Page 302 - DMTH502_LINEAR_ALGEBRA

Linear Algebra Richa Nandra, Lovely Professional University

Notes Unit 28: Positive Forms and More on Forms

CONTENTS
Objectives
Introduction

28.1 Positive Forms
28.2 More on Forms
28.3 Summary

28.4 Keywords
28.5 Review Questions
28.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand when a form f on a real or complex vector space v is non-negative. If the form
f is Hermitian and f( , ) > 0 for every in v, the form f is positive.
 Know that f is a positive form if and only if A = A* and the principal minors of the matrix
A of f are all positive.
 See that if A is the matrix of the form f in the ordered basis { , ..., } of v and the principal
1 n
minors of A are all different from 0, then there is a unique upper triangular matrix P with
P = 1(1 k n) such that P*AP is upper triangular.
kk
Introduction

In this unit the form f on a real or complex vector space is studied and seen under what conditions
the form f is positive.
On the basis of the principal minors of A being all different from 0, the positive form f, it is seen
that there is an upper-triangular matrix P with P = 1 (1 k n) such that B = AP is lower
kk
triangular.
28.1 Positive Forms

In this unit we study non-negative (sesqui) forms and their relation to a given inner product on
the given finite vector space.
A form f on a real or complex vector space v is non-negative if it is Hermitian and f( , ) 0 for
every in v. The form f is positive if it is Hermitian and f( , ) > 0 for all 0.
A positive form on v is simply an inner production v. Let f be a form on the finite dimensional
space. Let = ( , , ... ) be an ordered basis of v, and let A be the matrix of f on the basis , i.e.,
1 2 n
A = f( , ). If = x + ...... + x , then
jk k j 1 1 n n
f( , ) = ( f x a j , x a )
k k
j
j k

296 LOVELY PROFESSIONAL UNIVERSITY