Canonical forms for normal matrices that commute with their complex conjugate

We consider the class of normal complex matrices that commute with their complex conjugate. We show that such matrices are real orthogonally similar to a canonical direct sum of 1-by-1 and certain 2-by-2 matrices. A canonical form for quasi-real normal matrices is obtained as a special case. We also exhibit a special form of the spectral theorem for normal matrices that commute with their conjugate.     

Entropy of orthogonal matrices and minimum distance orthostochastic matrices from the uniform van der Waerden matrices

In this article we formulate an optimization problem of minimizing the distance from the uniform van der Waerden matrices to orthostochastic matrices of different orders. We find a lower bound for the number of stationary points of the minimization problem, which is connected to the number of possible partitions of a natural number. The existence of Hadamard matrices ensures the existence of global minimum orthostochastic matrices for such problems. The local minimum orthostochastic matrices have been obtained for all other orders except for 11 and 19. We explore the properties of Hadamard, conference and weighing matrices to obtain such minimizing orthostochastic 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.     

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.     

Invertible Matrices over Finite Additively Idempotent Semirings

We investigate invertible matrices over finite additively idempotent semirings. The main result provides a criterion for the invertibility of such matrices. We also give a construction of the inverse matrix and a formula for the number of invertible matrices.     

Interlacing properties of the eigenvalues of some matrix classes

We establish the eigenvalue interlacing property (i.e. the smallest real eigenvalue of a matrix is less than the smallest real eigenvalue of any of its principal submatrices) for the class of matrices introduced by Kotelyansky (all principal and almost principal minors of these matrices are positive). We show that certain generalizations of Kotelyansky and totally positive matrices possess this property. We also prove some interlacing inequalities for the other eigenvalues of Kotelyansky matrices.     

Infinite triangular matrices, q-Pascal matrices, and determinantal representations

We use basic properties of infinite lower triangular matrices and the connections of Toeplitz matrices with generating-functions to obtain inversion formulas for several types of q-Pascal matrices, determinantal representations for polynomial sequences, and identities involving the q-Gaussian coefficients. We also obtain a fast inversion algorithm for general infinite lower triangular matrices.     

The Laplacian of a Graph as a Density Matrix: A Basic Combinatorial Approach to Separability of Mixed States

We study entanglement properties of mixed density matrices obtained from combinatorial Laplacians. This is done by introducing the notion of the density matrix of a graph. We characterize the graphs with pure density matrices and show that the density matrix of a graph can be always written as a uniform mixture of pure density matrices of graphs. We consider the von Neumann entropy of these matrices and we characterize the graphs for which the minimum and maximum values are attained. We then discuss the problem of separability by pointing out that separability of density matrices of graphs does not always depend on the labelling of the vertices. We consider graphs with a tensor product structure and simple cases for which combinatorial properties are linked to the entanglement of the state. We calculate the concurrence of all graphs on four vertices representing entangled states. It turns out that for these graphs the value of the concurrence is exactly fractional. Received July 28, 2004     

On algebras of Toeplitz fuzzy matrices

In this paper, necessary and sufficient conditions are given for a product of Toeplitz fuzzy matrices to be Toeplitz. As an application, a criterion for normality of Toeplitz fuzzy matrices is derived and conditions are deduced for symmetric idempotency of Toeplitz fuzzy matrices. We discuss similar results for Hankel fuzzy matrices. Keywords: Fuzzy matrix, Toeplitz and Hankel matrices.     

Separating doubly nonnegative and completely positive matrices

The cone of Completely Positive (CP) matrices can be used to exactly formulate a variety of NP-Hard optimization problems. A tractable relaxation for CP matrices is provided by the cone of Doubly Nonnegative (DNN) matrices; that is, matrices that are both positive semidefinite and componentwise nonnegative. A natural problem in the optimization setting is then to separate a given DNN but non-CP matrix from the cone of CP matrices. We describe two different constructions for such a separation that apply to 5 × 5 matrices that are DNN but non-CP. We also describe a generalization that applies to larger DNN but non-CP matrices having block structure. Computational results illustrate the applicability of these separation procedures to generate improved bounds on difficult problems.     

Sums of nilpotent matrices

We study matrices over general rings which are sums of nilpotent matrices. We show that over commutative rings all matrices with nilpotent trace are sums of three nilpotent matrices. We characterize 2-by-2 matrices with integer entries which are sums of two nilpotents via the solvability of a quadratic Diophantine equation. Some exemples in the case of matrices over noncommutative rings are given.     

A recursive least-squares algorithm for pairwise comparison matrices

Pairwise comparison matrices are commonly used for setting priorities among competing objects. In a leading decision making method called the analytic hierarchy process the principal right eigenvector components represent the weights of the alternatives. The direct least-squares method extracts the weight vector by first finding a rank-one matrix which minimizes the Euclidean distance from the original ratio matrix. We develop a recursive least-squares algorithm and reveal a striking correspondence between these two approaches for these matrices. The recursion applies for merely positive matrices also. We prove that a convergent iteration leads to matrices by which the Perron-eigenvectors and the Perron approximation of the original matrix may be produced. We show that certain useful properties of the recursion advance the development of reliable measures of perturbations of transitive matrices. Numerical analysis is included for a macroeconomic problem taken from the literature.     

On the eigenvalues of quaternion matrices

This article is a continuation of the article [F. Zhang, Ger?gorin type theorems for quaternionic matrices, Linear Algebra Appl. 424 (2007), pp. 139–153] on the study of the eigenvalues of quaternion matrices. Profound differences in the eigenvalue problems for complex and quaternion matrices are discussed. We show that Brauer's theorem for the inclusion of the eigenvalues of complex matrices cannot be extended to the right eigenvalues of quaternion matrices. We also provide necessary and sufficient conditions for a complex square matrix to have infinitely many left eigenvalues, and analyse the roots of the characteristic polynomials for 2?×?2 matrices. We establish a characterisation for the set of left eigenvalues to intersect or be part of the boundary of the quaternion balls of Ger?gorin.     

Group generalized inverses of M-matrices associated with periodic and nonperiodic jacobi matrices

We consider two types of irreducible stochastic matrices —tridiagonal matrices and periodic jacobi matrices —which can be viewed as transition matrices of interval and circular random walks,respcetively. For both types of matrices we give a formula for the group inverse of the associated singular M-matrix. We discuss both the sign patterns and the relative sizes of the entries in these group inverses and apply our results to give qualitative information about random walks on an interval and on a circle.     

On a class of quaternary complex Hadamard matrices

We introduce a class of regular unit Hadamard matrices whose entries consist of two complex numbers and their conjugates for a total of four complex numbers. We then show that these matrices are contained in the Bose–Mesner algebra of an association scheme arising from skew Paley matrices.     

Gauss-Manin connections for arrangements, III Formal connections

We study the Gauss-Manin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a complex rank one local system. We define formal Gauss-Manin connection matrices in the Aomoto complex and prove that, for all arrangements and all local systems, these formal connection matrices specialize to Gauss-Manin connection matrices.

     

On the finiteness property for rational matrices

We analyze the periodicity of optimal long products of matrices. A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. It was conjectured a decade ago that all finite sets of real matrices have the finiteness property. This “finiteness conjecture” is now known to be false but no explicit counterexample is available and in particular it is unclear if a counterexample is possible whose matrices have rational or binary entries. In this paper, we prove that all finite sets of nonnegative rational matrices have the finiteness property if and only if pairs of binary matrices do and we state a similar result when negative entries are allowed. We also show that all pairs of 2×2 binary matrices have the finiteness property. These results have direct implications for the stability problem for sets of matrices. Stability is algorithmically decidable for sets of matrices that have the finiteness property and so it follows from our results that if all pairs of binary matrices have the finiteness property then stability is decidable for nonnegative rational matrices. This would be in sharp contrast with the fact that the related problem of boundedness is known to be undecidable for sets of nonnegative rational matrices.     

Stanley's zrank conjecture on skew partitions

We present an affirmative answer to Stanley's zrank conjecture, namely, the zrank and the rank are equal for any skew partition. We show that certain classes of restricted Cauchy matrices are nonsingular and furthermore, the signs are determined by the number of zero entries. We also give a characterization of the rank in terms of the Giambelli-type matrices of the corresponding skew Schur functions. Our approach also applies to the factorial Cauchy matrices and the inverse binomial coefficient matrices.

     

Expansion of Weighted Pseudoinverse Matrices in Matrix Power Products

On the basis of the Euler identity, we obtain expansions for weighted pseudoinverse matrices with positive-definite weights in infinite matrix power products of two types: with positive and negative exponents. We obtain estimates for the closeness of weighted pseudoinverse matrices and matrices obtained on the basis of a fixed number of factors of matrix power products and terms of matrix power series. We compare the rates of convergence of expansions of weighted pseudoinverse matrices in matrix power series and matrix power products to weighted pseudoinverse matrices. We consider problems of construction and comparison of iterative processes of computation of weighted pseudoinverse matrices on the basis of the obtained expansions of these matrices.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1539–1556, November, 2004.     

Perfect elimination orderings for symmetric matrices

We introduce a new class of structured symmetric matrices by extending the notion of perfect elimination ordering from graphs to weighted graphs or matrices. This offers a common framework capturing common vertex elimination orderings of monotone families of chordal graphs, Robinsonian matrices and ultrametrics. We give a structural characterization for matrices that admit perfect elimination orderings in terms of forbidden substructures generalizing chordless cycles in graphs.