Positive Eigenvalues of Hadamard Square Roots on Finite Abelian Groups

Abstract

For a finite abelian group $G$, let ${\mu }_2(G)$ 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 ${\mu }_2(G)\le 5$ precisely when $|G|\le 15$, and that ${\mu }_2(G)=6$ 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 $\{0,\pm r_1,\dots ,\pm r_d\}$, where $d\ge 1$ and all $2d+1$ displayed elements are distinct, then there are integers ${\alpha }_1,\dots ,{\alpha }_d$, not all zero, such that ${\sum }_i{\alpha }_i{r}_i=0$ and ${\sum }_i|{\alpha }_i|\le \max \{5,d+2\}$. The bound five is sharp for $d=2$ and $d=3$.

arXiv comment34 pages, 0 figures. Revised and expanded version with a new title. Extended to finite abelian groups, with a complete classification of the groups for which the minimum number of positive eigenvalues is six