We present a new transform for EEG signals whose basis functions are well suited to represent the large-scale dynamics associated with event related potentials. The method involves instantiating an approximate model of the electrical properties of the brain as a conductor medium and then studying the free vibrational modes of the model. These form a set of basis functions, which we call eigenbrains, that can be used to meaningfully re-represent the brain's electrical activity. Eigenbrains are compared to principal component analysis and independent component analysis to highlight differences and similarities.