We propose a method for analyzing extremal behavior through the lens of a most-efficient basis of vectors. The method is analogous to principal component analysis, but is based on methods from extreme value analysis. Specifically, rather than decomposing a covariance or correlation matrix, we obtain our basis vectors by performing an eigendecomposition of a matrix which describes pairwise extremal dependence. We apply the method to precipitation observations over the contiguous US. We find that the time series of large coefficients associated with the leading eigenvector shows very strong evidence of a positive trend, and there is evidence that large coefficients of other eigenvectors have relationships with the El Niño-Southern Oscillation.