Chapter 7 - Complex Vector Spaces and General Inner Products

Abstract

In Chapter 7, we generalize the results of previous chapters to Cn, the set of complex n-vectors, and consider its similarities to and differences from Rn. In Section 7.1, we introduce Cn, and its basic operations, including the complex dot product of vectors, along with complex matrices, and the conjugate transpose, while introducing Hermitian, skew-Hermitian, and normal matrices. In Section 7.2, we examine complex linear systems and complex eigenvalues and eigenspaces, and diagonalization. Section 7.3 compares the properties of general complex vector spaces and linear transformations with their real counterparts. In Section 7.4, we study the complex analog of the Gram-Schmidt Process along with unitary matrices, a generalization of orthogonal matrices. Unitarily diagonalizable matrices and self-adjoint operators are discussed. Finally, in Section 7.5, we discuss inner product spaces, which possess an additional operation analogous to the dot product on Rn. We also study orthogonal bases, the Generalized Gram-Schmidt Process, and orthogonal complements in inner product spaces.