiagonal Companion Matrices

Abstract

Matrix and discrepancy view of generalized random and quasirandom graphs

Abstract

The Smith normal form of product distance matrices

Abstract

Nonlinear maps preserving Lie products on triangular algebras

Abstract

Sufficient conditions to be exceptional

Abstract

Orthogonal diagonalization for complex skew-persymmetric anti-tridiagonal Hankel matrices

Abstract

Preserving zeros of Lie product on alternate matrices

Abstract

On some characterizations of strong power graphs of finite groups

Abstract

On the cardinality of complex matrix scalings

Abstract

Ranks of permutative matrices

Abstract

Zero-one completely positive matrices and the A(R, S) classes

Abstract

A class of cyclic (v; k1, k2, k3; λ) difference families with v ≡ 3 (mod 4) a prime

Abstract

Antieigenvalue analysis for continuum mechanics, economics, and number theory

Abstract

Unified speed estimation of various stabilities

Abstract

Studying the various properties of MIN and MAX matrices - elementary vs. more advanced methods

Abstract

Generation of all magic squares of order 5 and interesting patterns finding

Abstract

Matrix rank/inertia formulas for least-squares solutions with statistical applications

Abstract

Accurate calculations of Stationary Distributions and Mean First Passage Times in Markov Renewal Processes and Markov Chains

Abstract

Why the Kemeny Time is a constant

Abstract

On a criterion of D-stability for P-matrices

Abstract

Regularization for high-dimensional covariance matrix

Abstract

Nonlinear Markov processes in big networks

Abstract

Positive semidefiniteness of estimated covariance matrices in linear models for sample survey data

Abstract

Sensitivity analysis in linear models

Abstract

Professor Haruo Yanai and multivariate analysis

Abstract

Moore-Penrose inverse of a hollow symmetric matrix and a predistance matrix

Abstract

Editors’ Summary for the Special Issue: Proceedings of the 24th International Workshop on Matrices and Statistics

Essential sign change numbers of full sign pattern matrices

Abstract

A Lie product type formula in Euclidean Jordan algebras

Abstract

Some norm inequalities for special Gram matrices

Abstract

An update on a few permanent conjectures

Abstract

On numerical range of sp(2n, C)

Abstract

Derivatives of orbital function and an extension of Berezin-Gel’fand’s theorem

Abstract

Characterizations of the distribution of the Demmel condition number of real Wishart matrices

Abstract

Editors’ Summary of the Special Issue dedicated to the 5th International Conference on Matrix Analysis and Applications

A class of cyclic (v; k₁, k₂, k₃; λ) difference families with v ≡ 3 (mod 4) a prime