Quadratic Forms - Part One

With FAQ

12.12.12

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Abstract Algebra 24

Quadratic Forms - Part One

Objectives

From this section a learner is expected to achieve the following

Learn the concept of quadratic form.
Study the matrix form associated with quadratic form
Familiarize with canonical form.
Study the nature of quadratic forms
Learn the method of reducing to canonical form by diagonalising the matrix

Sections

1. Introduction

2. Quadratic form

3. Matrix form of quadratic form

4. Matrix of quadratic form in symmetric form

5. Canonical form

6. Nature of quadratic forms

7.Reducing to canonical form by diagonalising the matrix

1. Introduction

The study of homogeneous polynomials of second degree is very important in mathematics; we call such polynomials by the name quadratic forms. In this session we describe the definition and examples of quadratic form. Matrix of quadratic form in symmetric form will be discussed. Canonical form associated with a quadratic form and the method of reducing to canonical form by diagonalizing the matrix will be discussed. A brief discussion on the nature of quadratic forms will be made in this session.

2. Quadratic form

Definition (quadratic form) A homogeneous polynomial of second degree of the form

. . . (1)

whose coefficients are elements of a field F is called a quadratic form over the field F in n variables . If the field under consideration is the field of real numbers, then the quadratic form is a real quadratic form.

Example 1

is a quadratic form in the two variables
is a quadratic form in the variables x, y.
is a quadratic form in three variables
is a quadratic form in the variables x, y, z.

3. Matrix form of quadratic form

If we take

and

equation (1) takes the matrix form

. . . (2)

where is the transpose of the matrix X.

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4. Matrix of quadratic form in symmetric form

The coefficient of in Eq. (2) is . It is customary to take , so that the matrix of a quadratic form in n variables be symmetric matrix of order n.

If the quadratic form is in two variables then the matrix of the quadratic form is of the following form:

If the quadratic form is in three variables then the matrix of the quadratic form is of the following form:

Example 2Obtain the quadratic form associated with the matrix

Solution

As we are given with a 2 × 2 symmetric matrix A, the required a quadratic form will be in two variables, say and

Take

Then the required quadratic form is

Aliter:

The standard quadratic form in two variables is

…(3)

Here

Putting these values in Eq. (3), we obtain the quadratic form

Example 3 Give the matrix associated with the quadratic form

Solution

Given is a quadratic form in three variables and.

Comparing with the standard quadratic form

we have

a = 6, b = 17, c = 3, f = 7, g = 1, h = 10

so the matrix associated with the quadratic form is

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5. Canonical Form

We know that a quadratic form

can be written in the form XTAX, where

Suppose q is changed to the form

where are linear expressions of ; that is, q is reduced to a linear combination of only squares of the new variables .

In general, the linear transformation

X = BY

over the fieldof real numbers carries the real quadratic form

with symmetric matrix A into the quadratic form

. . . (4)

with symmetric matrix .

Definition If a real quadratic form be expressed as a sum of squares of new variables by means of a real non-singular transformation, then the new form obtained is called the canonical form of the given transformation. The number of square variables in the canonical form is the rank of the quadratic form.

Definition Two quadratic forms in the same variables are called equivalent if and only if there exists a non-singular linear transformation which, together with , carries one of the forms into the other.

Remark By the definition of congruent matrices, is congruent to A, so that we have the following:

1.The rank of a quadratic form is invariant under a non-singular transformation of the variables.

2.Two quadratic forms are equivalent if and only if their matrices are congruent.

Since every symmetric matrix of rank r can be reduced to a diagonal matrix having exactly r non-zero elements in the diagonal, it follows that a quadratic form of rank r can be reduced to the form

. . . (5)

in which only terms in the squares of the variables occur, by a non-singular linear transformation .

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6. Nature of quadratic forms

Positive definite quadratic form

If and are all-positive, then the canonical expression can take only positive values for real non-zero values of . Then the quadratic form

and the expression

are said to be positive definite.

Negative definite quadratic form

If and are all negative, then the quadratic form

and the expression

are said to be negative definite.

Positive semi definite quadratic form

If at least one of and is zero and others are positive, then the quadratic form

and the expression

are said to be positive semi definite.

Negative semi definite quadratic form

If at least one of and is zero and others are negative, then the quadratic form

and the expression

are said to be negative semi definite.

Indefinite quadratic form

If at least one of and is negative and at least one of them is positive, then the quadratic form

and the expression

are called indefinite since it can take positive as well as negative values.

Remarks

1.If q is negative definite (resp., negative semi-definite), then –q is positive definite (resp., positive semi-definite).

2.If is positive definite, then |A| > 0.

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7.Reducing to canonical form by diagonalising the matrix

In this method we diagonalize the matrixA of the quadratic form and then the canonical form is obtained by the transformation , where .

To determine D and for a given matrix A, we proceed as follows:

We start with where I is the identity matrix of order same as that of A. Then change A on the right hand side to a diagonal matrix using the row and column transformations. For every transformation, make the corresponding changes in I also. When A changes to D, I changes to .

We will illustrate this method through example. We note that the row operations are allowed on the right side of dashed line (treating AI as a single matrix), but column operations are not allowed to the right side of dashed line (here also we treat AI as a single matrix).

Example 4 By indicating the transformation, reduce

to canonical form. Specify the nature of the quadratic form.

Solution

We take the given quadratic form to be where

and Ais the symmetric matrix

Comparing with the standard form

we have

Hence the real symmetric quadratic matrix is

Now to reduce to the canonical form, we proceed as follows:

where D is the diagonal matrix

and

Hence the linear transformation

carries the quadratic form

to the canonical form

given by

Since and are positive and is negative, the given quadratic form is indefinite.

Example 5 Prove that the quadratic form

is positive definite.

Solution

The matrix of the given quadratic form is

Now

Hence

Hence the given quadratic form is

Since are all positive, the given quadratic form is positive definite

Example 6 Show that the quadratic form

is positive semi-definite.

Solution

The matrix of the given quadratic form is

Hence we have

Thus the transformation reduces q to the canonical form

Since , the quadratic form q is positive semi-definite.

Index

The number of positive square terms in a canonical form of the quadratic form is called index of the quadratic form and is denoted by p.

Signature

The difference of number of positive and negative square terms in a canonical form of the quadratic form is called its signature and is denoted by s.

Example 7 Reduce the quadratic form

to canonical form and find the rank, index and signature. Also discuss the nature of the quadratic form.

Solution

The matrix of the given quadratic form is

Now to reduce to the canonical form, we proceed as follows:

Hence

Thus, we have

Hence, the quadratic form is

Since and , the given quadratic form is indefinite. Hence,

rank= 3

Index= number of positive square terms = 2.

Signature = The difference of number of positive and negative square

terms

Summary

In this session the definition and examples of quadratic form have been given. Matrix of quadratic form in symmetric form have been discussed. Canonical form associated with a quadratic form and the method of reducing to canonical form by diagonalizing the matrix have been discussed. A brief discussion on the nature of quadratic forms have been made.

Assignments

1.Write down the quadratic form corresponding to the symmetric matrix .

2.Write the quadratic form

in matrix notation.

3.Write down the matrix of the quadratic form

and verify that it can be written as .

4. Discuss the nature of the following quadratic form:

5. Prove that the quadratic form is positive definite.

Quiz

1. The quadratic form

and the expression

are negative definite if ______

(a) If and are all-positive,

(b) If at least one of and is zero and others are positive

(d) If at least one of and is negative and at least one of them is positive

Ans. (c) If and are all negative,

2. The quadratic form

and the expression

are positive semi definite if ______

(a) If and are all-positive,

(b) If at least one of and is zero and others are positive

(d) If at least one of and is negative and at least one of them is positive

Ans. (b) If at least one of and is zero and others are positive

3. The number of positive square terms in a canonical form of the quadratic form is called ______

(a) order of the quadratic form

(b) determinant of the quadratic form

(d) none of the above

Ans. (c) index of the quadratic form

4. The difference of number of positive and negative square terms in a canonical form of the quadratic form is called

(a) order

(b) determinant

(d) signature

Ans. (d) signature

FAQ

1. Is always a symmetric matrix, where A and B are square matrices of same order?

Solution No. However, if A and B are square matrices of same order and if A symmetric (B need not be symmetric), then is symmetric. This is because and and hence

2. When we say that two matrices are congruent?

Solution Two matrices A and B over a field are called congruent if there exists an invertible matrix P over the same field such that

Matrix congruence is an equivalence relation.

Glossary

Quadratic Form: A homogeneous polynomial of second degree of the form

Canonical form: If a real quadratic form be expressed as a sum of squares of new variables by means of a real non-singular transformation, then the new form obtained is called the canonical form of the given transformation.

Rank: The number of square variables in the canonical form is the rank of the quadratic form.

Equivalent quadratic forms: Two quadratic forms in the same variables are called equivalent if and only if there exists a non-singular linear transformation which, together with , carries one of the forms into the other.

Index: The number of positive square terms in a canonical form of the quadratic form is called index of the quadratic form.

Signature: The difference of number of positive and negative square terms in a canonical form of the quadratic form is called its signature.

References

1J. B. Fraleigh, A First Course in Abstract Algebra, Fifth Edition, Addison-Wesley, California, 1999.

2J. H. Kwak and S. Hong, Linear Algebra, Second Edition, Birkhauser, Boston.

3K. Hoffman and R. Kunze, Linear Algebra, PHI Learning Private Ltd., New Delhi, 2011