This document discusses bilinear and quadratic forms. It begins by defining a bilinear form on a vector space as a function that is linear in each argument. An example given is that an inner product is a bilinear form. It then defines a symmetric bilinear form and notes that an inner product is symmetric. A theorem is presented that the matrix of a bilinear form is symmetric if and only if the form is symmetric. Finally, it defines the quadratic form associated with a symmetric bilinear form and presents a theorem relating bilinear and quadratic forms.

1. Bilinear forms
By
G.Nagalakshmi M.Sc., M.Phil.,
Assistant Professor,
V.V.Vanniaperumal College for Women,
Virudhunagar.

2. Definition:
Let V be a vector space over a field F. A bilinear form on V is a function
f : V x V→ F such that i) f ( v) = , v ) + ,v )
ii) f( u , ) =
Where
Example:
Let V be a vector space over R. Then an inner product on V is a bilinear
form on V.

3. Definition
Let V be a vector space over a field F. A bilinear form
on V is said to be symmetric bilinear form if f( u,v) = f( v,u )
for all u ,v in V.
Example:
Let V be a vector space over R. Then an inner product
on V is a symmetric bilinear form on V.

4. Theorem
A bilinear form f defined on V is symmetric iff its matrix
( aij ) with respect to any one basis { v1, v2,………vn } is
symmetric.
Proof :
Let f be a symmetric bilinear form.
Now ,
aij = f ( vi, vj )
= f (vj ,vi ) ( since f is symmetric )
= aji
(aij ) is a symmetric matrix.
Conversely , let (aij ) be a symmetric matrix.
Hence A = AT

5. Then f ( u,v ) = X A YT
= (X A YT )T ( since X A YT is a 1x1 matrix )
=Y AT XT
= Y A XT
= f( v,u )
f is a symmetric bilinear form.

6. Definition:
Let f be a symmetric bilinear form defined by V. Then the
quadratic form associated with f is the mapping q : V → F
defined by q(v) = f(v,v ). The matrix of the bilinear form f is
called the matrix of the associated quadratic form q.

7. Example
•Consider the symmetric matrix
A =
The quadratic form q determined by A with respect to the
standard basis for V3 ( R ) is given by
q(v) = X A XT
= ( x1 x2 x3 )
q= + 4 +6
is the quadratic form
+ 4
+ 4 x1 x2 +14 x2 x3 + 6 x1 x3

8. Find the quadratic form for the diagonal matrix
A =
Solution:
The quadratic form q determined by A with respect to the
standard basis for V3 ( R ) is given by
q(v) = X A XT
= ( x1 x2 x3 )
q= +2 +3

9. Theorem:
Let f be a symmetric bilinear form defined by V. Let q be
the associated quadratic form.
• f( u ,v ) =
•f( u ,v ) =
Proof:
i)Consider RHS { q( u+v)- q( u-v) }
= { f( u+v, ,u+v) – f( u-v, u-v )}
{ q( u+v)- q( u-v) }
{ q( u+v)- q( u) – q(v) }

10. = { f( u,u) + f(u,v) +f(v,u) + f( v,v) - f(u,u) + f(u,v)
+ f(v,u)- f(v,v)}
= { 4 f (u,v )}
= f ( u , v ) = LHS
Similarly (ii) also can be proved.

11. Thank You