  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.
