Linear Operators on Inner Product Spaces

Abstract

In this chapter, we are going to study linear operators on finite-dimensional inner product spaces. In the last chapter, we introduced adjoints of linear maps between possibly different inner product spaces. Here we shall see how the adjoint can be used to understand linear operators on a fixed inner product space. The important operators we study here are the self-adjoint, skew-adjoint, normal, orthogonal, and unitary operators. We shall spend several sections on the existence of eigenvalues, diagonalizability, and canonical forms for these special but important linear operators. Having done that, we go back to the study of general linear maps and operators and establish the singular value and polar decompositions. We also show Schur’s theorem to the effect that complex linear operators have upper triangular matrix representations. This result does not depend on the spectral theorem. It is also possible to give a quick proof of the spectral theorem using only the material covered in Sect. 4.1. The chapter finishes with a section on quadratic forms and how they tie in with the theory of self-adjoint operators.