Chapter 7 - Complex Vector Spaces and General Inner Products

Abstract

This chapter examines properties of the eigenspaces of matrices with complex entries and compares the properties of general complex vector spaces and linear transformations with their real counterparts. One reason for generalizing to the complex number system is that it can take advantage of the “Fundamental Theorem of Algebra” that states that every nth-degree polynomial can be factored completely when complex roots are permitted. The way this permits to find additional (non-real) solutions to eigenvalue problems is revealed. The complex analogs of the Gram-Schmidt Process and orthogonal matrices are also highlighted. However, the complex numbers are used in the chapter to define and study complex n -vectors and matrices for emphasizing their differences with real vectors and matrices. Addition and scalar multiplication of matrices are defined entry wise in the usual manner. In particular, the Fundamental Theorem of Algebra states that any complex polynomial of degree n factors into a product of n linear factors. Thus, for every n × n matrix A, p A (x) can be expressed as a product of n linear factors. Therefore, the algebraic multiplicities of the eigenvalues of A must add up to n. This eliminates one of the two reasons that some real matrices are not diagonalizable.