Page 341 - DMTH502_LINEAR_ALGEBRA

Unit 30: Bilinear Forms and Symmetric Bilinear Forms

The number Notes
+
dim V – dim V –
+
is often called the signature of f. It is introduced because the dimensions of V and V are easily
–
determined from the rank of f and the signature of f.
Perhaps we should make one final comment about the relation of symmetric bilinear forms on
real vector spaces to inner products. Suppose V is a finite-dimensional real vector space and that
V , V , V are subspaces of V such that
1 2 3
V = V V V
1 2 3
Suppose that f is an inner product on V , and f is an inner product on V . We can then define a
1 I 2 2
symmetric bilinear form f on V as follows: If , are vectors in V, then we can write
= + + and = + +
l 2 3 1 2 3
with . and in V . Let
j j j
f( , ) = f ( + ) – f ( + )
l 1 1 2 2 2
+
The subspace V for f will be V , V is a suitable V for f, and V is a suitable V . One part of the
–
3 1 2
statement of Theorem 5 is that every symmetric bilinear form on V arises in this way. The
additional content of the theorem is that an inner product is represented in some ordered basis
by the identity matrix.
Self Assessment

3. Let V be a finite-dimensional vector space over a subfield F of the complex numbers and
let S be the set of all symmetric bilinear forms in V. Show that S is a subspace of L(V, V, F).
4. The following expressions define quadratic forms q on R . Find the symmetric bilinear
2
form f corresponding to each q.
(a) ax 2
1
(b) x + 9x 2
2
1 2
(c) bx x
1 2
30.3 Summary

 In this unit concept of bilinear form is introduced.
 It is seen that there a strong relation between bilinear forms and inner products.

 The isomorphism between the space of bilinear forms and the space of n × n matrices is
established.
 The rank of a bilinear form is defined and non-degenerate bilinear forms are introduced.

30.4 Keywords

A Bilinear Form: A bilinear form on V is a function f, which assigns to each pair of vectors, ,
in V a scalar f( , ) in F, and satisfies linear relations.
A non-degenerate bilinear form on a vector space V is a bilinear form if for each non-zero in
V, there is in V such that f( , ) 0 as well as for each non-zero in V, there is and in V such
that f( , ) 0.

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