Page 337 - DMTH502_LINEAR_ALGEBRA

Unit 30: Bilinear Forms and Symmetric Bilinear Forms

Self Assessment Notes

2
1. Which of the following functions f, defined on vectors = (x , x ) and (y , y ) in R , are
1 2 1 2
bilinear forms?
(a) f( , ) = (x – y ) + x y
2
1 I 2 2
(b) f( , ) = (x + y ) + (x – y ) 2
2
1 I 1 1
(c) f( , ) = x y – x y
1 2 2 1
2. Let f be any bilinear form on a finite-dimensional space V. Let W be the subspace of all
such that f( , ) = 0 for every . Show that
rank f = dim V – dim W.

30.2 Symmetric Bilinear Forms

In dealing with a bilinear form sometimes it is asked when is there an ordered basis for V in
which f is represented by a diagonal matrix. It will be seen in this part of the unit that if f is a
symmetric bilinear form, i.e., f( , ) = f( , ) then f will be represented by a diagonal matrix in
an ordered basis of the space V.
If V is a finite-dimensional, the bilinear form f is symmetric if and only if the matrix A in some
ordered basis is symmetric, A = A.
t
To see this, one enquires when the bilinear form
t
f(X, Y) = X AY
is symmetric.

This happens if and only if X AY = Y AX for all column matrices X and Y. Since X AY is a 1 1
t
t
t
t
t
t
t
matrix, we have X AY = Y A X. Thus f is symmetric if and only if Y A X = Y AX for all X, Y. Clearly
t
t
t
this just means that A = A . In particular, one should note that if there is an ordered basis for V in
which f is represented by a diagonal matrix, then f is symmetric, for any diagonal matrix is a
symmetric matrix.
If f is a symmetric bilinear form, the quadratic form associated with f is the function q from V into
F defined by
q( ) = f( , )
If F is a subfield of the complex numbers, the symmetric bilinear form f is completely determined
by its associated quadratic form, according to the polarization identity
1 1
f( , ) = q( + ) – q( – ) ...(5)
4 4
If f is the bilinear form of Example 4, the dot product, the associated quadratic form is
( q x 1 , ... x n ) = x + ... + x 2 n
2
1
t
In other words, q( ) is the square of the length of . For the bilinear form f (X, Y) = X AY, the
A
associated quadratic form is
A x x
q (X) = X AX = ij i j
t
A
, i j

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