We present the first principal component analysis (PCA) applied to a sample of 119 Spitzer Infrared Spectrograph (IRS) spectra of local ultraluminous infrared galaxies (ULIRGs) at z < 0.35. The purpose of this study is to objectively and uniquely characterize the local ULIRG population using all information contained in the observed spectra. We have derived the first three principal components (PCs) from the covariance matrix of our data set which account for over 90 per cent of the variance. The first PC is characterized by dust temperatures and the geometry of the mix of source and dust. The second PC is a pure star formation component. The third PC represents an anticorrelation between star formation activity and a rising active galactic nucleus (AGN). Using the first three PCs, we are able to accurately reconstruct most of the spectra in our sample. Our work shows that there are several factors that are important in characterizing the ULIRG population, dust temperature, geometry, star formation intensity, AGN contribution, etc. We also make comparison between PCA and other diagnostics such as ratio of the 6.2 μm PAH emission feature to the 9.7 μm silicate absorption depth and other observables such as optical spectral type. An electronic version of the first three PCs of the local ULIRG population is available at http://astronomy.sussex.ac.uk/~lw94/PCA/.