When a topological space is known only from sampling, persistence provides a useful tool to study its homological properties. In many applications one can sample not only the space, but also a map acting on the space. The understanding of the topological features of the map is often of interest, in particular in time series analysis. We consider the concept of persistence in finite dimensional vector spaces and combine it with a graph approach to computing homology of maps in order to study the persistence of eigenspaces of maps induced in homology. This is research in progress, joint with Herbert Edelsbrunner.