Positive Eigenvalues of Hadamard Square Roots on Finite Abelian Groups

Abstract

For a finite abelian group $G$, let $μ_2(G)$ be the minimum number of positive eigenvalues of an entrywise positive real symmetric group-circulant matrix whose entrywise square is positive definite. We prove that $μ_2(G)\le5$ precisely when $|G|\le15$, and that $μ_2(G)=6$ precisely when $|G|=16$ or $G$ is cyclic of order $18$, $20$ or $22$. For the noncyclic groups of order at most fifteen, the minimum is three for $C_2^2$ and five otherwise. If the positive Fourier support of such a matrix is $\{0,\pm r_1,\ldots,\pm r_d\}$, where $d\ge1$ and all $2d+1$ displayed elements are distinct, then there are integers $α_1,\ldots,α_d$, not all zero, such that $\sum_iα_i r_i=0$ and $\sum_i|α_i|\le\max\{5,d+2\}$. The bound five is sharp for $d=2$ and $d=3$.