Nonnegative Integer Matrices Research Articles

The spectra of nonnegative integer matrices via formal power series

We characterize the possible nonzero spectra of primitive integer matrices (the integer case of Boyle and Handelman’s Spectral Conjecture). Characterizations of nonzero spectra of nonnegative matrices over Z {\mathbb Z} and Q {\mathbb Q} follow from this result. For the proof of the main theorem we use polynomial matrices to reduce the problem of realizing a candidate spectrum ( λ 1 , λ 2 , … , λ d ) (\lambda _1,\lambda _2,\ldots ,\lambda _d) to factoring the polynomial ∏ i = 1 d ( 1 − λ i t ) \prod _{i=1}^d (1-\lambda _it) as a product ( 1 − r ( t ) ) ∏ i = 1 n ( 1 − q i ( t ) ) (1-r(t))\prod _{i=1}^n (1-q_i(t)) where the q i q_i ’s are polynomials in t Z + [ t ] t{\mathbb Z}_+[t] satisfying some technical conditions and r r is a formal power series in t Z + [ [ t ] ] t{\mathbb Z}_+[[t]] . To obtain the factorization, we present a hierarchy of estimates on coefficients of power series of the form ∏ i = 1 d ( 1 − λ i t ) / ∏ i = 1 n ( 1 − q i ( t ) ) \prod _{i=1}^d (1-\lambda _it)/\prod _{i=1}^n (1-q_i(t)) to ensure nonpositivity in nonzero degree terms.

Another Form of Matrix Nim

A new form of 2-dimensional nim is investigated. Positions are rectangular matrices of non-negative integers. Moves consist of chosing a positive integer and a row or column and subtracting the integer from every element of the chosen row or column. Last to move wins. The $2\times1$ case is just Wythoff's Game. The outcomes of all $2\times2$ positions are found in both the impartial and partizan cases. Some hope is given of being able to solve sums of $2\times2$ games in the partizan case.

On the Volume of the Polytope of Doubly Stochastic Matrices

We study the calculation of the volume of the polytope Bn of n × n doubly stochastic matrices (real nonnegative matrices with row and column sums equal to one). We describe two methods. The first involves a decompos ition of the polytope into simplices. The second involves the enumeration of “magic squares”, that is, n × n nonnegative integer matrices whose rows and columns all sum to the same integer. We have used the first method to confirm the previously known values through n = 7. This method can also be used to compute the volumes of faces of Bn For example, we have observed that the volume of a particular face of Bn appears to be a product of Catalan numbers. We have used the second method to find the volume for n = 8, which we believe was not previously known.

Structural Identification in Large Scale Systems: Generic McMillan Degree Computation

The McMillan degree of a transfer fimction model is one of the most important structural characteristics and plays an important role in deciding about the quality of a system model, as far as aollowing solvability of certain control ptoblems. In this paper the problem of identifying the generic McMillan degree of a rational matrix is considered The transfer function matrices of interest are those referred to as Structured Transfer Function (STF) matrices and have certain elements fixed to zero, some elements being constant and other elements expressing some identified dominant dynamics of the system. For the family of STF matrices the problem of determining the generic McMillan degree is considered using genericity arguments and an optimisation procedure based on path properties of nonnegative integer matrices. A novel approach is introduced using the notion of irreducibility of nonnegative matrices which leads to an efficient new algorithm for computation of the generic value of the McMillan degree. Links to standard problems of graph theory are made. The problem examined here belongs to the general area of Structural Identification where the evaluation of structural characteristics of STF models is under investigation with robust computational methods. Such problems are of specific interest to large scale systems.

Invariants of twist-wise flow equivalence

Flow equivalence of irreducible nontrivial square nonnegative integer matrices is completely determined by two computable invariants, the Parry-Sullivan number and the Bowen-Franks group. Twist-wise flow equivalence is a natural generalization that takes account of twisting in the local stable manifold of the orbits of a flow. Two new invariants in this category are established.

Linear operators that preserve maximal column ranks of nonnegative integer matrices

The maximal column rank of an m m by n n matrix over a semiring is the maximal number of the columns of A A which are linearly independent. We characterize the linear operators which preserve the maximal column ranks of nonnegative integer matrices.

The bijection between plane partitions and nonnegative integer matrices

The bijection between plane partitions and nonnegative integer matrices

The bijection between plane partitions and nonnegative integer matrices

The one-to-one correspondence between the set of plane partitions withr rows andm columns and the set of matrices of nonnegative integers with the same numbers of rows and columns has been constructed.

On classes of normalized matrices

The integer m × n matrices A = ( a ij ), B = ( b ij ) are said to be equivalent if a ij = u i + b ij + v j for all i = 1,…, m, j = 1,…, n, for some u 1,…, u m , v 1,…, v n ∈ Z. For an integer matrix X the symbol S( X) denotes the set of all nonnegative integer matrices equivalent to X having a zero element in each row and each column. We develop algorithms to solve the problems of the following type for a given class S( X) : decide whether A ∈ S( X); find the largest possible value for each position among all matrices in S( X); find a matrix in S( X) with prescribed values of a specified entry or row (column); find a matrix in S( X) with term rank 2. A complete description of those S( X) containing only zero-one matrices is presented.

SU( N) lattice integrable models associated with graphs

We explore the construction of RSOS critical integrable models attached to a graph, trying to extend Pasquier's construction from SU(2) to SU( N), with main emphasis on the case of SU(3): the heights are the nodes of a graph, which encodes the allowed configurations. A class of graphs that are natural candidates for this construction is defined. In the case N = 3, they all seem to be related to finite subgroups of SU(3). For any N, they are associated with arbitrary representations of the SU(N) fusion algebra over matrices of non-negative integers. It is argued that these graphs should support a representation of the Hecke algebra.

Semiring fank versus column rank

This paper concerns two notions of rank of matrices over semirings: semiring rank and column rank. These two rank functions are the same over fields and Euclidean rings, but differ for matrices over many combinatorially interesting semirings including the nonnegative integer matrices, the fuzzy matrices, and the Binary Boolean matrices. We investigate the largest value of r for which the column rank and semiring rank of all m× n matrices over a given semiring are both r. This value is determined for the semirings mentioned above as well as many others.

Operators that preserve semiring matrix functions

Characterizations are obtained of those linear operators over certain semirings that preserve (1) the rth coefficient of the rook polynomial, those that preserve (2) the term rank of matrices with term rank r, and those that preserve (3) the rth elementary symmetric permanental function. Our results apply to many algebraic systems of combinatorial interest, including the nonnegative integer matrices, Boolean matrices, and fuzzy matrices.

Strong shift equivalence and shear adjacency of nonnegative square integer matrices

For two square matrices A, B of possibly different sizes with nonnegative integer entries, write A ≈ 1 B if A = RS and B = SR for some two nonnegative integer matrices R, S. The transitive closure of this relation is called strong shift equivalence and is important in symbolic dynamics, where it has been shown by R.F. Williams to characterize the isomorphism of two topological Markov chains with transition matrices A and B. One invariant is the characteristic polynomial up to factors of λ. However, no procedure for deciding strong shift equivalence is known, even for 2×2 matrices A, B. In fact, for n × n matrices with n > 2, no nontrivial sufficient condition is known. This paper presents such a sufficient condition: that A and B are in the same component of a directed graph whose vertices are all n × n nonnegative integer matrices sharing a fixed characteristic polynomial and whose edges correspond to certain elementary similarities. For n > 2 this result gives confirmation of strong shift equivalences that previously could not be verified; for n = 2, previous results are strengthened and the structure of the directed graph is determined.

Term-rank, permanent, and rook-polynomial preservers

Characterizations are obtained of those linear operators on the m × n matrices over an arbitrary semiring that preserve term rank. We also present characterizations of permanent and rook-polynomial preserving operators on matrices over certain types of semirings. Our results apply to many combinatorially interesting algebraic systems, including nonnegative integer matrices, matrices over Boolean algebras, and fuzzy matrices.

Handle bases and bounds on the number of subgraphs

We define a handle basis for strongly connected digraphs. Such bases are similar to cycle bases, except that all but the first element is a simple path. We show that a subgraph of a diagraph is uniquely characterized by an integer valued function on the vertices (i.e., its boundary) and by the elements of a handle basis which it intersects. We then use this support-boundary uniqueness theorem to conclude that the number of subgraphs with a given boundary is at most 2 d − 1, where d is the cyclomatic number of the digraph. If we require the subgraphs H to be cyclically simple, i.e., roughly, that no cycle has access to any other, then the number of such H with given boundary is at most 2 d − 1 . This generalizes the bound due to Foregger on the permanent of fully indecomposable nonnegative integer matrices. Studying a natural partial order on the handles yields a better bound on the permanent in a special case: when the handles are totally ordered, the permanent is bounded by a Fibonacci number.

Shorter Notes: Character Values of Finite Groups as Eigenvalues of Nonnegative Integer Matrices

Shorter Notes: Character Values of Finite Groups as Eigenvalues of Nonnegative Integer Matrices

A generalization of a theorem of König

We generalize a theorem of König concerning nonnegative integer matrices with constant line sums.

Character values of finite groups as eigenvalues of nonnegative integer matrices

Let C 1 {C_1} , C 2 , … {C_2}, \ldots , C k {C_k} be the conjugacy classes of the finite group G G and choose x i ∈ C i {x_i} \in {C_i} , for i = 1 i = 1 , 2 , … 2, \ldots , k k . For every complex character θ \theta of G G there is a k × k k \times k matrix M ( θ ) M(\theta ) whose entries are nonnegative integers such that X − 1 M ( θ ) X = diag ( θ ( x 1 ) , θ ( x 2 ) , … , θ ( x k ) ) {X^{ - 1}}M(\theta )X = {\text {diag}}(\theta ({x_1}),\theta ({x_2}), \ldots ,\theta ({x_k})) where X X is the character table matrix of G G . Some consequences are shown.

A proof of the isomorphism of wxyz-transformals and 2 x 2 integer matrices

This paper proves the conjecture that the family of transformals in Gosper's calculus of series rearrangements is isomorphic to 2 x 2 non-negative integer matrices under multiplication. A natural mapping is exhibited, and shown to be an isomorphism. Splitting pairs are defined as a weakening of the notion of commutativity. The proof does not rely on series rearrangement, and can be applied to splitting pairs over any monoid.

The asymptotic number of labeled graphs with given degree sequences

Asymptotics are obtained for the number of n × n symmetric non-negative integer matrices subject to the following constraints: (i) each row sum is specified and bounded, (ii) the entries are bounded, and (iii) a specified “sparse” set of entries must be zero. The result can be interpreted in terms of incidence matrices for labeled graphs.