Structure of Bilinear Forms
There are three major types of bilinear forms: hermitian (or symmetric), unitary, and alternating (skew-symmetric). In this chapter, we give structure theorems giving normalized expressions for these forms with respect to suitable bases. The chapter also
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XV
Structure of Bilinear Forms
There are three major types of bilinear forms : hermitian (or symmetric), unitary, and alternating (skew-symmetric) . In this chapter, we give structure theorems giving normalized expressions for these forms with respect to suitable bases . The chapter also follows the standard pattern of decomposing an object into a direct sum of simple objects , insofar as possible .
§1.
PRELIMINARIES, ORTHOGONAL SUMS
The purpose of this chapter is to go somewhat deeper into the structure theory for our three types of forms. To do this we shall assume most of the time that our ground ring is a field, and in fact a field of characteristic -=1= 2 in the symmetric case. We recall our three definitions. Let E be a module over a commutative ring R. Let g : E x E -+ R be a map. If g is bilinear, we call g a symmetric form if g(x, y) = g(y, x) for all x, y E E. We call g alternating if g(x , x) = 0, and hence g(x, y) = - g(y, x) for all x, y E E. If R has an automorphism of order 2, written a ~ 5, we say that 9 is a hermitian form if it is linear in its first variable, antilinear in its second, and g(x , y)
= g(y, x).
We shall write g(x, y) = 0 and b, < 0 for j > s. We shall prove that r = s. Indeed, it will suffice to prove th at Proof
are linearl y independent, fo r then we get r r = s by symmetry. Suppose that
+n-
s
~
n, whence r
~
s, and
Then
Squ aring both sides yields
The left-hand side is ~ 0, and the right-hand side is ~ O. Hence both sides are equ al to 0, and it follows th at Xi = Yj = 0, in other words that our vectors are linearly independent. Corollary 4.2. A ssume that every positive element of k is a square. Then there ex ists an orth ogonal basis {Vb " " vn } of E such that vf = I for i ~ r and vf = - I f or i > r, and r is uniqu ely determined. Proof We di vide each vecto r in a n o rthogo na l basis by the sq ua re root of the absolute value of its sq ua re.
A basis having the property of the corollary is called orthonormal. If X is an element of E ha ving coord inate s (XI" '" x n ) with respect to thi s basis, then
x 2 = xi + ... + x; -
x;+
I -
.. . -
x;.
578
XV, §4
STRUCTURE OF BILINEAR FORMS
We say that a symmetric form 9 is positive definite if X 2 > 0 for all X E E, X O. This is the case if and only if r = n in Theorem 4 .1. We say that 9 is negative definite if X2 < 0 for all X E E , X O.
*
*
Corollary 4.3. The vector space E admits an orthogonal decomposition E = E+ 1- E- such that g is positive definite on E + and negative definite on E - . The dimension of E+ (or E-) is the same in all such decompositions.
Let us now assume that the form g is positive definite and that every positive element of k is a square. We define the norm of an element V E E by
IVI= ~ . Then we have Ivi > 0 if v =F O. We also have the Schwarz inequality Iv ,wl~lvllwl
for all v, WEE. This is proved in the usual way , expanding
o ~ (av ± bW)2 = by bilinearity, and letting b =
(av
± bw) . (av ± bw)
Iv I and a = Iw I.
=+= 2ab v . w
One then gets
~ 21 v 121 w 12 .
If Iv
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