  We analyze the spectrum of harmonic flow, $v_n(p_T)$ for $n=0\text{--}5$, in
event-by-event hydrodynamic simulations of Pb+Pb collisions at the CERN Large
Hadron Collider ($\sqrt{s_{NN}}=2.76\,{\text{TeV}}$) with principal component
analysis (PCA). The PCA procedure finds two dominant contributions to the
two-particle correlation function. The leading component is identified with the
event plane $v_n(p_T)$, while the subleading component is responsible for
factorization breaking in hydrodynamics. For $v_0$, $v_1$, and $v_3$ the
subleading flow is a response to the radial excitation of the corresponding
eccentricity. By contrast, for $v_2$ the subleading flow in \emph{peripheral
collisions} is dominated by the nonlinear mixing between the leading elliptic
flow and radial flow fluctuations. In the $v_2$ case, the sub-sub-leading mode
more closely reflects the response to the radial excitation of $\varepsilon_2$.
A consequence of this picture is that the elliptic flow fluctuations and
factorization breaking change rapidly with centrality, and in central
collisions (where the leading $v_2$ is small and nonlinear effects can be
neglected) the subsub-leading mode becomes important. Radial flow fluctuations
and nonlinear mixing also play a significant role in the factorization breaking
of $v_4$ and $v_5$. We construct good geometric predictors for the orientation
and magnitudes of the leading and subleading flows based on a linear response
to the geometry, and a quadratic mixing between the leading principal
components. Finally, we suggest a set of measurements involving three point
correlations which can experimentally corroborate the nonlinear mixing of
radial and elliptic flow and its important contribution to factorization
breaking as a function of centrality.
