Invariant factors of integral quaternion matrices

Existence of a diagonal form under unimodular equivalence is proved for matrices with entries from the Hurwitz ring of integral quaternions. The diagonal elements satisfy certain divisibility relations with an unexpected character, and these force a degree of uniqueness to the diagonal form. Connections between the so obtained invariant factors of a full matrix and those of a submatrix are then established.     

Invariant factors and generalized Schur matrices

The invariant factors of a generalization of the Schur matrix are found.     

Invariant factors of integral quaternion matrices II

In the author's previous paper a diagonal form was found under unimodular equivalence for matrices over the Hurwitz ring of integral quaternions, and uniqueness was established for the norms of certain constituents of the diagonal elements. In the present paper it is shown that the odd, primitive, parts of the all but one of the diagonal elements may be freely chosen provided that the norm constraint is met.     

Invariant factors, Wiener-Hopf factorization indices, and invariant factors at infinity of rational matrices

The relationship between the finite structure, the infinite structure, and the Wiener-Hopf factorization indices of any rectangular rational matrix is studied.     

Invariant functionals for random matrices

A new approach to the study of the Lyapunov exponents of random matrices is presented. It is proved that, under general assumptions, any family of nonnegative matrices possesses a continuous concave positively homogeneous invariant functional (“antinorm”) on ℝ₊^d. Moreover, the coefficient corresponding to an invariant antinorm equals the largest Lyapunov exponent. All conditions imposed on the matrices are shown to be essential. As a corollary, a sharp estimate for the asymptotics of the mathematical expectation for logarithms of norms of matrix products and of their spectral radii is derived. New upper and lower bounds for Lyapunov exponents are obtained. This leads to an algorithm for computing Lyapunov exponents. The proofs of the main results are outlined.     

Type-II matrices and combinatorial structures

Type-II matrices are a class of matrices used by Jones in his work on spin models. In this paper we show that type-II matrices arise naturally in connection with some interesting combinatorial and geometric structures.     

Invariant polynomials of sums of matrices

Some recent results on invariant polynomials of sums of two matrices are examined.     

Some combinatorial properties of centrosymmetric matrices

The elements in the group of centrosymmetric n×n permutation matrices are the extreme points of a convex subset of n²-dimensional Euclidean space, which we characterize by a very simple set of linear inequalities, thereby providing an interesting solvable case of a difficult problem posed by L. Mirsky, as well as a new analogue of the famous theorem on doubly stochastic matrices due to G. Birkhoff. Several further theorems of a related nature also are included.     

Invariant theoretic characterization of subdiscriminants of matrices

An invariant theoretic characterization of subdiscriminants of matrices is given. The structure as a module over the special orthogonal group of the minimal degree non-zero homogeneous component of the vanishing ideal of the variety of real symmetric matrices with a bounded number of different eigenvalues is investigated. These results are applied to the study of sum of squares presentations of subdiscriminants of real symmetric matrices.     

Factored matrices can generate combinatorial identities

By identifying the terms in the LU decomposition of various matrices, one produces combinatorial identities. Examples are given with formulas involving binomial coefficients and other numbers arising from simple recurrence formulas, number-theoretic functions, q-series, and orthogonal polynomials.     

Triangular matrices and combinatorial inversion formulas

The paper is devoted to the construction of the matrix inverse of an infinite triangular matrix and to finding the connection coefficients between polynomial sequences and general combinatorial inversion formulas.     

Self-dual codes from orbit matrices and quotient matrices of combinatorial designs

In this paper we give constructions of self-orthogonal and self-dual codes, with respect to certain scalar products, with the help of orbit matrices of block designs and quotient matrices of symmetric (group) divisible designs (SGDDs) with the dual property. First we describe constructions from block designs and their extended orbit matrices, where the orbit matrices are induced by the action of an automorphism group of the design. Further, we give some further constructions of self-dual codes from symmetric block designs and their orbit matrices. Moreover, in a similar way as for symmetric designs, we give constructions of self-dual codes from SGDDs with the dual property and their quotient matrices.     

A combinatorial property for semigroups of matrices

In this paper we study the semigroups of matrices over a commutative semiring. We prove that a semigroup of matrices over a tropical semiring satisfies a combinatorial property called weak permutation property. We consider an application of this result to the Burnside problem for groups.     

Invariant polynomials of matrices with prescibed entries

We study the invariant polynomials of an n by n matrix when n-1 arbitrary entries are prescribed.     

Integral and rational completions of combinatorial matrices

Earlier results by Marshall Hall on integral completions of matrices satisfying orthogonality conditions are extended as far as possible, with special attention given to the Hadamard case. A result on restricting the denominators of rational completions to a power of 2 is also given.     

Incidence matrices and inequalities for combinatorial designs

In this paper we use incidence matrices of block designs and row–column designs to obtain combinatorial inequalities. We introduce the concept of nearly orthogonal Latin squares by modifying the usual definition of orthogonal Latin squares. This concept opens up interesting combinatorial problems and is expected to be useful in planning experiments by statisticians. © 2002 John Wiley & Sons, Inc. J Combin Designs 10: 17–26, 2002     

A combinatorial problem involving graphs and matrices

In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x₁,x₂,…,x_k?n^k there exist k distinct integers j₁,j₂,…,j_k so that x_i= a_ii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k).