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Page 1
Bilinear, Quadratic and Hermitian Forms
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Bilinear, Quadratic and Hermitian Forms
Lesson Developer: Vivek N Sharma
College / Department: Department of Mathematics,
S.G.T.B. Khalsa College, University of Delhi
Page 2
Bilinear, Quadratic and Hermitian Forms
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Bilinear, Quadratic and Hermitian Forms
Lesson Developer: Vivek N Sharma
College / Department: Department of Mathematics,
S.G.T.B. Khalsa College, University of Delhi
Bilinear, Quadratic and Hermitian Forms
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents
1. Learning Outcomes
2. Introduction
3. Definition of a Bilinear Form on a Vector Space
4. Various Types of Bilinear Forms
5. Matrix Representation of a Bilinear Form on a Vector Space
6. Quadratic Forms on R
??
7. Hermitian Forms on a Vector Space
8. Summary
9. Exercises
10. Glossary and Further Reading
11. Solutions/Hints for Exercises
1. Learning Outcomes:
After studying this unit, you will be able to
? define the concept of a bilinear form on a vector space.
? explain the equivalence of bilinear forms with matrices.
? represent a bilinear form on a vector space as a square matrix.
? define the concept of a quadratic form on R
?? .
? explain the notion of a Hermitian form on a vector space.
Page 3
Bilinear, Quadratic and Hermitian Forms
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Bilinear, Quadratic and Hermitian Forms
Lesson Developer: Vivek N Sharma
College / Department: Department of Mathematics,
S.G.T.B. Khalsa College, University of Delhi
Bilinear, Quadratic and Hermitian Forms
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents
1. Learning Outcomes
2. Introduction
3. Definition of a Bilinear Form on a Vector Space
4. Various Types of Bilinear Forms
5. Matrix Representation of a Bilinear Form on a Vector Space
6. Quadratic Forms on R
??
7. Hermitian Forms on a Vector Space
8. Summary
9. Exercises
10. Glossary and Further Reading
11. Solutions/Hints for Exercises
1. Learning Outcomes:
After studying this unit, you will be able to
? define the concept of a bilinear form on a vector space.
? explain the equivalence of bilinear forms with matrices.
? represent a bilinear form on a vector space as a square matrix.
? define the concept of a quadratic form on R
?? .
? explain the notion of a Hermitian form on a vector space.
Bilinear, Quadratic and Hermitian Forms
Institute of Lifelong Learning, University of Delhi pg. 3
2. Introduction:
Bilinear forms occupy a unique place in all of mathematics. The study of linear
transformations alone is incapable of handling the notions of orthogonality in geometry,
optimization in many variables, Fourier series and so on and so forth. In optimization
theory, the relevance of quadratic forms is all the more. The concept of dot product is a
particular instance of a bilinear form. Quadratic forms, in particular, play an all important
role in deciding the maxima-minima of functions of several variables. Hermitian forms
appear naturally in harmonic analysis, communication systems and representation theory.
The theory of quadratic forms derives much motivation from number theory. In short, there
are enough reasons to undertake a basic study of bilinear and quadratic forms.
We first remark that in this lesson, we shall deal with the fields F =Q,R,C only. That is, for
our purposes, ?? ?? ?? ?? (F)? 2. Now, let us begin with definitions and examples.
3. Definition of a Bilinear Form on a Vector Space:
We know that a linear functional is a scalar-valued linear transformation on a vector space.
In a similar spirit, a bilinear form on a vector space is also a scalar-valued mapping of the
vector space. The difference lies in the fact that while a linear functional is a function of a
single vector variable, a bilinear form is a function of two vector variables. In other words,
while a linear functional on a vector space ?? has the domain set ?? , a bilinear form on ?? has
the domain set the Cartesian product ?? ×?? . A bilinear form is linear in both the variables.
Hence, the name bears the adjective ‘bilinear’.
3.1. Definition: Let ?? be a vector space over a field F. A bilinear form on a vector space ??
over a field F is a mapping ?? :?? ×?? ?F such that for all ?? 1
,?? 2
,?? ,?? ?? ?? and ?? ?? F, we have,
1. ?? ?? 1
+?? 2
,?? =?? ?? 1
,?? +?? ?? 2
,?? ;
2. ?? ?? ,?? 1
+?? 2
=?? ?? ,?? 1
+?? ?? ,?? 2
;
3. ?? ?? ?? ,?? =?? ?? ?? ,?? ;
4. ?? ?? ,?? ?? =?? ?? ?? ,?? .
Thus, a bilinear form on a vector space ?? is a function on ?? ×?? such that it is linear in both
coordinates.
I.Q.1
Page 4
Bilinear, Quadratic and Hermitian Forms
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Bilinear, Quadratic and Hermitian Forms
Lesson Developer: Vivek N Sharma
College / Department: Department of Mathematics,
S.G.T.B. Khalsa College, University of Delhi
Bilinear, Quadratic and Hermitian Forms
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents
1. Learning Outcomes
2. Introduction
3. Definition of a Bilinear Form on a Vector Space
4. Various Types of Bilinear Forms
5. Matrix Representation of a Bilinear Form on a Vector Space
6. Quadratic Forms on R
??
7. Hermitian Forms on a Vector Space
8. Summary
9. Exercises
10. Glossary and Further Reading
11. Solutions/Hints for Exercises
1. Learning Outcomes:
After studying this unit, you will be able to
? define the concept of a bilinear form on a vector space.
? explain the equivalence of bilinear forms with matrices.
? represent a bilinear form on a vector space as a square matrix.
? define the concept of a quadratic form on R
?? .
? explain the notion of a Hermitian form on a vector space.
Bilinear, Quadratic and Hermitian Forms
Institute of Lifelong Learning, University of Delhi pg. 3
2. Introduction:
Bilinear forms occupy a unique place in all of mathematics. The study of linear
transformations alone is incapable of handling the notions of orthogonality in geometry,
optimization in many variables, Fourier series and so on and so forth. In optimization
theory, the relevance of quadratic forms is all the more. The concept of dot product is a
particular instance of a bilinear form. Quadratic forms, in particular, play an all important
role in deciding the maxima-minima of functions of several variables. Hermitian forms
appear naturally in harmonic analysis, communication systems and representation theory.
The theory of quadratic forms derives much motivation from number theory. In short, there
are enough reasons to undertake a basic study of bilinear and quadratic forms.
We first remark that in this lesson, we shall deal with the fields F =Q,R,C only. That is, for
our purposes, ?? ?? ?? ?? (F)? 2. Now, let us begin with definitions and examples.
3. Definition of a Bilinear Form on a Vector Space:
We know that a linear functional is a scalar-valued linear transformation on a vector space.
In a similar spirit, a bilinear form on a vector space is also a scalar-valued mapping of the
vector space. The difference lies in the fact that while a linear functional is a function of a
single vector variable, a bilinear form is a function of two vector variables. In other words,
while a linear functional on a vector space ?? has the domain set ?? , a bilinear form on ?? has
the domain set the Cartesian product ?? ×?? . A bilinear form is linear in both the variables.
Hence, the name bears the adjective ‘bilinear’.
3.1. Definition: Let ?? be a vector space over a field F. A bilinear form on a vector space ??
over a field F is a mapping ?? :?? ×?? ?F such that for all ?? 1
,?? 2
,?? ,?? ?? ?? and ?? ?? F, we have,
1. ?? ?? 1
+?? 2
,?? =?? ?? 1
,?? +?? ?? 2
,?? ;
2. ?? ?? ,?? 1
+?? 2
=?? ?? ,?? 1
+?? ?? ,?? 2
;
3. ?? ?? ?? ,?? =?? ?? ?? ,?? ;
4. ?? ?? ,?? ?? =?? ?? ?? ,?? .
Thus, a bilinear form on a vector space ?? is a function on ?? ×?? such that it is linear in both
coordinates.
I.Q.1
Bilinear, Quadratic and Hermitian Forms
Institute of Lifelong Learning, University of Delhi pg. 4
3.2. An Important Example of a Bilinear Form:
Every square matrix, having entries from a field F (=R or C), gives rise to a bilinear form. Let
?? be an ?? ×?? matrix over a field F. Then, the function
?? :F
?? ×F
?? ?F
defined by
?? ?? ,?? = ?? ?? ?? ??
is a bilinear form, on the vector space ?? = F
?? , for every pair of vectors ?? ,?? ?? F
?? . One can
easily verify the bilinearity of the mapping ?? using simple properties of matrix addition,
matrix multiplication and matrix transpose. This example demonstrates that every square
matrix over a field produces a bilinear form. Mere demonstration is not enough. In the
section 5, we shall prove that every square matrix over a field determines a bilinear form.
Let us now look at a specific instance of the mapping ?? just defined.
Example 1: Let F =Q and let ?? ?? ?? ?? 3
(Q) be the matrix ?? =
1 0 2
0 0 3
1 0 0
??
??
??
??
?
??
. Construct the
corresponding Q-bilinear form on Q
3
.
Solution: The desired bilinear form ?? :Q
3
×Q
3
?Q is
?? ?? ,?? = ?? ?? ?? ?? = ? ? u v w
1 0 2
0 0 3
1 0 0
??
??
??
??
?
??
r
s
t
??
??
??
??
??
=?? ?? + 3?? ?? -?? ?? + 2?? ?? ;
for all ?? =
u
v
w
??
??
??
??
??
?? Q
3
and ?? =
r
s
t
??
??
??
??
??
?? Q
3
.
I.Q.2
Just as there are various types of matrices (like symmetric, diagonal, upper-triangular,
skew-symmetric), there are various kinds of bilinear forms. We shall now study them.
Page 5
Bilinear, Quadratic and Hermitian Forms
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Bilinear, Quadratic and Hermitian Forms
Lesson Developer: Vivek N Sharma
College / Department: Department of Mathematics,
S.G.T.B. Khalsa College, University of Delhi
Bilinear, Quadratic and Hermitian Forms
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents
1. Learning Outcomes
2. Introduction
3. Definition of a Bilinear Form on a Vector Space
4. Various Types of Bilinear Forms
5. Matrix Representation of a Bilinear Form on a Vector Space
6. Quadratic Forms on R
??
7. Hermitian Forms on a Vector Space
8. Summary
9. Exercises
10. Glossary and Further Reading
11. Solutions/Hints for Exercises
1. Learning Outcomes:
After studying this unit, you will be able to
? define the concept of a bilinear form on a vector space.
? explain the equivalence of bilinear forms with matrices.
? represent a bilinear form on a vector space as a square matrix.
? define the concept of a quadratic form on R
?? .
? explain the notion of a Hermitian form on a vector space.
Bilinear, Quadratic and Hermitian Forms
Institute of Lifelong Learning, University of Delhi pg. 3
2. Introduction:
Bilinear forms occupy a unique place in all of mathematics. The study of linear
transformations alone is incapable of handling the notions of orthogonality in geometry,
optimization in many variables, Fourier series and so on and so forth. In optimization
theory, the relevance of quadratic forms is all the more. The concept of dot product is a
particular instance of a bilinear form. Quadratic forms, in particular, play an all important
role in deciding the maxima-minima of functions of several variables. Hermitian forms
appear naturally in harmonic analysis, communication systems and representation theory.
The theory of quadratic forms derives much motivation from number theory. In short, there
are enough reasons to undertake a basic study of bilinear and quadratic forms.
We first remark that in this lesson, we shall deal with the fields F =Q,R,C only. That is, for
our purposes, ?? ?? ?? ?? (F)? 2. Now, let us begin with definitions and examples.
3. Definition of a Bilinear Form on a Vector Space:
We know that a linear functional is a scalar-valued linear transformation on a vector space.
In a similar spirit, a bilinear form on a vector space is also a scalar-valued mapping of the
vector space. The difference lies in the fact that while a linear functional is a function of a
single vector variable, a bilinear form is a function of two vector variables. In other words,
while a linear functional on a vector space ?? has the domain set ?? , a bilinear form on ?? has
the domain set the Cartesian product ?? ×?? . A bilinear form is linear in both the variables.
Hence, the name bears the adjective ‘bilinear’.
3.1. Definition: Let ?? be a vector space over a field F. A bilinear form on a vector space ??
over a field F is a mapping ?? :?? ×?? ?F such that for all ?? 1
,?? 2
,?? ,?? ?? ?? and ?? ?? F, we have,
1. ?? ?? 1
+?? 2
,?? =?? ?? 1
,?? +?? ?? 2
,?? ;
2. ?? ?? ,?? 1
+?? 2
=?? ?? ,?? 1
+?? ?? ,?? 2
;
3. ?? ?? ?? ,?? =?? ?? ?? ,?? ;
4. ?? ?? ,?? ?? =?? ?? ?? ,?? .
Thus, a bilinear form on a vector space ?? is a function on ?? ×?? such that it is linear in both
coordinates.
I.Q.1
Bilinear, Quadratic and Hermitian Forms
Institute of Lifelong Learning, University of Delhi pg. 4
3.2. An Important Example of a Bilinear Form:
Every square matrix, having entries from a field F (=R or C), gives rise to a bilinear form. Let
?? be an ?? ×?? matrix over a field F. Then, the function
?? :F
?? ×F
?? ?F
defined by
?? ?? ,?? = ?? ?? ?? ??
is a bilinear form, on the vector space ?? = F
?? , for every pair of vectors ?? ,?? ?? F
?? . One can
easily verify the bilinearity of the mapping ?? using simple properties of matrix addition,
matrix multiplication and matrix transpose. This example demonstrates that every square
matrix over a field produces a bilinear form. Mere demonstration is not enough. In the
section 5, we shall prove that every square matrix over a field determines a bilinear form.
Let us now look at a specific instance of the mapping ?? just defined.
Example 1: Let F =Q and let ?? ?? ?? ?? 3
(Q) be the matrix ?? =
1 0 2
0 0 3
1 0 0
??
??
??
??
?
??
. Construct the
corresponding Q-bilinear form on Q
3
.
Solution: The desired bilinear form ?? :Q
3
×Q
3
?Q is
?? ?? ,?? = ?? ?? ?? ?? = ? ? u v w
1 0 2
0 0 3
1 0 0
??
??
??
??
?
??
r
s
t
??
??
??
??
??
=?? ?? + 3?? ?? -?? ?? + 2?? ?? ;
for all ?? =
u
v
w
??
??
??
??
??
?? Q
3
and ?? =
r
s
t
??
??
??
??
??
?? Q
3
.
I.Q.2
Just as there are various types of matrices (like symmetric, diagonal, upper-triangular,
skew-symmetric), there are various kinds of bilinear forms. We shall now study them.
Bilinear, Quadratic and Hermitian Forms
Institute of Lifelong Learning, University of Delhi pg. 5
4. Special Types of Bilinear Forms:
Bilinear forms of significant importance include: symmetric, skew-symmetric, and
alternating bilinear forms. The forms are conceptually inter-linked. We begin with their
definitions.
Definitions of Symmetric, Skew-Symmetric and Alternating Bilinear Forms: A
bilinear form ?? :?? ×?? ?F is symmetric if
?? ?? ,?? =?? ?? ,?? ? ?? ,?? ?? ?? ;
skew-symmetric if
?? ?? ,?? =-?? ?? ,?? ? ?? ,?? ?? ?? ;
and alternating if
?? ?? ,?? = 0 ? ?? ?? ?? .
4.1 Examples and Non-Examples of Symmetric, Skew-Symmetric and
Alternating Bilinear Forms:
(1) The usual dot product of vectors in R
?? defines a symmetric bilinear form on R
?? . The
mapping
?? : R
?? ×R
?? ?R
defined by
?? ?? ,?? =?? .?? = ?? 1
?? 1
+?? 2
?? 2
+? +?? ?? ?? ??
is a bilinear form on R
?? for every vector ?? = (?? ?? )
?? =1 1 ?? and ?? = (?? ?? )
?? =1 1 ??
in R
?? and satisfies the symmetry property
?? ?? ,?? =?? (?? ,?? ),
since the dot product is commutative.
(2) Let ?? = R
2
and its elements be viewed as column vectors. Then, the determinant map
?? ?? ?? : R
2
×R
2
?R
given by
?? ?? ??
a
b
??
??
??
,
c
d
??
??
??
=?? ?? ?? ac
bd
??
??
??
=?? ?? -?? ??
is a skew-symmetric and alternating bilinear form on R
2
. Skew-symmetry follows
because interchanging the two columns of the matrix changes the sign of the
determinant; and it is alternating because whenever the two columns are identical, the
determinant is zero.
Read More
FAQs on Lecture 2 - Bilinear, Quadratic and Hermitian Forms - Linear Algebra - Engineering Mathematics
| 1. What is a bilinear form? |  |
Ans. A bilinear form is a function that takes two vectors as input and produces a scalar as output. It is linear in each of its arguments separately. In other words, if we fix one vector and vary the other, the function behaves linearly.
| 2. What is a quadratic form? | |
Ans. A quadratic form is a function that takes a vector as input and produces a scalar as output. It is defined by a quadratic polynomial in the components of the vector. In other words, it is a homogeneous polynomial of degree two.
| 3. What is a hermitian form? | |
Ans. A hermitian form is a generalization of a bilinear form to complex vector spaces. It takes two vectors as input and produces a complex number as output. It is linear in the first argument and conjugate-linear in the second argument.
| 4. How are bilinear forms and quadratic forms related? | |
Ans. A quadratic form can be expressed as a special case of a bilinear form, where both input vectors are the same. By fixing one vector and varying the other, we can obtain various values of the quadratic form. In this sense, a quadratic form is a bilinear form that is symmetric.
| 5. What are the applications of bilinear, quadratic, and hermitian forms in engineering? | |
Ans. Bilinear, quadratic, and hermitian forms have numerous applications in engineering. They are used in optimization problems, such as finding the minimum or maximum of a function subject to certain constraints. They are also used in physics, particularly in the study of fields and potentials. Additionally, they find applications in computer graphics, signal processing, and image analysis, among other areas.
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