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.

Keywords:

• Power-product matrix
• nonsingularity
• sparsity
• determinant

AMS:

• 15B36
• 15A09
• 15A15
• 05A19
• 11Cxx

Acknowledgments

The authors would like to express their sincere thanks to the referee for his/her careful reading of the manuscript and helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Let $x\in {\mathbb{R}}^n$. Then the zero norm $\Vert x{\Vert }_0$ is the number of nonzero elements in x.

Additional information

Funding

The first author is supported by the National Natural Science Foundation of China [grant number 11601327].