Operator theory in finite-dimensional vector spaces

Abstract

This chapter is preliminary to the following one where perturbation theory for linear operators in a finite-dimensional space is presented. We assume that the reader is more or less familiar with elementary notions of linear algebra. In the beginning sections we collect fundamental results on linear algebra, mostly without proof, for the convenience of later reference. The notions related to normed vector spaces and analysis with vectors and operators (convergence of vectors and operators, vector-valued and operator-valued functions, etc.) are discussed in somewhat more detail. The eigenvalue problem is dealt with more completely, since this will be one of the main subjects in perturbation theory. The approach to the eigenvalue problem is analytic rather than algebraic, depending on function-theoretical treatment of the resolvents. It is believed that this is a most natural approach in view of the intended extension of the method to the infinite-dimensional case in later chapters.