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.

在本文中，我们对一类特殊的整数矩阵感兴趣，即幂积矩阵，它由两个正整数n和d定义。每个矩阵元素都由d到n的两个弱组合的幂积计算部分。幂积矩阵有几个有趣的应用，例如多项式的幂和表示和多项式的凸差平方和分解。我们研究了该矩阵的一些属性，包括：非奇异性、稀疏性和行列式。基于枚举组合学中的技术，我们证明了幂积矩阵是非奇异的，并且可以精确计算非零项的数量。该矩阵显示出稀疏结构，这在某些应用程序所需的逆矩阵的数值计算中是一个很好的特征。特别注意计算n  = 2 的行列式，获得了明确的公式。