Nonnegative Integral Matrices Research Articles

Non-negative integral matrices with given spectral radius and controlled dimension

Abstract A celebrated theorem of Douglas Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number p, we prove that there is an integral irreducible matrix with spectral radius p, and with dimension bounded above in terms of the algebraic degree, the ratio of the first two largest Galois conjugates, and arithmetic information about the ring of integers of its number field. This arithmetic information can be taken to be either the discriminant or the minimal Hermite-like thickness. Equivalently, given a Perron number p, there is an irreducible shift of finite type with entropy $\log (p)$ defined as an edge shift on a graph whose number of vertices is bounded above in terms of the aforementioned data.

A canonical construction for nonnegative integral matrices with given line sums

Let p be a positive integer and let A(p)(R,S) be the class of nonnegative integral matrices with entries less than or equal to p, with row–sum partition R, and column–sum partition S.In this paper we state a new necessary and sufficient condition for A(p)(R,S)≠∅. This condition generalizes the well known Gale–Ryser theorem. We also present a canonical construction for matrices in A(p)(R,S).

NONNEGATIVE INTEGRAL MATRICES HAVING GENERALIZED INVERSES

For an <TEX>$m{\times}n$</TEX> nonnegative integral matrix A, a generalized inverse of A is an <TEX>$n{\times}m$</TEX> nonnegative integral matrix G satisfying AGA = A. In this paper, we characterize nonnegative integral matrices having generalized inverses using the structure of nonnegative integral idempotent matrices. We also define a space decomposition of a nonnegative integral matrix, and prove that a nonnegative integral matrix has a generalized inverse if and only if it has a space decomposition. Using this decomposition, we characterize nonnegative integral matrices having reflexive generalized inverses. And we obtain conditions to have various types of generalized inverses.

RSK correspondence and classically irreducible Kirillov–Reshetikhin crystals

We give a new combinatorial model of the Kirillov–Reshetikhin crystals of type An(1) in terms of non-negative integral matrices based on the classical RSK algorithm, which has a simple description of the affine crystal structure without using the promotion operator. We have a similar description of the Kirillov–Reshetikhin crystals associated to exceptional nodes in the Dynkin diagrams of classical affine or non-exceptional affine type, which are called classically irreducible together with those of type An(1).

Constructing integral matrices with given line sums

D. Gale, in 1957 and H.J. Ryser, in 1963, independently proved the famous Gale–Ryser theorem on the existence of (0, 1)–matrices with prescribed row and column sums. Around the same time, in 1968, Mirsky solved the more general problem of finding conditions for the existence of a nonnegative integral matrix with entries less than or equal to p and prescribed row and column sums. Using the results of Mirsky, Brualdi shows that a modified version of the domination condition of Gale–Ryser is still necessary and sufficient for the existence of a matrix under the same constraints. In this article we prove another extension of Gale–Ryser’s domination condition. Furthermore we present a method to build nonnegative integral matrices with entries less than or equal to p and prescribed row and column sums.

Generalized Bowen–Franks groups of integral matrices with the same zeta function

In this paper the relation between the zeta function of an integral matrix and its generalized Bowen–Franks groups is studied. Suppose that A and B are nonnegative integral matrices whose invertible part is diagonalizable over the field of complex numbers and A and B have the same zeta function. Then there is an integer m , which depends only on the zeta function, such that, for any prime q such that gcd ( q , m ) = 1 , for any g ( x ) ∈ Z [ x ] with g ( 0 ) = 1 , the q -Sylow subgroup of the generalized Bowen–Franks group BF g ( x ) ( A ) and BF g ( x ) ( B ) are the same. In particular, if m = 1 , then zeta function determines generalized Bowen–Franks groups.

A generating function for all semi-magic squares and the volume of the Birkhoff polytope

We present a multivariate generating function for all n×n nonnegative integral matrices with all row and column sums equal to a positive integer t, the so called semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope B n of n×n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for the volumes of B n and any of its faces.

Approximation algorithms for minimizing segments in radiation therapy

Intensity modulated radiation therapy (IMRT) is one of the most effective modalities for modern cancer treatment. The key to successful IMRT treatment hinges on the delivery of a two-dimensional discrete radiation intensity matrix using a device called a multileaf collimator (MLC). Mathematically, the delivery of an intensity matrix using an MLC can be viewed as the problem of representing a non-negative integral matrix (i.e., the intensity matrix) by a linear combination of certain special non-negative integral matrices called segments, where each such segment corresponds to one of the allowed states of the MLC. The problem of representing the intensity matrix with the minimum number of segments is known to be NP-complete. In this paper, we present two approximation algorithms for this matrix representation problem. To the best of our knowledge, these are the first algorithms to achieve non-trivial performance guarantees for multi-row intensity matrices.

Minimal nonnegative integral matrices and uniquely determined (0,1)-matrices

In this note we give an elementary proof of a theorem that characterizes those three-dimensional (0,1)-matrices that are determined by their plane sum vectors.

Strong shift equivalence of 2 by 2 non‐negative integral matrices

It is known that if A and B are nontriangular 2 × 2 non-negative integral matrices similar over the integers and –tr A ≤det A, then A and B are strongly shift equivalent. Suppose that A and B are 2 × 2 non-negative integral matrices similar over the integers. In this article it is shown that if −2 tr A≤det A <– tr A and if | det A | is not a prime, then A and B are strongly shift equivalent.

On the Classification of Reflexive Polyhedra

Reflexive polyhedra encode the combinatorial data for mirror pairs of Calabi-Yau hypersurfaces in toric varieties. We investigate the geometrical structures of circumscribed polytopes with a minimal number of facets and of inscribed polytopes with a minimal number of vertices. These objects, which constrain reflexive pairs of polyhedra from the interior and the exterior, can be described in terms of certain non-negative integral matrices. A major tool in the classification of these matrices is the existence of a pair of weight systems, indicating a relation to weighted projective spaces. This is the corner stone for an algorithm for the construction of all dual pairs of reflexive polyhedra that we expect to be efficient enough for an enumerative classification in up to 4 dimensions, which is the relevant case for Calabi-Yau compactifications in string theory.

On a Special Class of Generalized Doubly Stochastic Matrices and Its Relation to Bézier Polygons

We introduce and study a new set $\mathcal{B}$ of $m \times n$ nonnegative integral matrices $B_{mn} $, $m \geq n \geq 1$. They have column-rhomboidal form and are closely related to the confluent Vandermonde matrices $V_{mn} $ with n-tuple node 1 and to Bézier polygons.

On commuting squares and subfactors

After showing (see Theorem 2.8) how commuting squares lead, in the presence of a certain additional “rotational symmetry” condition, to irreducible subfactors, with pleasant additional features, of the hyperfinite II 1 factor R, it is shown (see Theorem 3.1) that such rotationally symmetric commuting squares can be constructed, starting from each member of a certain class of symmetric non-negative integral matrices. Specialisations of the matrix show (see Sect. 4), for instance, that for each positive integer N, (N + √N 2 + 4) 2 and (N + √N 2 + 8) 2 belong to the set I R 0 of index-values of irreducible subfactors of R, that (N + 1 N ) 2 is an accumulation point of I R 0, and that 9 is an accumulation point of accumulation points of I R 0, 16 is an accumulation point of accumulation points of accumulation points of I R 0, and so on.

On the Perron-Frobenius eigenvector for nonnegative integral matrices whose largest eigenvalue is integral

We first derive the bound |det(λI − A)|⩽λk − λk0 (λ0⩽λ), where A is a k × k nonnegative real matrix and λ0 is the spectral radius of A. If A is irreducible and integral, and its largest nonnegative eigenvalue is an integer n, then we use this inequality to derive the upper bound nk−1 on the components of the smallest integer eigenvector corresponding to n. Finer information on the components is also derived.

The entropies of topological Markov shifts and a related class of algebraic integers

AbstractWe give an algebraic characterization of the class of spectral radii of aperiodic non-negative integral matrices, and describe a method of constructing all such matrices with given spectral radius. The logarithms of the numbers in are the entropies of mixing topological Markov shifts. There is an arithmetic structure to , including factorization into irreducibles in only finitely many ways. This arithmetic structure has dynamical consequences, such as the impossibility of factoring the p-shift into a direct product of nontrivial homeomorphisms for prime p.

Strong shift equivalence of 2 × 2 matrices of non-negative integers

AbstractThe concept of strong shift equivalence of square non-negative integral matrices has been used by R. F. Williams to characterize topological isomorphism of the associated topological Markov chains. However, not much has been known about sufficient conditions for strong shift equivalence even for 2×2 matrices (other than those of unit determinant). The main theorem of this paper is: If A and B are positive 2×2 integral matrices of non-negative determinant and are similar over the integers, then A and B are strongly shift equivalent.

The number of symmetric nonnegative integral matrices with prescribed row sums

The number of symmetric nonnegative integral matrices with prescribed row sums

Semigroups of non-negative integral matrices

In [1], Pall proved an interesting result on a certain class of 2 × 2 integral matrices. He showed that the semigroup of 2 × 2 matrices of determinant 1 and non-negative entries contains exactly 2 primes , and every other non-unit is expressible uniquely as products of these primes. Before formally stating this result, we need some notation. Let Gn denote the semigroup of n × n matrices with determinant 1 and nonnegative integral entries, In the n × n identity matrix, the n × n matrix with a 1 as its (i, j) element and zeros elsewhere, and let . When the dimension is clear, we shall drop the superscripts.

Group characters and nonnegative integral matrices

Group characters and nonnegative integral matrices

Combinatorial designs and related systems

Abstract : The incidence matrix A of a (v,k,lambda)-design satisfies A(A superscript T) + (k - lambda)I + lambda J, where (A superscript T) denotes the transpose of A. The matrix I is the identity matrix and the matrix J is the matrix of 1's. This equation occurs repeatedly in one form or another throughout the literature on combinatorial designs. In the present paper we alter the left side of the equation drastically and investigate XY = (k - lambda)I + lambda J, where X and Y are nonnegative integral matrices of sizes n by m and m by n, respectively. We take n > 1 and k not equal to lambda. The new equation is still open to a purely set theoretic interpretation. (Author)