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Sachin Kaushal, Lovely Professional University Unit 31: Skew-symmetric Bilinear Forms

Unit 31: Skew-symmetric Bilinear Forms Notes

CONTENTS
Objectives
Introduction

31.1 Skew-symmetric Bilinear Forms
31.2 Summary
31.3 Keywords

31.4 Review Questions
31.5 Further Readings

Objectives

After studying this unit, you will be able to:
 See that skew-symmetric bilinear form is studied in a similar way as the symmetric
bilinear form was studied.
1
 Know that here the quadratic form is given by the difference h( , ) = [f( , ) –f[ , ]]
2
 Understand that the space L(V, V, F) is the direct sum of the subspace of symmetric forms
and the subspace of skew-symmetric forms.

Introduction

In this unit a bilinear form f on V called skew-symmetric form i.e. f( , ) = –f( , ) is studied.
Close on the steps of symmetric bilinear form of the unit 30 the skew-symmetric form is developed.
It is seen that in the case of a skew-symmetric form, its matrix A in some (or every) ordered basis
is skew-symmetric, A = –A.
t
31.1 Skew-symmetric Bilinear Forms

After discussing symmetric bilinear forms we can deal with the skew-symmetric forms with
ease. Here again we are dealing wth finite vector space over a subfield F of the field of complex
numbers.
A bilinear form f on V is called skew-symmetric if f( , ), –f( , ) for all , and in V. It means
that f( , ) = 0. So we now need to introduce two different quadratic forms as follows:

If we let
1
g( , ) = [f( , ) + f( , )]
2
1
h( , ) = [f( , ) + f( , )]
2
So it is seen that g is a symmetric bilinear form on V and h is a skew-symmetric form on V. Also
f = g + h. These expressions for V, as the symmetric and skew-symmetric form is unique. So the
space L(V, V, F) is the direct sum of the subspace of symmetric forms and the subspace of
skew-symmetric forms.

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