Indefinite Inner Products

Abstract

In traditional linear algebra the concepts of length, angle, and orthogonality are defined by a definite inner product. Here, the definite inner product is replaced by an indefinite one and this produces substantial changes in the geometry of subspaces. Thus, the geometry of subspaces in this context is fundamental for our subject, and is the topic of this chapter.

As in the definite case, when an inner product is introduced on Cⁿ, then certain n x n matrices (seen as linear transformations of Cⁿ) have symmetries defined by the inner product. If the inner product is definite this leads to the usual classes of hermitian, unitary, and normal matrices. If the inner product is indefinite, then analogous classes of matrices are defined and will be investigated in subsequent chapters.