Real matrices whose columns have equal modulus coordinates

Abstract

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