Open Access Published by De Gruyter Open Access

Volume 7 Issue 1

January 2019

Issue of Special Matrices

Contents
Journal Overview

Open Access January 1, 2019

Self-dual Leonard pairs

Kazumasa Nomura, Paul Terwilliger

Page range: 1-19

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Abstract

Let F denote a field and let V denote a vector space over F with finite positive dimension. Consider a pair A, A* of diagonalizable F-linear maps on V, each of which acts on an eigenbasis for the other one in an irreducible tridiagonal fashion. Such a pair is called a Leonard pair. We consider the self-dual case in which there exists an automorphism of the endomorphism algebra of V that swaps A and A*. Such an automorphism is unique, and called the duality A ↔ A*. In the present paper we give a comprehensive description of this duality. In particular,we display an invertible F-linearmap T on V such that the map X → TXT −1 is the duality A ↔ A*. We express T as a polynomial in A and A*. We describe how T acts on 4 flags, 12 decompositions, and 24 bases for V.

Open Access January 18, 2019

Bidiagonalization of (k, k + 1)-tridiagonal matrices

S. Takahira, T. Sogabe, T.S. Usuda

Page range: 20-26

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Abstract

In this paper,we present the bidiagonalization of n-by-n (k, k+1)-tridiagonal matriceswhen n < 2k. Moreover,we show that the determinant of an n-by-n (k, k+1)-tridiagonal matrix is the product of the diagonal elements and the eigenvalues of the matrix are the diagonal elements. This paper is related to the fast block diagonalization algorithm using the permutation matrix from [T. Sogabe and M. El-Mikkawy, Appl. Math. Comput., 218, (2011), 2740-2743] and [A. Ohashi, T. Sogabe, and T. S. Usuda, Int. J. Pure and App. Math., 106, (2016), 513-523].

Open Access January 8, 2019

Some matrix properties preserved by generalized matrix functions

Michele Benzi, Ru Huang

Page range: 27-37

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Abstract

Generalized matrix functions were first introduced in [J. B. Hawkins and A. Ben-Israel, Linear and Multilinear Algebra, 1(2), 1973, pp. 163-171]. Recently, it has been recognized that these matrix functions arise in a number of applications, and various numerical methods have been proposed for their computation. The exploitation of structural properties, when present, can lead to more efficient and accurate algorithms. The main goal of this paper is to identify structural properties of matrices which are preserved by generalized matrix functions. In cases where a given property is not preserved in general, we provide conditions on the underlying scalar function under which the property of interest will be preserved by the corresponding generalized matrix function.

Open Access April 12, 2019

Some Hermite-Hadamard type inequalities for operator convex functions and positive maps

S. S. Dragomir

Page range: 38-51

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Abstract

In this paper we establish some inequalities of Hermite-Hadamard type for operator convex functions and positive maps. Applications for power function and logarithm are also provided.

Open Access May 22, 2019

Explicit determinants, inverses and eigenvalues of four band Toeplitz matrices with perturbed rows

Maoyun Zhang, Xiaoyu Jiang, Zhaolin Jiang

Page range: 52-66

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Abstract

In this paper, four-band Toeplitz matrices and four-band Hankel matrices of type I and type II with perturbed rows are introduced. Explicit determinants, inverses and eigenvalues for these matrices are derived by using a nice inverse formula of block bidiagonal Toeplitz matrices.

Open Access July 15, 2019

Infinite dimensional generalizations of Choi’s Theorem

Shmuel Friedland

Page range: 67-77

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Abstract

In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.

Open Access August 12, 2019

Best linear unbiased estimation for varying probability with and without replacement sampling

Stephen Haslett

Page range: 78-91

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Abstract

When sample survey data with complex design (stratification, clustering, unequal selection or inclusion probabilities, and weighting) are used for linear models, estimation of model parameters and their covariance matrices becomes complicated. Standard fitting techniques for sample surveys either model conditional on survey design variables, or use only design weights based on inclusion probabilities essentially assuming zero error covariance between all pairs of population elements. Design properties that link two units are not used. However, if population error structure is correlated, an unbiased estimate of the linear model error covariance matrix for the sample is needed for efficient parameter estimation. By making simultaneous use of sampling structure and design-unbiased estimates of the population error covariance matrix, the paper develops best linear unbiased estimation (BLUE) type extensions to standard design-based and joint design and model based estimation methods for linear models. The analysis covers both with and without replacement sample designs. It recognises that estimation for with replacement designs requires generalized inverses when any unit is selected more than once. This and the use of Hadamard products to link sampling and population error covariance matrix properties are central topics of the paper. Model-based linear model parameter estimation is also discussed.

Open Access September 3, 2019

Determinant of binary circulant matrices

M. Hariprasad

Page range: 92-94

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Abstract

This article gives a closed-form expression for the determinant of binary circulant matrices.

Open Access September 3, 2019

On computing the minimum singular value of a tensor sum

A. Ohashi, T. Sogabe

Page range: 95-106

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Abstract

Recently, the Lanczos bidiagonalization method over tensor space has been proposed for computing the maximum and minimum singular values of a tensor sum T . The method over tensor space is practical in memory and has a simple implementation due to recent developments in tensor computations; however, there is still room for improvement in the convergence to the minimum singular value. This study reconstructed an invert Lanczos bidiagonalization method from vector space to tensor space. The resulting algorithm requires solving linear systems at each iteration step. Using standard direct methods, such as the LU decomposition for solving the linear systems requires a huge memory of O ( n 6 ). Therefore, this paper proposes a tensor-structure-preserving direct methods of T whose memory requirements are of O ( n 3 ), which is equivalent to the order of iterative methods. Numerical examples indicate that the number of iterations tends to be much smaller than that of the conventional method.

Open Access September 13, 2019

Construction of generalized rotations and quasi-orthogonal matrices

Luis Verde-Star

Page range: 107-113

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Abstract

We propose some methods for the construction of large quasi-orthogonal matrices and generalized rotations that may be used in applications in data communications and image processing. We use certain combinations of constructions by blocks similar to the one used by Sylvester to build Hadamard matrices. The orthogonal designs related with the matrix representations of the complex numbers, the quaternions, and the octonions are used in our construction procedures.

Open Access September 24, 2019

A note on multilevel Toeplitz matrices

Lei Cao, Selcuk Koyuncu

Page range: 114-126

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Abstract

Chien, Liu, Nakazato and Tam proved that all n × n classical Toeplitz matrices (one-level Toeplitz matrices) are unitarily similar to complex symmetric matrices via two types of unitary matrices and the type of the unitary matrices only depends on the parity of n . In this paper we extend their result to multilevel Toeplitz matrices that any multilevel Toeplitz matrix is unitarily similar to a complex symmetric matrix. We provide a method to construct the unitary matrices that uniformly turn any multilevel Toeplitz matrix to a complex symmetric matrix by taking tensor products of these two types of unitary matrices for one-level Toeplitz matrices according to the parity of each level of the multilevel Toeplitz matrices. In addition, we introduce a class of complex symmetric matrices that are unitarily similar to some p -level Toeplitz matrices.

Open Access September 26, 2019

Computational methods for difference families in finite abelian groups

Dragomir Ž. Ðoković, Ilias S. Kotsireas

Page range: 127-141

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Abstract

Our main objective is to show that the computational methods, developed previously to search for difference families in cyclic groups, can be fully extended to the more general case of arbitrary finite abelian groups. In particular the power spectral density test and the method of compression can be used to help the search.

Open Access October 30, 2019

On relationships between two linear subspaces and two orthogonal projectors

Yongge Tian

Page range: 142-212

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Abstract

Sum and intersection of linear subspaces in a vector space over a field are fundamental operations in linear algebra. The purpose of this survey paper is to give a comprehensive approach to the sums and intersections of two linear subspaces and their orthogonal complements in the finite-dimensional complex vector space. We shall establish a variety of closed-form formulas for representing the direct sum decompositions of the m -dimensional complex column vector space 𝔺 m with respect to a pair of given linear subspaces 𝒨 and 𝒩 and their operations, and use them to derive a huge amount of decomposition identities for matrix expressions composed by a pair of orthogonal projectors onto the linear subspaces. As applications, we give matrix representation for the orthogonal projectors onto the intersections of a pair of linear subspaces using various matrix decomposition identities and Moore–Penrose inverses; necessary and su˚cient conditions for two linear subspaces to be in generic position; characterization of the commutativity of a pair of orthogonal projectors; necessary and su˚cient conditions for equalities and inequalities for a pair of subspaces to hold; equalities and inequalities for norms of a pair of orthogonal projectors and their operations; as well as a collection of characterizations of EP-matrix.

Special Issue Dedicated to Charles R. Johnson

Open Access December 2, 2019

Frobenius normal forms of doubly stochastic matrices

Pietro Paparella

Page range: 213-217

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Abstract

An elementary proof of a fundamental result on doubly stochastic matrices in Frobenius normal form is given. This result is used to establish several well-known results concerning permutations, including a theorem due to Ruffini.

Open Access December 2, 2019

A note on the eigenvalue free intervals of some classes of signed threshold graphs

Milica Anđelić, Tamara Koledin, Zoran Stanić

Page range: 218-225

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Abstract

We consider a particular class of signed threshold graphs and their eigenvalues. If Ġ is such a threshold graph and Q ( Ġ ) is a quotient matrix that arises from the equitable partition of Ġ , then we use a sequence of elementary matrix operations to prove that the matrix Q ( Ġ ) – xI ( x ∈ ℝ) is row equivalent to a tridiagonal matrix whose determinant is, under certain conditions, of the constant sign. In this way we determine certain intervals in which Ġ has no eigenvalues.

Open Access December 2, 2019

A note on completing quasi-distance and distance matrices

Yulin Zhang, A. Veloso da Costa, J. M. S. Simões-Pereira

Page range: 226-229

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Abstract

We give a necessary and sufficient condition for the existence of a quasi-distance matrix where some positive off-diagonal entries have been prescribed. Moreover, we give an algorithm for obtaining such a matrix. We analyze also the case where distances instead of quasi-distances are considered.

Open Access December 13, 2019

Toeplitz nonnegative realization of spectra via companion matrices

Macarena Collao, Mario Salas, Ricardo L. Soto

Page range: 230-245

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Abstract

The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ 1 , . . ., λn} . If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A , and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ = {λ 1 , . . ., λn} of complex numbers in the left-half plane, that is, with Re λ i ≤ 0, i = 2, . . ., n , is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

Open Access December 13, 2019

Updating a map of sufficient conditions for the real nonnegative inverse eigenvalue problem

C. Marijuán, M. Pisonero, Ricardo L. Soto

Page range: 246-256

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Abstract

The real nonnegative inverse eigenvalue problem (RNIEP) asks for necessary and sufficient conditions in order that a list of real numbers be the spectrum of a nonnegative real matrix. A number of sufficient conditions for the existence of such a matrix are known. The authors gave in [11] a map of sufficient conditions establishing inclusion relations or independency relations between them. Since then new sufficient conditions for the RNIEP have appeared. In this paper we complete and update the map given in [11].

Open Access December 13, 2019

The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree

Kenji Toyonaga

Page range: 257-262

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Abstract

Given a combinatorially symmetric matrix A whose graph is a tree T and its eigenvalues, edges in T can be classified in four categories, based upon the change in geometric multiplicity of a particular eigenvalue, when the edge is removed. We investigate a necessary and sufficient condition for each classification of edges. We have similar results as the case for real symmetric matrices whose graph is a tree. We show that a g-2-Parter edge, a g-Parter edge and a g-downer edge are located separately from each other in a tree, and there is a g-neutral edge between them. Furthermore, we show that the distance between a g-downer edge and a g-2-Parter edge or a g-Parter edge is at least 2 in a tree. Lastly we give a combinatorially symmetric matrix whose graph contains all types of edges.

Open Access December 13, 2019

Numerical construction of structured matrices with given eigenvalues

Brian D. Sutton

Page range: 263-271

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Abstract

We consider a structured inverse eigenvalue problem in which the eigenvalues of a real symmetric matrix are specified and selected entries may be constrained to take specific numerical values or to be nonzero. This includes the problem of specifying the graph of the matrix, which is determined by the locations of zero and nonzero entries. In this article, we develop a numerical method for constructing a solution to the structured inverse eigenvalue problem. The problem is recast as a constrained optimization problem over the orthogonal manifold, and a numerical optimization routine seeks its solution.

Open Access December 17, 2019

The integer cp-rank of 2 × 2 matrices

Thomas Laffey, Helena Šmigoc

Page range: 272-275

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Abstract

We show the cp-rank of an integer doubly nonnegative 2 × 2 matrix does not exceed 11.

Open Access December 13, 2019

Achievable multiplicity partitions in the inverse eigenvalue problem of a graph

Mohammad Adm, Shaun Fallat, Karen Meagher, Shahla Nasserasr, Sarah Plosker, Boting Yang

Page range: 276-290

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Abstract

Associated to a graph G is a set 𝒮( G ) of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be chosen. If G has n vertices, then the multiplicities of the eigenvalues of any matrix in 𝒮 ( G ) partition n ; this is called a multiplicity partition. We study graphs for which a multiplicity partition with only two integers is possible. The graphs G for which there is a matrix in 𝒮 ( G ) with partitions [ n − 2, 2] have been characterized. We find families of graphs G for which there is a matrix in 𝒮 ( G ) with multiplicity partition [ n − k , k ] for k ≥ 2. We focus on generalizations of the complete multipartite graphs. We provide some methods to construct families of graphs with given multiplicity partitions starting from smaller such graphs. We also give constructions for graphs with matrix in 𝒮 ( G ) with multiplicity partition [ n − k , k ] to show the complexities of characterizing these graphs.

Open Access December 17, 2019

The almost semimonotone matrices

Megan Wendler

Page range: 291-303

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Abstract

A (strictly) semimonotone matrix A ∈ ℝ n × n is such that for every nonzero vector x ∈ ℝ n with nonnegative entries, there is an index k such that x k > 0 and ( Ax ) k is nonnegative (positive). A matrix which is (strictly) semimonotone has the property that every principal submatrix is also (strictly) semimonotone. Thus, it becomes natural to examine the almost (strictly) semimonotone matrices which are those matrices which are not (strictly) semimonotone but whose proper principal submatrices are (strictly) semimonotone. We characterize the 2 × 2 and 3 × 3 almost (strictly) semimonotone matrices and describe many of their properties. Then we explore general almost (strictly) semimonotone matrices, including the problem of detection and construction. Finally, we relate (strict) central matrices to semimonotone matrices.

Open Access December 13, 2019

Extensions of the Eneström-Kakeya theorem for matrix polynomials

A. Melman

Page range: 304-315

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Abstract

The classical Eneström-Kakeya theorem establishes explicit upper and lower bounds on the zeros of a polynomial with positive coefficients and has been generalized for positive definite matrix polynomials by several authors. Recently, extensions that improve the (scalar) Eneström-Kakeya theorem were obtained with a transparent and unified approach using just a few tools. Here, the same tools are used to generalize these extensions to positive definite matrix polynomials, while at the same time generalizing the tools themselves. In the process, a framework is developed that can naturally generate additional similar results.

Open Access December 13, 2019

Diagonalizable matrices whose graph is a tree: the minimum number of distinct eigenvalues and the feasibility of eigenvalue assignments

Carlos M. Saiago

Page range: 316-326

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Abstract

Considered are combinatorially symmetric matrices, whose graph is a given tree, in view of the fact recent analysis shows that the geometric multiplicity theory for the eigenvalues of such matrices closely parallels that for real symmetric (and complex Hermitian) matrices. In contrast to the real symmetric case, it is shown that (a) the smallest example (13 vertices) of a tree and multiplicity list (3, 3, 3, 1, 1, 1, 1) meeting standard necessary conditions that has no real symmetric realizations does have a diagonalizable realization and for arbitrary prescribed (real and multiple) eigenvalues, and (b) that all trees with diameter < 8 are geometrically di-minimal (i.e., have diagonalizable realizations with as few of distinct eigenvalues as the diameter). This re-raises natural questions about multiplicity lists that proved subtly false in the real symmetric case. What is their status in the geometric multiplicity list case?

Open Access December 23, 2019

Inertias of Laplacian matrices of weighted signed graphs

K. Hassani Monfared, G. MacGillivray, D. D. Olesky, P. van den Driessche

Page range: 327-342

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Abstract

We study the sets of inertias achieved by Laplacian matrices of weighted signed graphs. First we characterize signed graphs with a unique Laplacian inertia. Then we show that there is a sufficiently small perturbation of the nonzero weights on the edges of any connected weighted signed graph so that all eigenvalues of its Laplacian matrix are simple. Next, we give upper bounds on the number of possible Laplacian inertias for signed graphs with a fixed flexibility τ (a combinatorial parameter of signed graphs), and show that these bounds are sharp for an infinite family of signed graphs. Finally, we provide upper bounds for the number of possible Laplacian inertias of signed graphs in terms of the number of vertices.

About this journal

Special Issue:

Contemporary Spectral Graph Theory

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.

Scope:

Matrix theory
Spectral graph theory
Combinatorial matrix theory
Nonnegative matrices
Inequalities in matrices
Numerical matrix theory
Operator theory and matrices

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