CHAPTER 6. BILINEAR FORMS MODULE 1. DEFINITION AND BASIC PROPERTIES INDRANATH SENGUPTA Contents 1. Bilinear forms 1 2. Matrix representation of bilinear forms 2 Let V denote a vector space over a field F of characteristic other than 2. 1. Bilinear forms Definition 1. A bilinear form f is a map f : V × V ! F, such that f is bilinear if the following properties are satisfied (i) f(ax + bx0; y) = af(x; y) + bf(x0; y) for every x; x0; y 2 V and a; b 2 F. (ii) f(x; cy + dy0) = cf(x; y) + df(x; y0) for every x; y; y0 2 V and c; d 2 F. In other words, f is bilinear if it is separate linear in each variable. Notation. We usually write f(x; y) = hx; yi for x; y 2 V . Definition 2. The bilinear form f is said to be symmetric if f(x; y) = f(y; x). It is called skew-symmetric if f(x; y) = −f(y; x). Remark 1.1. Note that the characteristic of the field is 0 and hence −1 6= 1. n Example 1.2. The usual inner product on R is a symmetric bilinear n form on R . 1 2 Module 1 t Example 1.3. Let A = (aij) 2 Mn(F). Define hX; Y i = X AY , for n n X; Y 2 F . This is a bilinear form on F . Let F = R. n (i) If A = In, then it is the usual inner product on F . 1 −1 (ii) If A = , then −1 1 1 −1 y1 f((x1; x2); (y1; y2)) = (x1x2) = x1y1−x2y1−x1y2+x2y2: −1 1 y2 Proposition 1.4. Let A = (aij) 2 Mn(F). The bilinear form hX; Y i = XtAY is symmetric if and only if A is a symmetric matrix. Proof. Suppose that At = A. Then Y tAX = (Y tAX)t = XtAtY = n XtAY . Therefore, hX; Y i = hY; Xi, for every X; Y 2 F . n Conversely, suppose that hX; Y i = hY; Xi for every X; Y 2 F . t Then, hei; eji = eiAej = aij, and hej; eii = aji. Hence, A is a symmetric matrix. 2. Matrix representation of bilinear forms Let h; i be a bilinear form on a finite dimensional vector space V . Let B = fv1; : : : ; vng be a basis for V . Given X; Y 2 V , we can write Pn Pn X = i=1 xivi and Y = i=1 yivi, for xi; vi 2 F. By the bilinearity of h ; i, we get n n X X X hX; Y i = h xivi; yjvji = xiyjhvi; vji: i=1 j=1 1≤i;j≤n If we identify X and Y with the n-tuples 0x11 0y11 . X = @ . A 2 Fn and @ . A 2 Fn; xn yn then we can write hX; Y i = XtAY: Definition 3. The matrix A = (xij) is called the matrix of the bilinear form f = h ; i, with respect to the basis B = fv1; : : : ; vng. We use the notations [f]B and h ; iB to represent the matrix A associated with the bilinear form. Chapter 6. Bilinear Forms 3 The following theorem shows how matrices representing a bilinear form with respect to different bases are related. Theorem 2.1. Let A and A0 be matrices representing the bilinear form 0 h ; i on V , with respect to bases B = fv1; : : : ; vng and B = fw1; : : : ; wng 0 t respectively. Then, A = QAQ , for some Q 2 GLn(F). Proof. Let us write the bases as ordered tuples B = (v1; : : : ; vn) and 0 n B = (w1; : : : ; wn) of V . Then, we can write B = B0 · P; where P 2 GLn(F) is the basis change matrix. Let v; w 2 V be given by the ordered tuples X and Y respectively, with respect to the basis B. Similarly, let X0 and Y 0 be the ordered tuples representing v and w respectively, with respect to the basis B. Then, X0 = PX and Y 0 = PY . Therefore, hv; wi = XtAY with respect to the basis B0 and hv; wi = X0tA0Y 0 with respect to the basis B0. We get XtAY = hv; wi = X0tA0Y 0 = XtP tA0P Y: t 0 t −1 −1 t 0 t Hence, A = P A P . Let Q = (P ) = (P ) ; then A = QAQ . Let f1 and f2 be two bilinear forms on V and a 2 F. Define (i) (f1 + f2)(x; y) = f1(x; y) + f2(x; y); (ii) (af1)(x; y) = af1(x; y) for x; y 2 V . Then, f1 + f2 and af1 defined above are bilinear forms on V . Let B(V ) denote the collection of all bilinear forms on V . Then, with respect to the addition and scalar multiplication defined above the set B(V ) is a vector space over F. It is not difficult to prove that Theorem 2.2. Let V be a vector space of dimension n over F. Let B be an ordered basis for V . The map 'B : B(V ) ! Mn(F) defined as 'B(f) = [f]B is an isomorphism. t Definition 4. A matrix A 2 Mn(R) is called positive definite if X AX > n 0 for all 0 6= X 2 R . Theorem 2.3. A real symmetric matrix is positive definite if and only if det(Ai) > 0 for all i = 1; : : : ; n, where Ai is the upper left i × i submatrix of A. 4 Module 1 Theorem 2.4. Let A 2 Mn(R). The following statements are equiva- lent: (i) hX; Y i = XtAY , for X; Y 2 Rn represents an inner product on Rn; (ii) A is symmetric and positive definite. n Proof. If hX; Y i = XtAY represents an inner product on R , then A is n symmetric and hX; Xi = XtAX > 0 for all 0 6= X 2 R . Conversely, if A is symmetric and positive definite then hX; Y i = XtAY satisfies n all the conditions of an inner product on R . Definition 5. A symmetric bilinear form on a finite dimensional real vector space V is called positive definite if hv; vi > 0, for all 0 6= v 2 V . Example 2.5 (Lorentz form). A form h ; i such that hv; vi assumes both positive and negative values is called indefinite. The Lorentz form 2 hX; Y i = x1y1 + x2y2 + x3y3 − c x4y4; where 0 1 0 1 x1 y1 Bx2C By2C X = B C and Y = B C @x3A @y3A x4 y4 4 is a typical example of an indefinite form on space-time R . The coefficient c represents the speed of light and can be normalized to 1, and then the matrix of the form with respect to the given basis becomes 01 1 B 1 C B C : @ 1 A −1 We aim to classify all symmetric forms on finite dimensional real vector spaces. The main difficulty in this classification is created by the exis- tence of 0 6= v 2 V such that hv; vi = 0. For example, in the Lorentz 011 B0C form defined above v = B C is one such vector. @0A 1.