Again to Nandakumar on 29.1.2013

Abstract Algebra 26

18.12.2012

Subject Expert: Bijumon R

Content Editor: Nandakumar

E- Content in Mathematics

Bilinear Forms

Objectives

From this section a learner is expected to achieve the following

Learn the concept of bilinear forms, sesqui-linear forms and Hermitian forms.
Study the quadratic form associated with bilinear form
Familiarize with matrix form of bilinear form

Sections

1. Introduction

2. Bilinear Forms

3. Space of Bilinear Forms

4. Matrix Form of Bilinear Mapping

5. Quadratic Form Associated with Bilinear Form

6. Inner Product – A special Type of Bilinear form

7. Sesqui-Linear Forms

8. Space of Sesqui-linear Forms

9. Hermitian Forms

1. Introduction

In this session we describe bilinear forms. We will see that the set of all bilinear forms on a vector space V is a subspace of the space of all functions from into F. Matrix form of bilinear mapping and quadratic form associated with bilinear form will be discussed. We will see that inner product is a special type of bilinear form. Sesqui-linear formswill be introdued and will see that Hermitian form is a special type of Sesqui-linear form.

2. Bilinear Forms

Definition 1 Let be a vector space over the fieldF. A bilinear form (or bilinear mapping) on V is a function f, which assigns to each ordered pair of vectors in V a scalar in F , and which satisfies

(i)

and

(ii)

where are in V and are in F. The axioms (i) and (ii) are respectively called the linearity in the first and second variables.

Remark is a bilinear form on V if f is a function from into F which is linear as a function of either of its arguments when the other is fixed.

Example 1 Let be a vector space over a field F . The zero function from into F is clearly a bilinear form.

Example 2 Let where F is a field. The map defined by

is a bilinear form. This can be seen as follows:

Let and .

Then

and

Similarly axiom (ii) of the definition of bilinear form can be verified.

Example 3 Let V be a vector space over the field F and let and be linear functions on V. Define by

Then f is a bilinear form because of the following reason:

If we fix and regard f as a function of , then we simply have a scalar multiple of the linear functional . Similarly with fixed, f is a scalar multiple of .

Example 4 Let m and n be positive integers and F a field. Let V be the vector space of all matrices over F. Let A be a fixed matrix over F. Define by

Then is a bilinear form on V because of the following reason:

If and Z are matrices over F, then, since the transpose function and the trace function are linear, we have

Hence is linear in the first argument. It can be proved that is linear as a function of its second argument.

In the special case the matrix is , i.e., a scalar, and the bilinear form is simply

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3. Space of Bilinear Forms

Let be a vector space over the field F . Suppose f and g are bilinear forms on V and c a scalar. The sum and scalar product of the bilinear forms are defined by

and

With the vectors and scalars as in Definition 1, we have

Similarly, it can be proved that

Hence is a bilinear form. Thus the set of all bilinear forms on V is a subspace of the space of all functions from into F. The space of bilinear forms on V is denoted by

Definition 2 A bilinear mapping f is called symmetric if

A bilinear mapping f is called skew symmetric or anti-symmetric if

A bilinear mapping f is called alternatingif

Remark A bilinear mapping f can be expressed as the sum of a symmetric form g and a skew-symmetric form h, where

and

4. Matrix Form of Bilinear Mapping

Definition 3 Let f be a bilinear mapping on a finite dimensional vector space , and let be an arbitrary ordered basis of V. The matrix of f in the ordered basis B is the matrix A with entries We denote this matrix by

Theorem 1 Let V be a finite-dimensional vector space over the field F. For each ordered basis B of V, the function which associates with each bilinear form on V its matrix in the ordered basis B is an isomorphism of the space onto the space of matrices over the field F.

Proof. Let be an arbitrary ordered basis of V.

Let be vectors in V. Then they can be expressed as linear combinations of as follows:

and

Then

If we let then

…(1)

where and .

Since is the matrix of the bilinear form f in the ordered basis B, we write Eq. (1) as

…(2)

Eq.(2) shows that every bilinear form on V is of the type

…(3)

for some matrix A over F. Conversely, if we are given any matrix A, it is easy to see that Eq.(3)defines a bilinear form f on V, such that

The above discussion shows that the map is a one-one correspondence between the set of bilinear forms on V and the set of all matrices over F. Also, this map is a linear transformation because of the following reason:

for each i and j. This simply says that

This completes the proof.

Example 5 Let and let f be the bilinear form defined on and by

Then it can be seen that f is a bilinear form on . This bilinear form is called dot (or scalar) product. Referring to Eq.(2) in the proof of the previous theorem, we have

where B is an ordered basis for In particular, if we take B as the standard ordered basis for then

the identity matrix of order n. Hence the matrix of f in the standard ordered basis is the identity matrix of order n, and in that case

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5. Quadratic Form Associated with Bilinear Form

Definition 4 If f is a symmetric bilinear form, the quadratic form associated with f is the function q from V into F defined by

Example 6 If f is the bilinear form of Example 5, the associated quadratic form is

Example 7 For the bilinear form

the associated qudratic form is

6. Inner Product – A special Type of Bilinear form

If V is a vector space over the field of real numbers, an inner product on V is a symmetric bilinear form f on V which satisfies

A bilinear form which satisfies

is called positive definite.

Remark An inner product on a real vector space is a positive definite, symmetric bilinear form on that space.

7. Sesqui-Linear Forms

We now discuss sesqui-linear form on a real or complex vector space. We will see that if we deal with real vector space, sesqui-linear form is a bilinear form.

Definition 5 Let be a vector space over the field of real or complex numbers.. A sesqui-linear form (or simply form ) on V is a function f on with values in the field of scalars such that

(i)

and

(ii)

where are in V and are scalars. The axioms (i) and (ii) are respectively called the linearity in the first variable and conjugate linearity in the second variables. In the real case (ii) above takes the form

showing that in the real case a sesqui-linear form is a bilinear form.

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8. Space of Sesqui-linear Forms

We have seen in the section 3 that the set of bilinear forms is a subspace. By a similar argument, it can be seen that the set of sesqui-linear forms is also subspace of the space of all functions from into F.

Theorem 2 Let V be a finite-dimensional inner product space and f a sesqui-linear form on V. Then there exists a unique linear operator T on V such that

for all in V, and the map is an isomorphism of the space of sesqui-linearforms onto

Proof. Fix a vector in V. Then is a linear functional on V. Hence there is a unique vector in V such that

for every . We define a function U from V into V by setting

Thus for all in V

Hence, for in V and scalar c,

…(4)

Also, by the definition of sesqui-linear form,

…(5)

L.H.S. of Eqs. (4) and (5) are the same. Hence, we obtain

Hence, by the conjugate linearity in the second variable of the innerproduct, we have

Hence, for all in V and all scalars c,

and hence U is a linear operator on V and is an opeartor such that

for all in V.

To prove the uniqueness, we suppose there is another operator on V such that

for all in V. Then for all in V,

so that

for all in V. Thus for each sesqui-linearform f there is a unique linear operator such that

for all in V.

If f and g are sesqui-linearforms and c a scalar, then

…(6)

Also,

…(7)

for all in V. Now Equations (6) and (7) implies

so is a linear map. For each T in the equation

defines a sesqui-linearform f such that

Also, if and only if

Thus is an isomorphism.

Definition6 If f is a sesqui-linearform and is an arbitrary ordered basis of V, the matrix A with entries

is called the matrix of f in the ordered basis B.

We need the following fundamental result.

Theorem3 Let V be a finite-dimensional complex inner product space and let T be any linear operator on V. Then there is an orthonormal basis for V in which the matrix of T is upper triangular.

Theorem 4 Let f be a sesqui-linearform on a finite-dimensional complex inner product space V. Then there is an orthonormal basis for V in which the matrix of f is upper triangular.

Proof. Let T be the linear operator on V such that

for all in V. Then by Theroem 3, there is an orthonormal basis in which the matrix of T is upper-triangular. Hence,

when This completes the proof.

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9. Hermitian Forms

Definition7 A sesqui-linearform f on a real or complex vector space V is called Hermitian if

for all in V.

Theorem 5 A sesqui-linearform f on a complex inner product space V is Hermitian if and only if the linear operator T on V such that

for all in V is self-adjoint.

Proof. If T is a linear operator on a finite-dimensional inner product space V and f is the sesqui-linearform

then

Hence if and only if . This shows that f is Hermitian if and only if T is self-adjoint.

Remark We also note that when f is Hermitian so that is real for every The next theorem shows that on complex spaces this property characterizes Hermitian forms.

Theorem 6 Let V be a complex vector space and f a sesqui-linearform on V such that is real for every Then f is Hermitian.

Proof.

Let and be vectors in V. We must show that Now

…(8)

By the assumption of the theroem, and are real, hence by Eq. (8), the number is real. Looking at the same argument with instead of , we see that is real.

Now is real implies

implies

…(9)

and is real implies

implies

.…(10)

If we multiply Eq. (10) by i and add the result to Eq. (9), we obtain

Hence

This completes the proof.

The next result follows immediately.

Corollary Let T be a linear oparator on a complex finite-dimensional inner product space V. Then T is self-adjoint if and only if is real for every in V.

The following two results are needed in the proof of Principal Axis Theorem.

Theorem 7 Let V be a finite-dimensional inner product space and let T be a self-adjoint linear operator on V. Then there is an orthonormal basis for V, each vector of which is a characteristic vector for T.

Theorem 8 Let V be an inner product space and T a self-adjoint linear operator on V. Then each characteristic value of T is real.

Theorem 9 (Principal Axis Theorem) For every Hermitian form f on a finite-dimensional inner product space V, there is an orthonormal basis of V in which f is represented by a diagonal matrix with real entries.

Proof.

Let T be the linear operator such that for all in V. Then, since and , it follows that

for all hence Now T is self-adjoint implies by Theorem 7 that there is an orthonormal basis for V, each vector of which is a characteristic vector for T. Suppose is an orthonormal basis and that

for …(11)

By Theorem 8, each characteristic value of T is real.

By Eq.(11),

and hence the matrix of the Hermitian form f in the ordered basis B is a diagonal matrix with real entries given by

Summary

In this session we have described bilinear forms. We have seen that the set of all bilinear forms on a vector space V is a subspace of the space of all functions from into F. Matrix form of bilinear mapping and quadratic form associated with bilinear form have been discussed. We have seen that inner product is a special type of bilinear form. Sesqui-linear forms have been introdued and learnt that Hermitian form is a special type of Sesqui-linear form.

Assignments

1.Which of the followings functions f , defined on vectors in , are bilinear forms?

(a) .

(b) .

2.Describe explicitly all bilinear forms f on with the property that for all

3.Describe explicitly all bilinear forms f on with the property that for all

4.Describe explicitly all bilinear forms f on with the property that for all

5.Which of the followings functions f , defined on vectors in , are bilinear forms?

(a) .

(b) .

6.Let V be a complex vector space and let f be a sesqui-linear form on V which is symmetric. What is f ?

Quiz

1. Which of the following is correct?

(a) Every bilinear form is an inner product

(b) Every inner product is a bilinear form

(d) Every bilinear form is positive definite.

2. Which of the following is True?

(a) An inner product on a vector space over the field of real numbers is a symmetric bilinear form which is positive definite.

(b) An inner product on a vector space over the field of complex numbers is a symmetric bilinear form which is positive definite.

(d) None of the above.

3. Which of the following is True?

(a) An inner product over the field of real numbers is positive definite

(b) Every positive definite bilinear form is an inner product

(d) None of the above.

4. Fill in the blanks: If f is a symmetric bilinear form, the quadratic form q associated with f is ______

(a) the function from defined by

(b) the function from defined by

(d) the function from defined by

Quiz Answer

1. c2. a3. a4. d

FAQ

1. Is the inner product seen in this session coincides with the usual innerproudct?

Ans. It must be noted that the inner product mentioned in this session is defined only on real vector spaces. Usually inner products are defined over vector spaces over the field of real numbers or over the field of complex numbers. However, the inner product seen in this session coincides with the usual inner product defined over vector spaces over the field of real numbers.

2. Can we regard a bilinear form as a special case of sesqui-linear form?

Ans. In general, we cannot regard bilinear form as a special case of sesqui-linear form. However, sesqui-linear form defined over a real vector space is a bilinear form.

3. On complex spaces, it is possible to characterize Hermitian forms?

Ans. Yes. The following result shows that on complex spaces it is possible to characterize Hermitian forms: “Let V be a complex vector space and f a sesqui-linearform on V such that is real for every Then f is Hermitian.”

Glossary

Bilinear form (Bilinear mapping): Let be a vector space over the field F. A bilinear form (or bilinear mapping ) on V is a function f, which assigns to each ordered pair of vectors in V a scalar in F , and which satisfies

(ii)

and

(ii)

where are in V and are in F.

Space of bilinear forms: The set of all bilinear forms on V is a subspace of the space of all functions from into F and is the space of bilinear formson V denoted by

Symmetric bilinear form: A bilinear mapping f is called symmetric if

Skew-Symmetric bilinear form: A bilinear mapping f is called skew symmetric or anti-symmetric if

Alternating bilinear form: A bilinear mapping f is called alternating if

The matrix of a bilinear form in a given ordered basis: Let f be a bilinear mapping on a finite dimensional vector space , and let be an arbitrary ordered basis of V. The matrix of f in the ordered basis B is the matrix A with entries This matrix is denoted by

Quadratic form associated with a bilinaer form: If f is a symmetric bilinear form, the quadratic form associated with f is the function q from V into F defined by

Positive definite bilinear form: A bilinear form which satisfies

is called positive definite.

Inner product on a vector space over the field of real numbers: If V is a vector space over the field of real numbers, an inner product on V is a symmetric and positive definite bilinear form f on V.

Sesqui-linear form: Let be a vector space over the field of real or complex numbers.. A sesqui-linear form (or simply form ) on V is a function f on with values in the field of scalars such that

(ii)

and

(ii)

where are in V and are scalars.

Matrix of a Sesqui-linear form in a given ordered basis: If f is a sesqui-linearform and is an arbitrary ordered basis of V, the matrix A with entries

is called the matrix of f in the ordered basis B.

Hermitian form: A sesqui-linear form f on a real or complex vector space V is called Hermitian if

for all in V.

References

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

2Gunadhar Paria, Linear Algebra, New Central Book Agency, Calcutta, 1992.

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

4J. H. Kwak and S. Hong, Linear Algebra, Second Edition, Birkhauser, Boston.

5N. S. Gopalakrishnan, University Algebra, Wiley Eastern Limited, New Age International Limited, New Delhi 1995.