Page 351 - A Course in Linear Algebra with Applications

9.3: Bilinear Forms 335

The point that has emerged from the preceding discussion
is that matrices which represent the same bilinear form with
respect to different bases of the vector space are congruent.
This result is to be compared with the fact that the matrices
representing the same linear transformation are similar.
The conclusions of the the last few paragraphs are sum-
marized in the following basic theorem.

Theorem 9.3.1
(i) Let f be a bilinear form on an n-dimensional vector space
V over a field F and let B = {vi,..., v n } be an ordered basis
of V. Define A to be the n x n matrix whose (i,j) entry is
/(v», Vj); then

T
/(u,v) = ([u] B ) A[v] B ,

and A is the n x n matrix representing f with respect to B.
(ii) If B' is another ordered basis of V, then f is represented
with respect to B' by the matrix S T AS where S is the invertible
matrix describing the basis change B' —> B.
(iii) Conversely, if A is any n x n matrix over F, a bilinear
form on V is defined by the rule (u, v) = ([U]B) A[V]B- It
T
/
is represented by the matrix A with respect to the basis B.
Symmetric and skew-symmetric bilinear forms
A bilinear form / on a vector space V is called symmetric
if its values are unchanged by reversing the arguments, that
is, if
/(u,v) = /(v,u)

for all vectors u and v. Similarly, / is said to be skew-
symmetric if
/(u,v) = - / ( v , u )

/
is always valid. Notice the consequence, (u, u) = 0 for all
vectors u. For example, any real inner product is a symmetric