Linear Algebra, 2/e, 2nd Edition

7

Bilinear Forms

7.1  INTRODUCTION

As we have seen in Chapter 5, the idea of orthogonality in ℝⁿ or ℂⁿ, introduced through dot products, was the key to establish an important result about real symmetric matrices (that they can always be diagonalized). The concept of bilinear forms generalizes dot products and provides a way for introducing the idea of orthogonality in arbitrary vector spaces. As such, these forms are basic in various areas of advanced mathematics as well as in many applications. Along with bilinear forms, we also introduce alternating bilinear forms in this chapter; such forms and associated symplectic groups are also important in diverse areas of mathematics and physical sciences.

7.2  BASIC CONCEPTS

Recall that the dot product ...