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Lesson: Bilinear, Quadratic and Hermitian Forms Lesson Developer: Vivek N Sharma College / Department: Department of Mathematics, S.G.T.B

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Lesson: Bilinear, Quadratic and Hermitian Forms Lesson Developer: Vivek N Sharma College / Department: Department of Mathematics, S.G.T.B Bilinear, Quadratic and Hermitian Forms 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 Institute of Lifelong Learning, University of Delhi pg. 1 Bilinear, Quadratic and Hermitian Forms 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 ℝ푛 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 ℝ푛 . explain the notion of a Hermitian form on a vector space. Institute of Lifelong Learning, University of Delhi pg. 2 Bilinear, Quadratic and Hermitian Forms 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, 푐푕푎푟(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 푉 has the domain set 푉, a bilinear form on 푉 has the domain set the Cartesian product 푉 × 푉. A bilinear form is linear in both the variables. Hence, the name bears the adjective ‘bilinear’. 3.1. Definition: Let 푉 be a vector space over a field ℱ. A bilinear form on a vector space 푉 over a field ℱ is a mapping 푓: 푉 × 푉 → ℱ such that for all 푣1, 푣2, 푣, 푤 휀 푉 and 훼 휀 ℱ, we have, 1. 푓 푣1 + 푣2, 푤 = 푓 푣1, 푤 + 푓 푣2, 푤 ; 2. 푓 푣, 푤1 + 푤2 = 푓 푣, 푤1 + 푓 푣, 푤2 ; 3. 푓 훼푣, 푤 = 훼푓 푣, 푤 ; 4. 푓 푣, 훼푤 = 훼푓 푣, 푤 . Thus, a bilinear form on a vector space 푉 is a function on 푉 × 푉 such that it is linear in both coordinates. I.Q.1 Institute of Lifelong Learning, University of Delhi pg. 3 Bilinear, Quadratic and Hermitian Forms 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 퐴 be an 푛 × 푛 matrix over a field ℱ. Then, the function 푓: ℱ푛 × ℱ푛 → ℱ defined by 푓 푥, 푦 = 푥푡 퐴푦 is a bilinear form, on the vector space 푉 = ℱ푛 , for every pair of vectors 푥, 푦 휀 ℱ푛 . One can easily verify the bilinearity of the mapping 푓 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 푓 just defined. 1 0 2 Example 1: Let ℱ = ℚ and let 퐴 휀 퐺푙3(ℚ) be the matrix 퐴 = 0 0 3 . Construct the 1 0 0 corresponding ℚ-bilinear form on ℚ3. Solution: The desired bilinear form 푓: ℚ3 × ℚ3 → ℚ is r 푡 푓 푥, 푦 = 푥 퐴푦 = u v w s t = 푢푟 + 3푣푡 − 푤푟 + 2푢푡; u 3 3 for all 푥 = v 휀 ℚ and 푦 = 휀 ℚ . w 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. Institute of Lifelong Learning, University of Delhi pg. 4 Bilinear, Quadratic and Hermitian Forms 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 푓: 푉 × 푉 → ℱ is symmetric if 푓 푢, 푣 = 푓 푣, 푢 ∀ 푢, 푣 휀 푉; skew-symmetric if 푓 푢, 푣 = −푓 푣, 푢 ∀ 푢, 푣 휀 푉; and alternating if 푓 푣, 푣 = 0 ∀ 푣 휀 푉. 4.1 Examples and Non-Examples of Symmetric, Skew-Symmetric and Alternating Bilinear Forms: (1) The usual dot product of vectors in ℝ푛 defines a symmetric bilinear form on ℝ푛 . The mapping 푓: ℝ푛 × ℝ푛 ⟶ ℝ defined by 푓 푥, 푦 = 푥. 푦 = 푥1푦1 + 푥2푦2 + ⋯ + 푥푛 푦푛 푛 is a bilinear form on ℝ for every vector 푥 = (푥푖 )푖=1 1 푛 and 푦 = (푦푖 )푖=1 1 푛 in ℝ푛 and satisfies the symmetry property 푓 푥, 푦 = 푓(푦, 푥), since the dot product is commutative. (2) Let 푉 = ℝ2 and its elements be viewed as column vectors. Then, the determinant map 푑푒푡: ℝ2 × ℝ2 ⟶ ℝ given by a c ac 푑푒푡 , = 푑푒푡 = 푎푑 − 푏푐 b d bd is a skew-symmetric and alternating bilinear form on ℝ2. 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. Institute of Lifelong Learning, University of Delhi pg. 5 Bilinear, Quadratic and Hermitian Forms (3) The bilinear form 푓: ℂ × ℂ ⟶ ℂ given by 푓 푧, 푤 = 퐼푚 푧푤 ∀ 푧, 푤 휀 ℂ is a skew-symmetric bilinear form on ℂ because, ∀ 푧, 푤 휀 ℂ, we have 푓 푧, 푤 = 퐼푚 푧푤 = −퐼푚 푤푧 = −푓(푤, 푧). (4) The bilinear form 푓: ℚ3 × ℚ3 → ℚ (example 1) given by 푓 푥, 푦 = = 푢푟 + 3푣푡 − 푤푟 + 2푢푡 for all 푥 = 휀 ℚ3 and 푦 = 휀 ℚ3 is neither symmetric nor alternating on ℚ3. Value Addition : Remarks In all characteristics, an alternating bilinear form is skew-symmetric. However, in case of characteristic not 2, a bilinear form is skew-symmetric if, and only if, it is alternating. In characteristic not 2, every bilinear form 푓 푢, 푣 on 푉 can be written uniquely as a sum of a symmetric and an alternating bilinear form, as for every 푢, 푣 휀 푉, we have 푓 푢,푣 +푓(푣,푤) 푓 푢,푣 −푓(푣,푤) 푓 푢, 푣 = + . r 2 2 s In characteristic not 2, every symmetric bilinear form 푓 푢, 푣 on an 푛-dimensional t vector space 푉 is completely determined by its values 푓 푣, 푣 on the diagonal, as for every 푢, 푣 휀 푉, we have, using the symmetry of the bilinear form 푓, 푓 푢+푣,푣+푣 −푓 푣,푣 −푓(푢,푢) 1 푓 푢, 푣 = = 푓 푢, 푣 + 푓 푣, 푢 = 푓(푢, 푣). 2 2 u This is the famous polarization identity. v w Institute of Lifelong Learning, University of Delhi pg. 6 Bilinear, Quadratic and Hermitian Forms 5. Matrix Representation of a Bilinear Form on a Vector Space: We have seen in section 3 that every square matrix over a field gives rise to a bilinear form. So, it is natural to ask the converse: does a bilinear form on a vector space also produce a matrix? In the context of finite dimensional vector spaces, the answer is YES. Every bilinear form on a finite dimensional vector space can be represented as a matrix. This is what we shall now learn. Firstly, we define the notion of a matrix of a bilinear form on a finite dimensional vector space over a field. Definition of Matrix of a Bilinear Form: Let 푉 be an 푛-dimensional vector space over a field ℱ. Let 훽 = {푒1, 푒2, … , 푒푛 } be an ordered basis for 푉 over ℱ. Then, the matrix of 푓 relative to the ordered basis 훽 is the matrix 퐴 = [푎푖푗 ]푖,푗 =1 1 푛 defined by 푎푖푗 = 푓 푒푖 , 푒푗 ∀ 푖, 푗 = 1 1 푛. This matrix is also known as inner product matrix of 푓 relative to the ordered basis 훽. Now, let us look at some explicit examples. Example 2: Calculate the matrix of the bilinear form 푑푒푡 on ℝ2 relative to the standard basis for ℝ2. Solution: The bilinear form 푓 = 푑푒푡 on ℝ2 is given by c 푓 , = 푑푒푡 , = 푑푒푡 = 푎푑 − 푏푐; d 1 0 so that with respect to the standard basis 훽 = 푒1 = , 푒2 = , we have, 0 1 a c ac b d bd Institute of Lifelong Learning, University of Delhi pg.
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