Handbook of Linear Algebra, 2nd Edition

Chapter 13

Quadratic, Bilinear, and Sesquilinear Forms

Raphael Loewy

Technion

Bilinear forms are maps defined on V × V, where V is a vector space, and are linear with respect to each of their variables. There are some similarities between bilinear forms and inner products that are discussed in Chapter 5. Basic properties of bilinear forms, symmetric bilinear forms, and alternating bilinear forms are discussed. The latter two types of forms satisfy additional symmetry conditions.

Quadratic forms are obtained from symmetric bilinear forms by equating the two variables. They are widely used in many areas. A canonical representation of a quadratic form is given when the underlying field is ℝ or ℂ.

When the field is the complex numbers, it is standard ...