Page 350 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes for every in V. Conversely, since f is symmetric, the polarization identity
f( , ) = 1 q( + ) – 1 q( – )
4 4
shows us that T preserves f provided that q( T ) = q( ) for each in V. (We are assuming here that
the scalar field is a subfield of the complex numbers.)

n
Example 2: Let V be either the space R or the space C . Let f be the bilinear form
n
n
f( , ) = x y
i i
j 1
where = (x , ... , x ) and = (y , ... , y ). The group preserving f is called the n-dimensional (real
l n 1 n
or complex) orthogonal group. The name 'orthogonal group' is more commonly applied to the
associated group of matrices in the standard ordered basis. Since the matrix of f in the standard
basis is I, this group consists of the matrices M which satisfy M M = I. Such a matrix M is called an
t
n n (real or complex) orthogonal matrix. The two n n orthogonal groups are usually denoted
O(n, R) and O(n, C). Of course, the orthogonal group is also the group which preserves the
quadratic form
q(x , ... , x ) = x + ... + x n.
2
2
1 n 1
n
Example 3: Let f be the symmetric bilinear form on R with quadratic form
p n
q(x ..., x ) = x 2 j x 2 j
1 n
j 1 j p 1
Then f is non-degenerate and has signature 2p – n. The group of matrices preserving a form of
this type is called a pseudo-orthogonal group. When p = n, we obtain the orthogonal group
O( n, R) as a particular type of pseudo-orthogonal group. For each of the n + 1 values
p = 0, 1, 2, ... n, we obtain different bilinear forms f; however, for p = k and p = n – k the forms are
negatives of one another and hence have the same associated group. Thus, when n is odd, we
have (n + 1)/2 pseudo-orthogonal groups of n n matrices, and when n is even, we have
(n + 2)/2 such groups.
Theorem 2: Let V be an n-dimensional vector space over the field of complex numbers, and let f
be a non-degenerate symmetric bilinear form on V. Then the group preserving f is isomorphic
to the complex orthogonal group O(n, C).
Proof: Of course, by an isomorphism between two groups, we mean a one-one correspondence
between their elements which 'preserves' the group operation. Let G be the group of linear
operators on V which preserve the bilinear form f. Since f is both symmetric and non-degenerate,
Theorem 4 of unit 30 tells us that there is an ordered basis for V in which f is represented by the
n n identity matrix. Therefore, a linear operator T preserves f if and only if its matrix in the
ordered basis is a complex orthogonal matrix. Hence
T [T]
is an isomorphism of G onto O(n, C).

Theorem 3: Let V be an n-dimensional vector space over the field of real numbers, and let f be a
non-degenerate symmetric bilinear form on V. Then the group preserving f is isomorphic to an
n n pseudo-orthogonal group.
Proof: Repeat the proof of Theorem 2, using Theorem 5 of unit 30 instead of Theorem 4 of
unit 30.

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