Sippl M J, Scheraga H A
Proc Natl Acad Sci U S A. 1985 Apr;82(8):2197-201. doi: 10.1073/pnas.82.8.2197.
A solution of the problem of calculating cartesian coordinates from a matrix of interpoint distances (the embedding problem) is reported. An efficient and numerically stable algorithm for the transformation of distances to coordinates is then obtained. It is shown that the embedding problem is intimately related to the theory of symmetric matrices, since every symmetric matrix is related to a general distance matrix by a one-to-one transformation. Embedding of a distance matrix yields a decomposition of the associated symmetric matrix in the form of a sum over outer products of a linear independent system of coordinate vectors. It is shown that such a decomposition exists for every symmetric matrix and that it is numerically stable. From this decomposition, the rank and the numbers of positive, negative, and zero eigenvalues of the symmetric matrix are obtained directly.
报道了一种从点间距离矩阵计算笛卡尔坐标问题（嵌入问题）的解决方案。随后得到了一种将距离转换为坐标的高效且数值稳定的算法。结果表明，嵌入问题与对称矩阵理论密切相关，因为每个对称矩阵都通过一一变换与一个通用距离矩阵相关联。距离矩阵的嵌入会导致相关对称矩阵以坐标向量线性无关系统的外积之和的形式进行分解。结果表明，每个对称矩阵都存在这样的分解，并且它在数值上是稳定的。从这种分解中，可以直接得到对称矩阵的秩以及正、负和零特征值的数量。
