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 $N$. 

$A$ will definitely not be unique without some pretty serious restrictions.  The simplest case to think about might be to consider $M\times 1$ 'matrices', i.e. column vectors.  Then, $A^TA$ is simply the norm-squared of $A$, so for instance $A^TA=1$ would hold for any vector with norm $1$ (i.e. the unit sphere in $\Bbb{R}^M$). 
