The quadrature method of moments (QMOM), a promising new toot for aerosol dynamics simulation is extended to generally mixed multicomponent particle populations. This paper develops the mathematical and statistical foundation for a fully multivariate extension of the QMOM using principal components analysis (PCA). In essence, the full particle distribution function is systematically replaced by a set of lower-order mixed moments and corresponding multivariate quadrature points optimally assigned through PCA and back projection. The resulting PCA-QMOM is illustrated for a multivariate normal particle population in order to compare quadrature point assignments with analytic results, but the method is applicable to arbitrary distributions. Physical and optical properties can be reliably estimated by summation over the PCA-assigned quadrature points. Application of the PCA-QMOM to the dynamics of generally mixed particle populations evolving under condensation, coagulation, and sintering is described in the following paper (Part 11). Published by Elsevier Ltd.