Structural properties and recognition of restricted and strongly unimodular matrices

A (0, ±1) matrix A is restricted unimodular if every matrix obtained from A by setting to zero any subset of its entries is totally unimodular. Restricted unimodular matrices are also known as matrices without odd cycles. They have been studied by Commoner and recently Yannakakis has given a polynomial algorithm to recognize when a matrix belongs to this class. A matrix A is strongly unimodular if any matrix obtained from A by setting at most one of its entries to zero is totally unimodular. Crama et al. have shown that (0,1) matrix A is strongly unimodular if and only if any basis of (A, 1) is triangular, whereI is an identity matrix of suitable dimensions. In this paper we give a very simple algorithm to test whether a matrix is restricted unimodular and we show that all strongly unimodular matrices can be obtained by composing restricted unimodular matrices with a simple operation. Partially supported by a New York University Research Challenge Fund Grant.     

<Emphasis Type="Italic">k</Emphasis>-Sum decomposition of strongly unimodular matrices

Networks are frequently studied algebraically through matrices. In this work, we show that networks may be studied in a more abstract level using results from the theory of matroids by establishing connections to networks by decomposition results of matroids. First, we present the implications of the decomposition of regular matroids to networks and related classes of matrices, and secondly we show that strongly unimodular matrices are closed under k-sums for \(k=1,2\) implying a decomposition into highly connected network-representing blocks, which are also shown to have a special structure.     

Sums and k-sums in abelian groups of order k

Let G be an abelian group of order k. How is the problem of minimizing the number of sums from a sequence of given length in G related to the problem of minimizing the number of k-sums? In this paper we show that the minimum number of k-sums for a sequence a₁,…,ar that does not have 0 as a k-sum is attained at the sequence b₁,…,br−k+1,0,…,0, where b₁,…,br−k+1 is chosen to minimise the number of sums without 0 being a sum. Equivalently, to minimise the number of k-sums one should repeat some value k−1 times. This proves a conjecture of Bollobás and Leader, and extends results of Gao and of Bollobás and Leader.     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

Complement total unimodularity

A {0, 1} matrix U is defined to be complement totally unimodular (c.t.u.) if U as well as all matrices derived from U by certain complement operations are totally unimodular. Two decompositions of minimal violation matrices of total unimodularity are utilized to characterize the letter matrices by c.t.u. matrices. In addition properties of c.t.u. matrices are presented.     

Perfect zero–one matrices

A zero–one matrix is called perfect if the polytope of the associated set packing problem has integral vertices only. By this definition, all totally unimodular zero–one matrices are perfect. In this paper we give a characterization of perfect zero–one matrices in terms offorbidden submatrices. Perfect zero–one matrices are closely related to perfect graphs and constitute a generalization of balanced matrices as introduced by C. Berge. Furthermore, the results obtained here bear on an unsolved problem in graph theory, the strong perfect graph conjecture, also due to C. Berge.     

A relationship between stochastic matrices and eigenvalues of products

Let {B₁,…,B_n} be a set of n rank one n×n row stochastic matrices. The next two statements are equivalent: (1) If A is an n×n nonnegative matrix, then 1 is an eigenvalue ofB_kA for each k=1,…,n if and only if A is row stochastic. (2) The n×n row stochastic matrix S whose kth row is a row of B_k for k=1,…,n is nonsingular. For any set {B₁, B₂,…, B_p} of fewer than n row stochastic matrices of order n×n and of any rank, there exists a nonnegative n×n matrix A which is not row stochastic such that 1 is eigenvalue of every B_kA, k=1,…,p.     

A good submatrix is hard to find

Given a matrix, it is NP-hard to find a ‘large’ column, row, or arbitraty submatrix that satisfies property π, where π is nontrivial, holds for permutation matrices, and is hereditary on submatrices. Such properties include totally unimodular, transformable to a network matrix, permutable to consecutive ones, and many others. Similar results hold for properties such as positive definite, of bandwidth w, and symmetric.     

Convergence of Solutions to Equations Arising in Neural Networks

We study polynomial ordinary differential systems whereQ0 is an n×n matrix and M(t) is an n×k matrix. It is proven that, as t grows to infinity, the solution M(t) tends to a limit BU, where U is a k×k orthogonal matrix and B is an n×k matrix whose columns are k pairwise orthogonal, normalized eigenvectors of Q. Moreover, for almost every M ₀, these eigenvectors correspond to the k maximal eigenvalues of Q; for an arbitrary Q with independent columns, we provide a procedure of computing B by employing elementary matrix operations on M ₀. This result is significant for the study of certain neural network systems, and in this context it shows that M() provides a principal component analyzer.     

Real rank versus nonnegative rank

We consider the set of m×n nonnegative real matrices and define the nonnegative rank of a matrix A to be the minimum k such that A=BC where B is m×k and C is k×n. Given that the real rank of A is j for some j, we give bounds on the nonnegative rank of A and A².     

The rate of convergence for backwards products of a convergent sequence of finite Markov matrices

Recent papers have shown that Π^∞_{k = 1}P(k) = lim_m→∞ (P(m) ? P(1)) exists whenever the sequence of stochastic matrices {P(k)}^∞_{k = 1} exhibits convergence to an aperiodic matrix P with a single subchain (closed, irreducible set of states). We show how the limit matrix depends upon P(1).In addition, we prove that lim_m→∞lim_n→∞ (P(n + m) ? P(m + 1)) exists and equals the invariant probability matrix associated with P. The convergence rate is determined by the rate of convergence of {P(k)}^∞_{k = 1} towards P.Examples are given which show how these results break down in case the limiting matrix P has multiple subchains, with {P(k)}^∞_{k = 1} approaching the latter at a less than geometric rate.     

A sparse counterpart of Reichel and Gragg’s package QRUP

We describe how to maintain the triangular factor of a sparse QR factorization when columns are added and deleted and Q cannot be stored for sparsity reasons. The updating procedures could be thought of as a sparse counterpart of Reichel and Gragg’s package QRUP. They allow us to solve a sequence of sparse linear least squares subproblems in which each matrix B_k is an independent subset of the columns of a fixed matrix A, and B_k+1 is obtained by adding or deleting one column. Like Coleman and Hulbert [T. Coleman, L. Hulbert, A direct active set algorithm for large sparse quadratic programs with simple bounds, Math. Program. 45 (1989) 373-406], we adapt the sparse direct methodology of Björck and Oreborn of the late 1980s, but without forming A^TA, which may be only positive semidefinite. Our Matlab 5 implementation works with a suitable row and column numbering within a static triangular sparsity pattern that is computed in advance by symbolic factorization of A^TA and preserved with placeholders.     

On single and double Soules matrices

Until now the concept of a Soules basis matrix of sign patternN consisted of an orthogonal matrix R∈R^n,n, generated in a certain way from a positive n-vector, which has the property that for any diagonal matrix Λ = diag(λ₁, … , λ_n), with λ₁ ? ? ? λ_n ? 0, the symmetric matrix A = RΛR^T has nonnegative entries only. In the present paper we introduce the notion of a pair of double Soules basis matrices of sign patternN which is a pair of matrices (P, Q), each in R^n,n, which are not necessarily orthogonal and which are generated in a certain way from two positive vectors, but such that PQ^T = I and such that for any of the aforementioned diagonal matrices Λ, the matrix A = PΛQ^T (also) has nonnegative entries only. We investigate the interesting properties which such matrices A have.As a preamble to the above investigation we show that the iterates, , generated in the course of the QR-algorithm when it is applied to A = RΛR^T, where R is a Soules basis matrix of sign pattern N, are again symmetric matrices generated by the Soules basis matrices R_k of sign pattern N which are themselves modified as the algorithm progresses.Our work here extends earlier works by Soules and Elsner et al.     

On the asymptotic existence of Hadamard matrices

It is conjectured that Hadamard matrices exist for all orders 4t (t>0). However, despite a sustained effort over more than five decades, the strongest overall existence results are asymptotic results of the form: for all odd natural numbers k, there is a Hadamard matrix of order k²[a+blog₂k], where a and b are fixed non-negative constants. To prove the Hadamard Conjecture, it is sufficient to show that we may take a=2 and b=0. Since Seberry's ground-breaking result, which showed that we may take a=0 and b=2, there have been several improvements where b has been by stages reduced to 3/8. In this paper, we show that for all ?>0, the set of odd numbers k for which there is a Hadamard matrix of order k²2+[?log₂k] has positive density in the set of natural numbers. The proof adapts a number-theoretic argument of Erdos and Odlyzko to show that there are enough Paley Hadamard matrices to give the result.     

Relative perturbation bounds for the unitary polar factor

LetB be anm×n (m≥n) complex (or real) matrix. It is known that there is a uniquepolar decomposition B=QH, whereQ*Q=I, then×n identity matrix, andH is positive definite, providedB has full column rank. Existing perturbation bounds suggest that in the worst case, for complex matrices the change inQ be proportional to the reciprocal ofB's least singular value, or the reciprocal of the sum ofB's least and second least singular values if matrices are real. However, there are situations where this unitary polar factor is much more accurately determined by the data than the existing perturbation bounds would indicate. In this paper the following question is addressed: how much mayQ change ifB is perturbed to $\tilde B = D_1^* BD_2 $ , whereD ₁ andD ₂ are nonsingular and close to the identity matrices of suitable dimensions? It is shown that for a such kind of perturbation, the change inQ is bounded only by the distances fromD ₁ andD ₂ to identity matrices and thus is independent ofB's singular values. Such perturbation is restrictive, but not unrealistic. We show how a frequently used scaling technique yields such a perturbation and thus scaling may result in better-conditioned polar decompositions.     

Broadcasting from multiple originators

We begin an investigation of broadcasting from multiple originators, a variant of broadcasting in which any k vertices may be the originators of a message in a network of n vertices. The requirement is that the message be distributed to all n vertices in minimum time. A minimumk-originator broadcast graph is a graph on n vertices with the fewest edges such that any subset of k vertices can broadcast in minimum time. B_k(n) is the number of edges in such a graph. In this paper, we present asymptotic upper and lower bounds on B_k(n). We also present an exact result for the case when . We also give an upper bound on the number of edges in a relaxed version of this problem in which one additional time unit is allowed for the broadcast.     

Hessenberg matrix for sums of Hermitian positive definite matrices and weighted shifts

In this work, we introduce an algebraic operation between bounded Hessenberg matrices and we analyze some of its properties. We call this operation m-sum and we obtain an expression for it that involves the Cholesky factorization of the corresponding Hermitian positive definite matrices associated with the Hessenberg components.This work extends a method to obtain the Hessenberg matrix of the sum of measures from the Hessenberg matrices of the individual measures, introduced recently by the authors for subnormal matrices, to matrices which are not necessarily subnormal.Moreover, we give some examples and we obtain the explicit formula for the m-sum of a weighted shift. In particular, we construct an interesting example: a subnormal Hessenberg matrix obtained as the m-sum of two not subnormal Hessenberg matrices.     

Connected graphs with maximal Q-index: The one-dominating-vertex case

By the signless Laplacian of a (simple) graph G we mean the matrix Q(G)=D(G)+A(G), where A(G),D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. For every pair of positive integers n,k, it is proved that if 3?k?n-3, then H_n,k, the graph obtained from the star K_1,n-1 by joining a vertex of degree 1 to k+1 other vertices of degree 1, is the unique connected graph that maximizes the largest signless Laplacian eigenvalue over all connected graphs with n vertices and n+k edges.     

Sufficient enlargements of minimal volume for finite-dimensional normed linear spaces

Let BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite-dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection such that P(BY)⊂A. The main results of the paper: (1) Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. (2) If a finite-dimensional normed linear space has a minimal-volume sufficient enlargement which is not a parallelepiped, then it contains a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon.