For the first time a principle-component analysis is used to separate out different orthogonal modes of the two-particle correlation matrix from heavy ion collisions. The analysis uses data from $\sqrt{{s}_{{}_{NN}}}=2.76\phantom{\rule{0.28em}{0ex}}\text{TeV}$ PbPb and $\sqrt{{s}_{{}_{NN}}}=5.02\phantom{\rule{0.28em}{0ex}}\text{TeV}$ $p\text{Pb}$ collisions collected by the CMS experiment at the CERN Large Hadron Collider. Two-particle azimuthal correlations have been extensively used to study hydrodynamic flow in heavy ion collisions. Recently it was shown that the expected factorization of two-particle results into a product of the constituent single-particle anisotropies is broken. The new information provided by these modes may shed light on the breakdown of flow factorization in heavy ion collisions. The first two modes (``leading'' and ``subleading'') of two-particle correlations are presented for elliptical and triangular anisotropies in PbPb and $p\text{Pb}$ collisions as a function of ${p}_{\mathrm{T}}$ over a wide range of event activity. The leading mode is found to be essentially equivalent to the anisotropy harmonic previously extracted from two-particle correlation methods. The subleading mode represents a new experimental observable and is shown to account for a large fraction of the factorization breaking recently observed at high transverse momentum. The principle-component analysis technique was also applied to multiplicity fluctuations. These also show a subleading mode. The connection of these new results to previous studies of factorization is discussed.