On nonnegative matrices with given row and column sums

On nonnegative matrices with given row and column sums

Integral Matrices with Given Row and Column Sums

Let P = (p,,) and Q = (qij) be m x n integral matrices, R and S be integral vectors. Let Nf(R, S) denote the class of all m x n integral matrices A with row sum vector R and column sum vector S satisfying P < A < Q. For a wide variety of classes ‘%$I( R, S) satisfying our main condition, we obtain two necessary and sufficient conditions for the existence of a matrix in @(R, 5). The first charac...

Matrices with Prescribed Row and Column Sums

This is a survey of the recent progress and open questions on the structure of the sets of 0-1 and non-negative integer matrices with prescribed row and column sums. We discuss cardinality estimates, the structure of a random matrix from the set, discrete versions of the Brunn-Minkowski inequality and the statistical dependence between row and column sums.

Asymptotic enumeration of sparse nonnegative integer matrices with specified row and column sums

Let s = (s1, . . . , sm) and t = (t1, . . . , tn) be vectors of nonnegative integer-valued functions of m,n with equal sum S = ∑m i=1 si = ∑n j=1 tj. Let M(s, t) be the number of m × n matrices with nonnegative integer entries such that the ith row has row sum si and the jth column has column sum tj for all i, j. Such matrices occur in many different settings, an important example being the con...

Subdominant eigenvalues for stochastic matrices with given column sums

For any stochastic matrix A of order n, denote its eigenvalues as λ1(A), . . . , λn(A), ordered so that 1 = |λ1(A)| ≥ |λ2(A)| ≥ . . . ≥ |λn(A)|. Let cT be a row vector of order n whose entries are nonnegative numbers that sum to n. Define S(c), to be the set of n × n row-stochastic matrices with column sum vector cT . In this paper the quantity λ(c) = max{|λ2(A)||A ∈ S(c)} is considered. The ve...