Reversing the Gram Matrix Let $A$ be a $M\times N$  real matrix, then $B=A^TA$ is the gramian of $A$. Suppose $B$ is given, is $A$ unique? Can I say something on it depending on $M$ and $N$. 

$A$ can take the form of $U\lambda^{\frac{1}{2}}$ using the eigen decomposition of $B$. Now obviously you can multiply $U$ and $\lambda$ with two scalars such that the solution remains the same. The gram matrix preserves the norms and the inter point angles, because of the inner products. Now you can have infinite angle and distance preserving solutions of points, upon translation and rotation. So-now, you see that the solutions are not unique!
