Linear Algebra, 2/e, 2nd Edition

8

Inner Product Spaces

8.1  INTRODUCTION

Bilinear form, as we had seen in the last chapter, is a natural generalization of the dot products of ℝ² and ℝ³; the important geometric concept of perpendicularity can also be introduced for symmetric bilinear forms. For a symmetric bilinear form f, even the notion of the length of a vector can be introduced in case f(v, v) is positive real for any non-zero v; such a formis known as positive definite.

However, the theory of positive definite symmetric bilinear forms will be developed in the frame­work of hermitian forms in this chapter. Hermitian forms, in some sense, generalize real symmetric forms to the complex case. Most of the chapter will be devoted to positive definite hermitian forms, otherwise ...