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

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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.

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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

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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.

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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.

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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 u r  3  3 푓 푥, 푦 = = 푢푟 + 3푣푡 − 푤푟 + 2푢푡 for all 푥 = v 휀 ℚ and 푦 = s 휀 ℚ is neither w t   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 = + . 2 2  In characteristic not 2, every symmetric bilinear form 푓 푢, 푣 on an 푛-dimensional 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 This is the famous polarization identity.

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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 a c c ac 푓 ,  = 푑푒푡 ,  = 푑푒푡 = 푎푑 − 푏푐; b d d bd

1 0 so that with respect to the standard basis 훽 = 푒1 = , 푒2 =  , we have, 0 1

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Bilinear, Quadratic and Hermitian Forms

11 10 푎11 = 푓 푒1, 푒1 = 푑푒푡 = 0; 푎12 = 푓 푒1, 푒2 = 푑푒푡 = 1; 00 01 01 00 푎21 = 푓 푒2, 푒1 = 푑푒푡 = −1; 푎22 = 푓 푒2, 푒2 = 푑푒푡 = 0; 10 11 01 and therefore, the matrix of this bilinear form 푑푒푡 is 퐴 = . 10 Example 3: Compute the matrix of the ℚ-bilinear form 푓: ℚ3 × ℚ3 → ℚ given by 푓 푥, 푦 = 3 3 푥1푦2 + 푥3푦2 + 푥2푦1, ∀ 푥 = 푥1, 푥2, 푥3 , 푦 = 푦1, 푦2, 푦3 휀 ℚ relative to the standard basis for ℚ . Solution: The standard basis for ℚ3 is the usual basis

훽 = 푒1 = 1, 0, 0 , 푒2 = 0, 1, 0 , 푒3 = (0, 0, 1) ; and therefore, we have,

푎11 = 푓 푒1, 푒1 = 0; 푎12 = 푓 푒1, 푒2 = 1; 푎13 = 푓 푒1, 푒3 = 0;

푎21 = 푓 푒2, 푒1 = 1; 푎22 = 푓 푒2, 푒2 = 0; 푎23 = 푓 푒2, 푒3 = 0;

푎31 = 푓 푒3, 푒1 = 0; 푎32 = 푓 푒3, 푒2 = 1; 푎33 = 푓 푒3, 푒3 = 0. 0 1 0  Hence, the matrix of this bilinear form 푓 is 퐴 = 1 0 0 .  0 1 0 I.Q.3

We shall now formally prove that every square matrix is a matrix of a bilinear form; and that every bilinear form is completely determined by its matrix.

Theorem 1: Let 푉 be an 푛-dimensional vector space over a field ℱ. (i) Every 푛 × 푛 matrix 퐴 over ℱ is the matrix of some bilinear form 푓 on 푉 relative to some ordered basis of 푉. (ii) The matrix of any bilinear form 푓 on 푉 relative to any ordered basis of 푉 completely determines 푓 on 푉.

Proof: The proofs for both the parts are separate.

(i) Let 퐴 = [푎푖푗 ]푖,푗 =1 1 푛 be any 푛 × 푛 matrix over ℱ. The mapping 푓: 푉 × 푉 → ℱ defined by

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Bilinear, Quadratic and Hermitian Forms

y1  y 2 . 푡 푓 푥, 푦 = 푥 퐴푦 = 푥1, 푥2, … , 푥푛 . 퐴.  . .  yn defines a bilinear form on 푉. Now, let

훽 = 푒푖 = 0, 0, … , 1푖 , … , 0 푖 = 1 1 푛} be the canonical basis for 푉 over ℱ. Then, we can see that, ∀ 푖, 푗 = 1 1 푛,

푡 푓 푒푖, 푒푗 = 푒푖 . 퐴. 푒푗 0  . ..... .  ..... .  = 0, 0, … , 1푖 , … , 0 ....a 1 ij j ..... .   ..... . .  0

= 푎푖푗 and therefore, the matrix 퐵 of 푓 is 퐴 itself:

퐵 = [푎푖푗 ]푖,푗 =1 1 푛 = 퐴.

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Bilinear, Quadratic and Hermitian Forms

Thus, the square 푛 × 푛 matrix 퐴 is the matrix of the bilinear form 푓 relative to the standard basis of 푉 over ℱ. Since our choice of the matrix 퐴 was arbitrary, the assertion holds good in the general case. The proof is now complete. (ii) Let the square 푛 × 푛 matrix 퐴 given by

퐴 = [푎푖푗 ]푖,푗 =1 1 푛 be the matrix of a bilinear form 푓: 푉 × 푉 → ℱ relative to a basis

훽 = 푣1, 푣2, … , 푣푛 for 푉 over ℱ. Thus, ∀ 푖, 푗 = 1 1 푛, we have,

푎푖푗 = 푓(푣푖 , 푣푗 ). (A) We want to show that the value of this bilinear form 푓(푥, 푦) at any pair of vectors 푥, 푦 휀 푉 can be obtained from the entries of the matrix 퐴. Since 훽 is a basis for 푉 over ℱ

and 푥, 푦 휀 푉, there exist scalars 푥푖 , 푦푖 휀 ℱ for 푖 = 1 1 푛, such that,

n n 푥 = and 푦 = (B)  xiiv  yvii i1 i1 so that, we have,

푓 푥, 푦 = 푓( , 푦) (By (B))

n = (By linearity in first variable)  xii f(,) v y i1

nn = xii f(,) v yjj v (By (B)) ij11

nn = xi yj f(,) vi v j (By linearity in second variable) ij11

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Bilinear, Quadratic and Hermitian Forms

nn (By (A)); = xyi j 푎푖푗 ij11 and thus, knowing the matrix of a bilinear form relative to any basis completely determines the bilinear form. This completes the proof.

The moral of the above theorem is that the nature of the matrix reveals a lot about the behavior of the corresponding bilinear form.

Value Addition : Remarks  A bilinear form is symmetric if, and only if, its matrix relative to some basis is symmetric.  A bilinear form is alternating if, and only if, its matrix relative to some basis is skew- symmetric.

Now, a natural follow up question would be: what is the impact of basis change on the matrix representation of a bilinear form? The answer is contained in the following beautiful theorem.

Theorem 2: Let 푉 be an 푛-dimensional vector space over a field ℱ. Let 퐴 be the matrix of a bilinear form 푓: 푉 × 푉 → ℱ relative to some ordered basis 훽 for 푉 over ℱ. If 훽′ is another ordered basis for 푉 over ℱ, then the matrix representation of 푓 relative to the basis 훽′ is 푃푡 퐴푃 where 푃 is the transition matrix describing the basis change 훽′ ⟶ 훽.

′ Proof: Let 훽 = 푣1, 푣2, … , 푣푛 and 훽 = 푣1′, 푣2′, … , 푣푛 ′ . Since 푃 = [푝푖푗 ]푖,푗 =1 1 푛 is the transition matrix describing the basis change 훽′ ⟶ 훽, we have,

n 푣′ = ∀ 푗 = 1 1 푛. (A) 푗  pvij i i1

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′ Hence, the new matrix 퐴 = [푎푖푗 ′]푖,푗 =1 1 푛 representing the bilinear form relative the ordered basis 훽′ is given by ′ ′ ′ 푎푖푗 = 푓(푣푖 , 푣푗 )

n n = 푓( , ) (By (A))  pvki k  pvlj l k1 l1

nn = 푓 푣 , 푣 (By the bilinearity of 푓)  ppki lj 푘 푙 kl11

= 푎푘푙 (Because 푎푘푙 = 푓 푣푘 , 푣푙 ∀ 푘, 푙 = 1 1 푛); that is,

nn 푎′ = 푖푗  ppkiakl lj k11l so that, we get, ∀ 푖, 푗 = 1 1 푛, 푡 nn 푎′ = 푖푗  ppik akl lj k11l confirming the theorem that 퐴′ = 푃푡 퐴푃. The proof is now complete.

Let us see this theorem in actual practice.

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Example 4: Consider the ℝ-bilinear form on ℝ3, relative to the standard basis, given by

3 푥, 푦 = 2푥1푦1 − 3푥2푦2 + 푥3푦3 ∀ 푥 = 푥1, 푥2, 푥3 , 푦 = 푦1, 푦2, 푦3 휀 ℝ . Determine the matrix ′ representation of the equivalent form relative to the ordered basis 훽 = {푣1 = 1, 1, 1 , 푣2 =

−2, 1, 1 , 푣3 = 2, 1, 0 }. Solution: We first calculate the matrix 퐴 representing the given bilinear form relative to the standard basis for ℝ3. We are given that

3 푓 푥, 푦 = 2푥1푦1 − 3푥2푦2 + 푥3푦3 ∀ 푥 = 푥1, 푥2, 푥3 , 푦 = 푦1, 푦2, 푦3 휀 ℝ , from which we obtain

푎11 = 푓 푒1, 푒1 = 2; 푎12 = 푓 푒1, 푒2 = 0; 푎13 = 푓 푒1, 푒3 = 0;

푎21 = 푓 푒2, 푒1 = 0; 푎22 = 푓 푒2, 푒2 = 0; 푎23 = 푓 푒2, 푒3 = −3;

푎31 = 푓 푒3, 푒1 = 0; 푎32 = 푓 푒3, 푒2 = 1; 푎33 = 푓 푒3, 푒3 = 1. Hence, the matrix of this bilinear form 푓 relative to the standard basis for ℝ3 is 퐴 =

2 0 0  0 0 3 .  0 0 1 1 2 2  Now, the matrix 푃 describing the basis change 훽′ ⟶ 훽 is 푃 = 1 1 1 .  1 1 0 Hence, the required matrix 퐴′ for the new equivalent bilinear form is given by 퐴′ = 푃푡 퐴푃 푡

1 1 1  = 2 1 1  2 1 0

0 6 4  = 6 6 8 .  1 11 8

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6. Quadratic Forms on ℝ풏: Symmetric matrices and symmetric bilinear forms are fascinating in their own right. Both are equivalent and arise naturally in various contexts. Symmetric bilinear forms give rise to what are known as quadratic forms. In optimization theory, machine learning, applied probability, and above all, in number theory, quadratic forms have deep and serious applications. There are many open problems regarding quadratic forms.

Let us begin with the definition. In this section, we take ℱ = ℝ.

Definition of a Quadratic Form on a Vector Space: Let 푉 be an 푛-dimensional vector space over ℝ. Let 푓 be a symmetric bilinear form on 푉 with the matrix 퐴 relative to the standard basis for 푉. The mapping 푞: 푉 → ℝ defined by 푞 푥 = 푓 푥, 푥 = 푥푡 퐴푥 ∀ 푥 휀 푉 is called a quadratic form on the vector space 푉. Equivalently, we say that an expression of the form

n n 푞 푥 = ∀ 푥 = (푥 , 푥 , … , 푥 ) 휀 푉 aijxx i j 1 2 푛 i1 j1 defines a quadratic form 푞: 푉 → ℝ on the vector space 푉.

There is a special way to write the matrix for a quadratic form. We illustrate it through examples.

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Example 5: Determine the matrix of the quadratic form 푞: ℝ2 → ℝ given by 푞 푥, 푦 = 푥2 − 푥푦 + 푦2. Solution: We write the quadratic form 푞 푥, 푦 = 푥2 − 푥푦 + 푦2 as 1 1 푞 푥, 푦 = 푥푥 − 푥푦 − 푦푥 + 푦푦 2 2 and hence the matrix of this quadratic form is 1 1  2 퐴 = . 1  1 2 We may verify that ∀ 푢 = 푥, 푦 휀 ℝ2,

x 푢푡 퐴푢 = xy = 푥2 − 푥푦 + 푦2 = 푞(푥, 푦). y

Example 6: Determine the matrix of the quadratic form 푞: ℝ2 → ℝ given by 푞 푥, 푦 = 49푥2 − 72푥푦 + 121푦2. Solution: We write the quadratic form 푞 푥, 푦 = 49푥2 − 72푥푦 + 121푦2 as 푞 푥, 푦 = 49. 푥푥 − 36. 푥푦 − 36. 푦푥 + 121푦푦 and hence the matrix of this quadratic form is

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Bilinear, Quadratic and Hermitian Forms

49 36 퐴 = . 36 121 We may verify that ∀ 푢 = 푥, 푦 휀 ℝ2, x 푢푡 퐴푢 = xy = 49푥2 − 72푥푦 + 121푦2 = 푞(푥, 푦). y Example 7: Determine the matrix of the quadratic form 푞: ℝ3 → ℝ given by 푞 푥, 푦, 푧 = 푥2 + 2푦2 − 5푧2 − 푥푦 + 4푦푧 − 3푧푥. Solution: We write the quadratic form 푞 푥, 푦, 푧 = 푥2 + 2푦2 − 5푧2 − 푥푦 + 4푦푧 − 3푧푥 as 1 1 3 3 푞 푥, 푦, 푧 = 푥푥 + 2푦푦 − 5푧푧 − 푥푦 − 푦푥 + 2푦푧 + 2푧푦 − 푧푥 − 푥푧 2 2 2 2 and rearranging ‘row-wise’, we get

1 3 1 3 푞 푥, 푦, 푧 = 푥푥 − 푥푦 − 푥푧 − 푦푥 + 2푦푦 + 2푦푧 − 푧푥 + 2푧푦 − 5푧푧. 2 2 2 2

Hence, the matrix of this quadratic form is 13 1  22  1 퐴 =  22. 2 3 25 2 We may verify that ∀ 푢 = 푥, 푦, 푧 휀 ℝ3,

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Bilinear, Quadratic and Hermitian Forms

13 1  22 x 1  푢푡 퐴푢 = x y z  22y = 푥2 + 2푦2 − 5푧2 − 푥푦 + 4푦푧 − 3푧푥 2  z 3  25 2 = 푞(푥, 푦, 푧). 6.1 Canonical way of writing a quadratic form: How can we simplify complicated quadratic forms into the simple, manageable ones? This question was satisfactorily handled by Sylvester and Lagrange in 19th century. Their contributions are contained in the following theorems.

Theorem 3: Every square matrix of order 푛 is congruent to a unique matrix of the form

Ir  퐵 = I . s  O Theorem 4 (Sylvester): Let 푉 be an 푛-dimensional vector space over ℝ endowed with a quadratic form 푞. Then, there exists an ordered basis 훽 = 푣1, 푣2, … , 푣푛 of 푉 such that ∀ 푥 휀 푉, we have,

n 푥 = ⟹ 푞 푥 = (푥 )2 + (푥 )2 + ⋯ + (푥 )2 − (푥 )2 − ⋯ − (푥 )2. iiv 1 2 푟 푟+1 푠 i1 The integers 푟 and 푠 do not depend upon the choice of the basis. Their sum 푟 + 푠 is called the rank of the quadratic form and the difference 푟 − 푠 is called the signature of the quadratic form. Alternatively, the rank of a quadratic form is also defined to the rank of its matrix representation.

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Bilinear, Quadratic and Hermitian Forms

While Sylvester proved the existence of this canonical form, it was Lagrange who gave a simple algorithm of ‘completing the squares’ to reduce a quadratic form to its canonical form. We illustrate the process using examples. Example 8: Determine the rank and signature of the quadratic form 푞: ℝ3 → ℝ given by 푞 푥, 푦, 푧 = 4푥2 + 10푦2 + 11푧2 − 4푥푦 − 12푦푧 + 12푧푥 by reducing it to its canonical form. Solution: We see that 푞 푥, 푦, 푧 = 4푥2 + 10푦2 + 11푧2 − 4푥푦 − 12푦푧 + 12푧푥 1 3 1 = 4(푥 − 푦 + 푧)2 + 9(푦 − 푧)2 + 푧2 2 2 3 so that setting 1 3 1 푢 = 2(푥 − 푦 + 푧); 푣 = 3(푦 − 푧); and 푤 = 푧; 2 2 3 we obtain the desired canonical form as 푞 푢, 푣, 푤 = 푢2 + 푣2 + 푤2; and hence, signature is 3. Further, the matrix for the canonical form is

1 0 0  퐴 = 0 1 0 0 0 1  and therefore, rank is 3.

Example 9: Determine the rank and signature of the quadratic form 푞: ℝ3 → ℝ given by 푞 푥, 푦, 푧 = 2푥푦 + 2푦푧 by reducing it to its canonical form. Solution: We see that 푞 푥, 푦, 푧 = 2푥푦 + 2푦푧 1 = [ 푥 + 푦 + 푧 2 − (푥 − 푦 + 푧)2] 2

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Bilinear, Quadratic and Hermitian Forms

1 00 2  1 so that the matrix for the canonical form is 퐴 = 00 , and hence the 2 0 0 0   rank is 2. Further, the signature is zero. 7. Hermitian Forms on a Vector Space: Signals and systems in electrical engineering heavily depend upon Fourier Analysis. And a Fourier Transform is a typical example of a Hermitian Form on a complex vector space. Hermitian forms are defined only for complex vector spaces. They differ from the usual bilinear forms in a subtle way. We begin with the definition.

Definition of Hermitian Form: A Hermitian form on a complex vector space 푉 is a map

<, >: 푉 × 푉 ⟶ ℂ such that ∀ 푧1, 푧2, 푤1, 푤2 휀 푉 and 휇 휀 ℂ one has

1. < 푧1 + 푧2, 푤1 >=< 푧1, 푤1 > +< 푧2, 푤1 >;

2. < 푧1, 푤1 + 푤2 >=< 푧1, 푤1 > +< 푧1, 푤2 >;

3. < 휇푧1, 푤1 >= 휇 < 푧1, 푤1 >;

4. < 푧1, 휇푤1 >= 휇 < 푧1, 푤1 >. 7.1 Examples of Hermitian Forms: (1) On 푉 = ℂ푛 we have the standard Hermitian form given as

< 푧1, 푧2, … , 푧푛 , 푤1, 푤2, … , 푤푛 >= 푧 1푤1 + 푧 2푤2 + ⋯ + 푧 푛 푤푛

for every vector 푧1, 푧2, … , 푧푛 and 푤1, 푤2, … , 푤푛 휀 ℂ. (2) On the vector space of all square 푛 × 푛 matrices over ℂ, the trace map defines a Hermitian form: < 퐴, 퐵 >= 푡푟 퐴∗퐵 .

At last, we have the matrix representation for Hermitian forms.

7.2 Matrix Representation of Hermitian Forms: Let 푉 be an 푛-dimensional complex vector space endowed with a Hermitian form < , >. The matrix representing this form relative to a basis 훽 = 푣1, 푣2, … , 푣푛 of 푉 is

퐴 = [푎푖푗 ]푖,푗 =1 1 푛 , where, 푎푖푗 =< 푣푖 , 푣푗 > ∀ 푖, 푗 = 1 1 푛.

I.Q.4

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Bilinear, Quadratic and Hermitian Forms

8. Summary:

(i) A bilinear form on a vector space 푉 is a function 푉 × 푉 ⟶ ℱ such that it is linear in both coordinates. (ii) 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 . (iii) 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. (iv) Polarization Identity: In characteristic not 2, every symmetric bilinear form 푓 푢, 푣 on an 푛-dimensional 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 (v) Matrix Representation of a Bilinear Form: Let 푉 be an 푛-dimensional vector space over a field ℱ. Then, (a) Every 푛 × 푛 matrix 퐴 over ℱ is the matrix of some bilinear form 푓 on 푉 relative to some ordered basis of 푉. (b) The matrix of any bilinear form 푓 on 푉 relative to any ordered basis of 푉 completely determines 푓 on 푉. (vi) A bilinear form is symmetric if, and only if, its matrix relative to some basis is symmetric. (vii) Let 푉 be an 푛-dimensional vector space over a field ℱ. Let 퐴 be the matrix of a bilinear form 푓: 푉 × 푉 → ℱ relative to some ordered basis 훽 for 푉 over ℱ. If 훽′ is another ordered basis for 푉 over ℱ, then the matrix representation of 푓 relative to the basis 훽′ is 푃푡 퐴푃 where 푃 is the transition matrix describing the basis change 훽′ ⟶ 훽.

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Bilinear, Quadratic and Hermitian Forms

(viii) To every square 푛 × 푛 matrix 퐴 over ℝ, the mapping 푞: 푉 → ℝ given by 푞 푥 = 푓 푥, 푥 = 푥푡 퐴푥 ∀ 푥 휀 푉 defines a quadratic form on an 푛-dimensional real vector space 푉.

(ix) Every square matrix of order 푛 is congruent to a unique matrix of the form

Ir  퐵 = I . s  O (x) Sylvester’s Theorem: Let 푉 be an 푛-dimensional vector space over ℝ endowed

with a quadratic form 푞. Then, there exists an ordered basis 훽 = 푣1, 푣2, … , 푣푛 of 푉 such that ∀ 푥 휀 푉, we have,

n 푥 = ⟹ 푞 푥 = (푥 )2 + (푥 )2 + ⋯ + (푥 )2 − (푥 )2 − ⋯ − (푥 )2. iiv 1 2 푟 푟+1 푠 i1 (xi) The integers 푟 and 푠 do not depend upon the choice of the basis. Their sum 푟 + 푠 is called the rank of the quadratic form and the difference 푟 − 푠 is called the signature of the quadratic form. Alternatively, the rank of a quadratic form is also defined to the rank of its matrix representation. (xii) Hermitian forms are defined only for complex vector spaces. A Hermitian form on a complex vector space 푉 is a function < , >: 푉 × 푉 ⟶ ℱ such that it is linear in one coordinate and conjugate-linear in the other coordinate.

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Bilinear, Quadratic and Hermitian Forms

Exercises

1. Construct the bilinear form the following matrices:

1 2 5  (a) 퐴1 = 2 0 1  0 6 6

3 1 3  (b) 퐴2 = 2 3 0  0 0 0 2. Determine the matrix of the following bilinear forms relative to the standard basis of appropriate vector space:

(a) 2푥1푦1 − 3푥1푦3 + 2푥2푦2 − 5푥2푦3 + 4푥3푦1

(b) 2푥1푦1 + 푥1푦2 + 푥1푦3 + 3푥2푦1 − 2푥2푦3 + 푥3푦2 − 5푥3푦3

3. In each of the two cases below, a bilinear form is given relative to the standard basis of appropriate vector space. Determine the matrix representation relative to the new ordered basis given for each case:

′ (a) 푥1푦1 + 푥1푦2 + 푥2푦1 + 푥2푦2; 훽 = {푣1 = 1, −1 , 푣2 = 1, 1 } ′ (b) 2푥1푦1 + 푥2푦3 − 3푥3푦2 + 푥3푦3; 훽 = {푣1 = 1,2,3 , 푣2 = −1,1 2 , 푣3 = (1,2,1) }

4. Construct the quadratic form the following matrices:

0 5 1  (a) 퐴3 = 5 1 6  1 6 2

a h g  (b) 퐴4 = h b f  g f c

5. Reduce each of the following quadratic forms to their respective canonical forms. Hence, determine their rank and signature. (a) 2푦2 − 푧2 + 푥푦 + 푥푧 (b) 푦푧 + 푥푧 + 푥푦 + 푥푡 + 푦푡 + 푧푡

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Bilinear, Quadratic and Hermitian Forms

Glossary Bilinear form, symmetric bilinear form, alternating bilinear form, matrix of a bilinear form, quadratic form, rank, signature, Hermitian form