Power-product matrix: nonsingularity, sparsity and determinant

Abstract

In this paper, we are interested in a special class of integer matrices, namely the power-product matrix, defined with two positive integers n and d. Each matrix element is computed by a power-product of two weak compositions of d into n parts. The power-product matrix has several interesting applications such as the power-sum representation of polynomials and the difference-of-convex-sums-of-squares decomposition of polynomials. We investigate some properties of this matrix including: nonsingularity, sparsity and determinant. Based on techniques in enumerative combinatorics, we prove that the power-product matrix is nonsingular and the number of nonzero entries can be computed exactly. This matrix shows sparse structure which is a good feature in numerical computation of its inverse required in some applications. Special attention is devoted to the computation of the determinant for n = 2 whose explicit formulation is obtained.