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Sachin Kaushal, Lovely Professional University Unit 30: Bilinear Forms and Symmetric Bilinear Forms

Unit 30: Bilinear Forms and Symmetric Bilinear Forms Notes

CONTENTS
Objectives
Introduction

30.1 Bilinear Forms
30.2 Symmetric Bilinear Forms
30.3 Summary

30.4 Keywords
30.5 Review Questions
30.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand that the bilinear forms and inner products discussed in earlier units have a
strong relation.
 See the isomorphism between the space of bilinear forms and the space of n n matrices
is established.
 Know that the linear transformations from V into V* defined by (L ) ( ) = f( , ) = (R )
f f
( ) (where f is a bilinear form) are such that rank (L ) = rank (R ).
f f
Introduction

In this unit we are interested in studying a bilinear form f on a finite vector space of dimension n.
With the help of a number of examples it is shown how to get various forms of bilinear forms
including linear functionals, bilinear forms involving n 1 matrices.
It is also established that the rank of a bilinear form is equal to the rank of the matrix of the form
in any ordered basis.

30.1 Bilinear Forms

In this unit we treat bilinear forms on finite dimensional vector spaces. There are a few similarities
between the bilinear forms and the inner product spaces. Let V be a real inner product space and
suppose that A is a real symmetric linear transformation on V. The real valued function f(v)
defined on V by f(v) = (v, A, v) can also be called the quadratic form i.e. bilinear form associated
(n)
with A. If we assume A to be a real, n n symmetric matrix (a ) acting on F and for an arbitrary
ij
(n)
vector v = (x , x ..., x ) in F , then
1 2 n
2
2
2
f(v) = (v, A, v) = a x + a x + ... + a x + 2 a x x
11 1 22 2 nn n ij j j
iLj
In real n-dimensional Euclidean space such quadratic functions serve to define the quadratic
surfaces.

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