LECTURE 3
The Linear Theory
We will consider the vector space R^2 n with its standard symplectic form w 0.
This may be written in vector notation as
wo(v,w) = wT J 0 v ,
where wT is the transpose of the column vector w and J 0 is the block diagonal
matrixJ 0 = diag ( ( ~ ~l ) ,.. ., ( ~ ~l ) ).
The symplectic linear group Sp(2n,R) (sometimes written Sp(2n)) consists of all
matrices A such thatwo(Av, Aw) = wo(v, w),
or equivalently of all A such that
AT JoA =Jo.
Clearly Sp(2n, R) is a group. The identity J;f = -J 0 = J 01 gives rise to interesting
algebraic properties of this group. Firstly, it is closed under transpose, and secondly
every symplectic matrix is conjugate to its inverse transpose ( A-^1 ) T. The former
statement is proved by inverting the identity
(A-^1 f JoA-^1 =Jo,
and the second by multiplying the defining equation AT J 0 A = J 0 on the left by
(Jo)-l(AT)-1.Exercise 3.1. Show that if ,\ E C is an the eigenvalue of a symplectic matrix A
then so are 1/ ,\ , >., 1/5.. What happens when ,\ E R , or l>-1 = 1?
Recall that we are identifying en with R^2 n by setting Zj = Xj + iyj· Under
this identification, Jo corresponds to multiplication by i. Hence we may consider
GL(n, C) to be the subgroup of GL(2n, R) consisting of all matrices A such that
AJo = JoA.
Exercise 3.2. Given an n x n matrix A with complex entries, find a formula for
the corresponding real 2n x 2n matrix.
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