Page 349 - DMTH502_LINEAR_ALGEBRA

Unit 32: Groups Preserving Bilinear Forms

Notes
Example 1: Consider a three dimensional co-ordinates (x, y, z). Let us give a rotation
along z-direction by an angle Q so that the new co-ordinates are x’, y’, z’
then
x’ = x cos = y sin
y’ = x sin + y cos
z’ = z
We see that the square of the length becomes
2
2
2
2
2
x’ + y’ + z’ = (x cos – y sin ) + (x sin + y cos ) + z 2
2
2
= x + y + z .
2
So the rotation is a transformation that preserves the bilinear form of the length. For more
details see the next section.
32.2 Groups Preserving Bilinear Forms
We start this section with a few theorems and examples.
Theorem 1: Let f be a non-degenerate bilinear form on a finite-dimensional vector space V. The
set of all linear operators on V which preserve f is a group under the operation of composition.
Proof: Let G be the set of linear operators preserving f. We observed that the identity operator
is in G and that whenever S and T are in G the composition ST is also in G. From the fact that f is
non-degenerate, we shall prove that any operator T in G is invertible, and T is also in G.
–1
Suppose T preserves f. Let be a vector in the null space of T. Then for any in V we have
f( , ) = f(T , T ) = f(0, T ) = 0.
Since f is non-degenerate, = 0. Thus T is invertible. Clearly T also preserves f; for
–1
f(T –I , T –1 ) = f(TT –1 , TT –1 ) = f( , )
If f is a non-degenerate bilinear form on the finite-dimensional space V, then each ordered basis
for V determines a group of matrices 'preserving' f. The set of all matrices [T] , where T is a
linear operator preserving f, will be a group under matrix multiplication. There is an alternative
description of this group of matrices, as follows. Let A = [f] , so that if and are vectors in V
with respective coordinate matrices X and Y relative to , we shall have
f( , ) = X’AY.
Let T be any linear operator on V and M = [T] . Then
t
f(T , T ) = (MX) A (MY)
t
t
= X (M AM)Y.
t
Accordingly, T preserves f if and only if M A M = A. In matrix language then, Theorem 1 says the
t
following: If A is an invertible n n matrix, the set of all n n matrices M such that M AM = A is
a group under matrix multiplication. If A = [f] , then M is in this group of matrices if and only if
M = [T] , where T is a linear operator which preserves f.
Let f be a bilinear form which is symmetric. A linear operator T preserves f If and only if T
preserves the quadratic form
g( ) = f( , )
associated with f. If T preserves f, we certainly have
q(T ) = f(T , T ) = f( , ) = q( )

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