Page 344 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Thus a bilinear form f is skew-symmetric if and only if its matrix A is equal to –A in some
t
ordered basis.
When f is skew-symmetric, the matrix of f in any ordered basis will have all its diagonal entries
0. This just corresponds to the observation that f( , ) = 0 for every in V, since f( , ) = –f( , ).
Let us suppose f is a non-zero skew-symmetric bilinear form on V. Since f 0, there are vectors
, in V such that f( , ) 0. Multiplying by a suitable scalar, we may assume that f( , ) = 1.
Let be any vector in the subspace spanned by and , say = c + d . Then
f( , ) = f(c + d , ) = df( , ) = –d

f( , ) = f(c + d , ) = cf( , ) = c
and so
= f( , ) – f( , ) ...(1)
In particular, note that and are necessarily linearly independent; for, if = 0, then f( , ) =
f( , ) = 0.
Let W be the two-dimensional subspace spanned by and . Let W be the set of all vectors in
V such that f( , ) = f( , ) = 0, that is, the set of all such that f( , ) = 0 for every in the subspace
W. We claim that V = W W . For, let be any vector in V, and
= f( , ) – f( , )

= –
Then is in W, and is in W , for
f( , ) = f( – f( , ) + f( , ) , )
= f( , ) + f( , )f( , )

= 0
and similarly f( , ) = 0. Thus every in V is of the form = + , with in W and in W . From
(1) it is clear that W W = {0}, and so V = W W .

Now the restriction of f to W is a skew-symmetric bilinear form on W . This restriction may be
the zero form. If it is not, there are vectors ' and ' in W such that f( ', ') = 1. If we let W' be the
two-dimensional subspace spanned by ' and ', then we shall have
V = W W' W
0
where W is the set of all vectors in W such that f( ', ) = f( ', ) = 0. If the restriction of f to W
0 0
is not the zero form, we may select vectors ", " in W such that f( ", ") = 1, and continue.
0
In the finite-dimensional case it should be clear that we obtain a finite sequence of pairs of
vectors,

( , ), ( , ), ... , ( , )
1 1 2 2 k k
with the following properties:
(a) f( , ) = 1, j = 1, ... , k.
j j
(b) f( , ) = f( , ) = f[ , ) = 0, i j.
i j i j i j
(c) If W is the two-dimensional subspace spanned by and , then
j j j
V = W ... W W
1 k 0
where every vector in W is 'orthogonal' to all , and , and the restriction of f to W is the zero
0 j j 0
form.

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