Completing the rank identity for Hadamard powers of Euclidean distance matrices

Abstract

Horvat et al. (J. Math. Chem., 2014) showed that the rank of the $n$-th Hadamard power $D^{(n)}$ of a Euclidean distance matrix satisfies $\operatorname{rank}D^{(n)} \le R_d^n$, and proved that the inequality is strict whenever an annihilating polynomial exists. The converse - that the absence of annihilating polynomials forces $\operatorname{rank}D^{(n)} = R_d^n$ - was left as an open problem. We resolve it by exhibiting a kernel factorisation $D^{(n)} = Φ_V\, M\, Φ_V^T$, where $Φ_V$ is the evaluation matrix on the polynomial space $V$ and $M$ is a universal matrix independent of the point configuration. A trinomial expansion of the kernel reveals that $M$ has a block-diagonal structure whose blocks are sums of Gram matrices with positive coefficients; this yields the non-singularity of~$M$ and completes the rank identity.