Some Properties of Nonnegative Integral Matrices

Recently, M. Lewin proved a property of the sum of squares of row sums and column sums of an \(n \times n\) \((0, 1)\)-matrix, which has more \(1\)’s than \(0\)’s in the entries. In this article we generalize Lewin’s Theorem in several aspects. Our results are: (1)For \(m \times n\) matrices, where \(m\) and \(n\) can be different,(2) For nonnegative integral matrices as well as \((0, 1)\)-matrices,(3) For the sum of any positive powers of row sums and column sums,(4) and For any distributions of values in the matrix.In addition,we also characterize the boundary cases.