Power-product matrix: nonsingularity, sparsity and determinant

TL;DR

This paper proves the nonsingularity of power-product matrices V(n,d) for all positive integers n and d, using combinatorial and linear algebra techniques, and explores their determinant and sparsity properties.

Contribution

It establishes the universal nonsingularity of power-product matrices V(n,d) and analyzes their determinants, especially for the case V(2,d), advancing understanding of their algebraic structure.

Findings

V(n,d) matrices are nonsingular for all positive n and d

The matrices often exhibit sparse structure

Explicit computation of det V(2,d) provided

Abstract

We prove the nonsingularity of a class of integer matrices V(n,d), namely power-product matrix, for positive integers n and d. Some technical proofs are mainly based on linear algebra and enumerative combinatorics, particularly the generating function method and involution principle. We will show that the matrix V(n,d) is nonsingular for all positive integers n and d, and often with sparse structure. Special attention is given to the computation of the determinant V(2,d) with positive integer d.