Continuous Decompositions and Coalescing Eigenvalues for Matrices Depending on Parameters

This contribution is a summary of four lectures delivered by the first author at the CIME Summer school in June 2011 at Cetraro (Italy). Preparation of those lectures was greatly aided by the other authors of these lecture notes. Our goal is to present some classical, as well as some new, results related to decompositions of matrices depending on one or more parameters, with particular emphasis being paid to the case of coalescing eigenvalues (or singular values) for matrices depending on two or three parameters. There is an extensive literature on this subject, but a systematic collection of relevant results is lacking, and this provided the impetus for writing the lecture notes.