Page 352 - A Course in Linear Algebra with Applications

336 Chapter Nine: Advanced Topics

bilinear form; on the other hand, the form defined by the rule

zi \ Vi

2
is an example of a skew-symmetric bilinear form on R . As
the reader may suspect, there are connections with symmetric
and skew-symmetric matrices.
Theorem 9.3.2
Let f be a bilinear form on a finite-dimensional vector space
V and let A be a matrix representing f with respect to some
basis of V. Then f is symmetric if and only if A is symmetric
and f is skew-symmetric if and only if A is skew-symmetric.

Proof
T
Let A be symmetric. Then, remembering that [u] A[v] is
scalar, we have

T
T
T T
T
/(u, v) = [u] A[v] = ([u] A[v]) T = [v} A [u} = {v] A[u}
/
= (v,u).
Therefore / is symmetric. Conversely, suppose that / is sym-
metric, and let the ordered basis in question be {vi,..., v n } .
/
Then a^- = f(vi,Vj) = (VJ, Vj) = a^, so that A is symmet-
ric.
The proof of the skew-symmetric case is similar and is
left as an exercise.

Symmetric bilinear forms and quadratic forms
n
Let / be a bilinear form on R given by f(X, Y) =
T
X AY. Then / determines a quadratic form q where
T
q = f(X,X) = X AX.