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arXiv · 2609.22744

Positive Eigenvalues of Circulant Hadamard Square Roots

Wei Xie
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Abstract

Let $H$ be an entrywise positive real symmetric circulant matrix whose entrywise square is positive definite. We prove that $H$ has at least six positive eigenvalues if its order is at least $16$, and at least seven if its order is odd and at least $17$. The first threshold is sharp. The bounds are attained in order $16$ and in orders $17,19,21$, respectively. We also determine the minimum number of positive eigenvalues in every order from $5$ through $16$.

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Wei Xie. 2026-09-19. Positive Eigenvalues of Circulant Hadamard Square Roots. https://arxiv.org/abs/2609.22744

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