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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 $N$.

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You need more conditions to make it unique, even with scalars, eg, $(1)(1) = (-1)(-1) = 1$. –  copper.hat Apr 12 '13 at 17:09

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$A$ 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 $\Bbb{R}^M$).

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$A$ 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!

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