Many beginning university students struggle with the new approaches to mathematics that they find in their courses due to a shift in presentation of mathematical ideas, from a procedural approach to concept definitions and deductive derivations, and ideas building upon each other in quick succession. This paper highlights this struggle by considering some conceptual processes and difficulties students find in learning about eigenvalues and eigenvectors. We use the theoretical framework of Tall's three worlds of mathematical thinking, along with perspectives from Dubinsky's APOS (action, process, object, schema) theory and Thomas's representational versatility. The results of the study describe thinking about these concepts by several groups of first- and second-year university students. In particular the obstacles they faced, and the emerging links some were constructing between parts of their concept images formed from the embodied, symbolic, and formal worlds are presented. We also identify some fundamental problems with student understanding of the definition of eigenvectors that lead to implementation problems, and some of the concepts underlying such difficulties.