Real matrices whose columns have equal modulus coordinates

We study \(m \times n\) matrices whose columns are of the form \[\{(a_{1j},\ldots, a_{nj}): \quad a_{1j} = \lambda_j,\ a_{ij} = \pm\lambda_j\ , \ \lambda_j >0 ,\ j=1,2,\ldots,n\}.\] We explicitly construct for all \(a = (a_1,\ldots, a_{\frac{m(m- 1)}{2}}) \in \mathbb{R}^{\frac{m(m-1)}{2}}\) a mat...

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Bibliographic Details
Published in:arXiv.org 2023-03
Main Authors: Botelho-Andrade, Sara, Casazza, Peter G, Desai, Cheng, Tran, Tin, Tremain, Janet
Format: Article
Language:English
Online Access:Get full text
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Summary:We study \(m \times n\) matrices whose columns are of the form \[\{(a_{1j},\ldots, a_{nj}): \quad a_{1j} = \lambda_j,\ a_{ij} = \pm\lambda_j\ , \ \lambda_j >0 ,\ j=1,2,\ldots,n\}.\] We explicitly construct for all \(a = (a_1,\ldots, a_{\frac{m(m- 1)}{2}}) \in \mathbb{R}^{\frac{m(m-1)}{2}}\) a matrix of the above form whose rows have pairwise dot product equal to \(a\). Using Hardamard matrices constructed by Sylvester we classify all matrices of the above form whose rows have pairwise dot product equal to \(a\). We also use our results to reformulate the Hadamard conjecture.
ISSN:2331-8422