Chapter 10 - Spectral Decompositions

Abstract

This chapter discusses the question of diagonalization of a linear operator on a finite-dimensional vector space V. A linear operator T on V is diagonalizable, that is, can be represented by a diagonal matrix, only if there exists a basis of V that consists of eigenvectors of T. For a unitary vector space V, those linear operators that can be represented by a diagonal matrix relative to an orthonormal basis of V are the same as the normal linear operators on V. The chapter describes the diagonalizable linear operators in terms that are free of any reference to an inner product. The characterization obtained is in terms of a spectral decomposition. A certain acquaintance with projections is essential to a formulation of the concept of a spectral decomposition.