Suppose that matrix A is a symmetric positive definite matrix over the integers, i.e., $A \in Z^{n\times n}$, if B is a matrix over the real numbers, it is not difficult to find B such that $A = B \cdot B^T$. However, if B is restricted to be an integer and square matrix, in general, there is no solution. It is known that if we increase the number of columns of B (n $\times$ m matrix where m is much larger than n), it is possible to do an integer gram decomposition (for some general gram matrix A). I have the following questions:

  1. Is there any previous research paper that discusses this question (both from mathematical or computational respect ), or is it too difficult to tackle? Could anyone share some resources of the previous research?

  2. Is it possible to prove a lower bound on the number of columns $m$ s.t. B is an integer gram root? (say 2n, 3n)