Real matrices whose columns have equal modulus coordinates

We study $m\times n$ matrices whose columns are of the form $$\{(a_{1j},\dots ,a_{nj}):a_{1j}={\lambda }_j,a_{ij}=\pm {\lambda }_j,{\lambda }_j>0,j=1,2,\dots ,n\}\mathrm{.}$$ 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.