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Positive Eigenvalues of Hadamard Square Roots on Finite Abelian Groups
Abstract
For a finite abelian group G, let μ2(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 μ2(G)≤5 precisely when |G|≤15, and that μ2(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 C22 and five otherwise. If the positive Fourier support of such a matrix is {0,± r1,…,± rd}, where d≥1 and all 2d+1 displayed elements are distinct, then there are integers α1,…,αd, not all zero, such that ∑iαi ri=0 and ∑i|αi|≤max{5,d+2}. The bound five is sharp for d=2 and d=3.
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BibTeX
@misc{xie2026positive,
title = {{Positive Eigenvalues of Hadamard Square Roots on Finite Abelian Groups}},
author = {Wei Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.22744},
eprint = {2609.22744},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22744},
}
RIS
TY - UNPB
TI - Positive Eigenvalues of Hadamard Square Roots on Finite Abelian Groups
AU - Wei Xie
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2609.22744
UR - https://arxiv.org/abs/2609.22744
AB - 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)≤5 precisely when |G|≤15, 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,± r_1,…,± r_d}, where d≥1 and all 2d+1 displayed elements are distinct, then there are integers α_1,…,α_d, not all zero, such that ∑_iα_i r_i=0 and ∑_i|α_i|≤max{5,d+2}. The bound five is sharp for d=2 and d=3.
ER -