  The Event Horizon Telescope (EHT) is taking the first images of black holes
resolved at horizon scales to measure their shadows and probe accretion
physics. A promising avenue for testing the hypothesis that astrophysical black
holes are described by the Kerr solution to Einstein's equations is to compare
the size and shape of the shadow a black hole casts on the surrounding emission
to the predictions of the Kerr metric. We develop here an efficient parametric
framework to perform this test. We carry out ray-tracing simulations for
several parametrized non-Kerr metrics to create a large data set of non-Kerr
shadows that probe the allowed parameter space for the free parameters of each
metric. We then perform principal components analysis (PCA) on this set of
shadows and show that only a small number of components are needed to
accurately reconstruct all shadows within the set. We further show that the
amplitude of the PCA components are smoothly related to the free parameters in
the metrics, and therefore, that these PCA components can be fit to EHT
observations in order to place constraints on the free parameters of these
metrics that will help quantify any potential deviations from the Kerr
solution.
