Page 38 - Matrices theory and applications

2.3. Alternate Matrices and the Pfaffian

=
∈ M n (A) are similar (that is, M
Proposition 2.2.2 If M, M
−1
MP with P ∈ GL n (A)), then
P

det M =det M.
2.3 Alternate Matrices and the Pfaffian 21
The very simple structure of alternate forms is described in the following
statement.
Proposition 2.3.1 Let B be an alternate bilinear form on a vector space
E,of dimension n. Then there exists a basis
{x 1 ,y 1 ,... ,x k ,y k ,z 1,... ,z n−2k }
such that the matrix of B in this basis is block-diagonal, equal to
diag(J,... ,J, 0,... , 0),with k blocks J defined by

0 1
J = .
−10
Proof
We proceed by induction on the dimension n.If B = 0, there is nothing to
prove. If B is nonzero, there exist two vectors x 1 ,y 1 such that B(x 1 ,y 1 ) =0.
Multiplying one of them by B(x 1 ,y 1) −1 , one may assume that B(x 1 ,y 1 )=
1. Since B is alternate, {x 1 ,y 1 } is free. Let N be the plane spanned by x 1 ,y 1 .
The set of vectors x satisfying B(x, v) = 0 (or equivalently B(v, x)= 0,
since B must be skew-symmetric) for every v in N is denoted by N .The
⊥
formulas
B(ax 1 + by 1 ,x 1 )= −b, B(ax 1 + by 1,y 1 )= a

show that N ∩ N ⊥ = {0}. Additionally, every vector x ∈ E can be written
as x = y + n,where n ∈ N and y ∈ N ⊥ are given by
n = B(x, y 1 )x 1 − B(x, x 1 )y 1 , y := x − n.

Therefore, E = N ⊕ N . We now consider the restriction of B to the
⊥
subspace N ⊥ and apply the induction hypothesis. There exists a basis
{x 2 ,y 2 ,... ,x k ,y k ,z 1 ,... ,z n−2k } such that the matrix of the restriction of
B in this basis is block-diagonal, equal to diag(J,... ,J, 0,... , 0), with k−1
blocks J, which means that B(x j ,y j )= 1= −B(y j ,x j )and B(u, v)= 0
forevery otherchoiceof u, v in the basis. Obviously, this property extends
to the form B itself and the basis {x 1 ,y 1 ,... ,x k ,y k ,z 1 ,... ,z n−2k }.

We now choose an alternate matrix M ∈ M n (K) and apply Proposition
2.3.1 to the form defined by M. In view of Section 1.2.3, we have the
following.