Page 347 - DMTH502_LINEAR_ALGEBRA

Sachin Kaushal, Lovely Professional University Unit 32: Groups Preserving Bilinear Forms

Unit 32: Groups Preserving Bilinear Forms Notes

CONTENTS
Objectives
Introduction

32.1 Overview
32.2 Groups Preserving Bilinear Forms
32.3 Summary

32.4 Keywords
32.5 Review Questions
32.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand that there are certain classes of linear transformations including the identity
transformation that preserve the form f of bilinear forms.
 See that the collection of linear operators which preserve f, is closed under the formation
of operator products.
 Know that a linear operator T preserves the bilinear form f if and only if T preserves the
quadratic form
q( ) = f( , )
 See that the group preserving a non-degenerate symmetric bilinear form f on V is
isomorphism to an n n pseudo-orthogonal group.
Introduction

In this unit the groups preserving certain types of bilinear forms is studied.
It is seen that orthogonal groups preserve the length of a vector.
For non-degenerate symmetric bilinear form on V the group preserving f is isomorphic to n n
pseudo-orthogonal group.
4
For the symmetric bilinear form f on R with quadratic form
2
2
g(x, y, z, t) = t – x – y – z 2
2
4
a linear operator T on R preserving this particular bilinear form is called Lorentz transformation
and the group preserving f is called the Lorentz Group.
32.1 Overview

Here we shall be concerned with some groups of transformations which preserve the form of
the bilinear forms. Let T be a linear operator on V. We say that T preserves f if f(T , T ) = f( , ) for
all and in V. Consider a function g( , ) = f(T , T ). If T preserves f it simply means g = f. The
identity operator preserves every bilinear form. If S and T are two linear operators which

LOVELY PROFESSIONAL UNIVERSITY 341