Page 294 - Advanced Linear Algebra

278 Advanced Linear Algebra

/p > > of containing #, which contradicts the maximality of > . Hence,
=~ >.

This proves the following structure theorem for symplectic geometries.

Theorem 11.13
)
1 A symplectic geometry has an orthogonal basis if and only if it is totally
degenerate.
2 Any nonsingular symplectic geometry is a hyperbolic space, that is,
)
=

= ~ /p /p Ä p /
is a hyperbolic plane. Thus, there is a hyperbolic basis for
where each /
= , that is, a basis for which the matrix of the form is
8
v y
x c {
x {
x {
@ ~ x x c { {
x {
x Æ {
x {

w c z
In particular, the dimension of is even.
=
)
3 Any symplectic geometry has the form
=
=~ rad ²= ³ p >
>
where is a hyperbolic space and rad²= ³ is a totally degenerate space.
>
The rank of the form is dim²³ and = is uniquely determined up to
isometry by its rank and its dimension. Put another way, up to isometry,
there is precisely one symplectic geometry of each rank and dimension.
Symplectic forms are represented by alternate matrices, that is, skew-symmetric
matrices with zero diagonal. Moreover, according to Theorem 11.13, each
d alternate matrix is congruent to a matrix of the form

@
? ~ Á c > ?
c block
Since the rank of ? is Á c , no two such matrices are congruent.

Theorem 11.14 The set of d matrices of the form ? Á c is a set of
canonical forms for alternate matrices under congruence.

The previous theorems solve the classification problem for symplectic
geometries by stating that the rank and dimension of form a complete set of
=