Matrix decompositions of two-dimensional nuclear magnetic resonance spectra

Two-dimensional NMR spectra are rectangular arrays of real numbers, which are commonly regarded as digitized images to be analyzed visually. If one treats them instead as mathematical matrices, linear algebra techniques can also be used to extract valuable information from them. This matrix approach is greatly facilitated by means of a physically significant decomposition of these spectra into a product of matrices--namely, S = PAPT. Here, P denotes a matrix whose columns contain the digitized contours of each individual peak or multiple in the one-dimensional spectrum, PT is its transpose, and A is an interaction matrix specific to the experiment in question. The practical applications of this decomposition are considered in detail for two important types of two-dimensional NMR spectra, double quantum-filtered correlated spectroscopy and nuclear Overhauser effect spectroscopy, both in the weak-coupling approximation. The elements of A are the signed intensities of the cross-peaks in a double quantum-filtered correlated spectrum, or the integrated cross-peak intensities in the case of a nuclear Overhauser effect spectrum. This decomposition not only permits these spectra to be efficiently simulated but also permits the corresponding inverse problems to be given an elegant mathematical formulation to which standard numerical methods are applicable. Finally, the extension of this decomposition to the case of strong coupling is given.