The Role of Eigenvalues and Eigenvectors

This is one of the most important topics of linear algebra. In physics, the experimental results are usually numbers. In quantum mechanics, these numbers correspond to eigenvalues of observables which we describe with special linear operators in Hilbert spaces or in finite-dimensional subspaces thereof. Eigenvalues are also relevant for symmetries in physics. Eigenvalues and eigenvectors help significantly to clarify the structure of operators (endomorphisms). We recall that an operator in linear algebra is a linear map of a vector space to itself. We denote the set of operators on V by \({{\,\textrm{End}\,}}(V) \equiv {{\,\textrm{Hom}\,}}(V,V)\) . In this chapter, after some preliminaries and definitions, we will discuss the role of eigenvalues and eigenvectors of diagonalizable and nondiagonalizable operators. For any given operator f on a C vector space, we get to a corresponding direct sum decomposition of V , using only very elementary notions. This decomposition leads to the more refined Jordan decomposition of V . First, we consider the situation on an abstract vector space without other structures. Later, in Chap. 10, we shall discuss vector spaces with an inner product structure.