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 ∈ V and a, b ∈ F. (ii) f(x, cy + dy0) = cf(x, y) + df(x, y0) for every x, y, y0 ∈ V and c, d ∈ 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 ∈ 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) ∈ Mn(F). Define hX,Y i = X AY , for n n X,Y ∈ 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) ∈ 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 ∈ F . n Conversely, suppose that hX,Y i = hY,Xi for every X,Y ∈ 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 = {v1, . . . , vn} be a basis for V . Given X,Y ∈ V , we can write Pn Pn X = i=1 xivi and Y = i=1 yivi, for xi, vi ∈ 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

x1 y1 . . X =  .  ∈ Fn and  .  ∈ 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 = {v1, . . . , vn}. 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 = {v1, . . . , vn} and B = {w1, . . . , wn} 0 t respectively. Then, A = QAQ , for some Q ∈ 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 ∈ GLn(F) is the basis change matrix. Let v, w ∈ 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 tA0PY. 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 ∈ F. Define

(i) (f1 + f2)(x, y) = f1(x, y) + f2(x, y);

(ii) (af1)(x, y) = af1(x, y) for x, y ∈ 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 ϕB : B(V ) → Mn(F) defined as ϕB(f) = [f]B is an isomorphism.

t Definition 4. A matrix A ∈ Mn(R) is called positive definite if X AX > n 0 for all 0 6= X ∈ 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 ∈ Mn(R). The following statements are equiva- lent:

(i) hX,Y i = XtAY , for X,Y ∈ 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 ∈ 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 ∈ 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     x1 y1 x2 y2 X =   and Y =   x3 y3 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 1   1    .  1  −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 ∈ V such that hv, vi = 0. For example, in the Lorentz 1 0 form defined above v =   is one such vector. 0 1