TITLE: Recovering a set of vectors from their Gram matrix
QUESTION [6 upvotes]: 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

REPLY [4 votes]: Let $A = USV^*$ be the SVD solution for $A$, a symmetric matrix, which the Gram is. Then $X = U\sqrt{S}$ is an isometry for $X$, the original vectors. The rank of $S$ provides additional information about the input vectors $X$. To obtain the original vectors requires knowing 3 points of the original set so that the isometry can be rotated and/or reflected to match the original set.