Semi-Riemannian Geometry

Chapter 3 Bilinear Functions

The inner product is an example of a bilinear function that likely is familiar to the reader. In this book, we are more interested in a less restrictive type of bilinear function called a scalar product, but we will not ignore the inner product altogether. In order to decide whether a bilinear function is a scalar product, we need to determine whether it is symmetric and nondegenerate. These properties of bilinear functions will be the focus of the present chapter.

3.1 Bilinear Functions

Let V be a vector space. A function

is said to be bilinear (on V) if it is linear in both arguments; that is,

and

for all vectors u, v, w in V and all real numbers c. In the literature, a bilinear function is sometimes called a quadratic form. We often denote

writing 〈v, w〉 in place of b(v, w).

We say that b is:

symmetric

if 〈v, w〉 = 〈w, v〉 for all v, w in V.

alternating

if 〈v, w〉 =  − 〈w, v〉 for all v, w in V.

nondegenerate

if for all v in V, 〈v, w〉 = 0 for all w in V implies v = 0.

degenerate

if b is not nondegenerate.