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Suppose $\{ v_1, \ldots, v_k \}$ is a set of vectors in $\mathbb{R}^n$. The associated $k\times k$ Gram matrix is given by $$ G = [v_i \cdot v_j]_{i,j} $$ It's (apparently) well known that the Gram matrix of a set of vectors determines the vectors up to an isometry of $\mathbb{R}^n$ (e.g. [1])

My question is: does anyone know of a reference for an algorithm that performs the reconstruction? More precisely, I'm looking for an algorithm that takes $G$ as input and outputs $\{ Av_1, \ldots, Av_k \}$ for some $A \in SO(n)$.

[1] http://mathworld.wolfram.com/GramMatrix.html

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    It is the Cholesky factorization of $G$ that you are interested in.2012-11-28
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    Where is the $A \in SO(n)$ with Cholesky?2012-11-28
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    @adamW Perfect - thanks! copper.hat - the $A$ is implicit in the factorization that you find (e.g. the possible factorizations correspond to all the possible A's)2012-11-28
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    @David To finish this question I suggest you write up your own answer (expound on $A$ implicit in ...). I could answer, but as I am not familiar with the notation of $A \in SO(n)$ it would just be a restatement of my previous comment.2012-11-28

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