Page 137 - Matrices theory and applications

7. Exponential of a Matrix, Polar Decomposition, and Classical Groups
120
7.3 Structure of Classical Groups
Proposition 7.3.1 Let G be a subgroup of GL n (CC). We assume that G
∗
is stable under the map M → M and that for every M ∈ G ∩ HPD n ,the
√
square root M is an element of G.Then G is stable under polar decompo-
sition. Furthermore, polar decomposition is a homeomorphism between G
and
(G ∩ U n ) × (G ∩ HPD n ).
This proposition applies in particular to subgroups of GL n (IR)that are
stable under transposition and under extraction of square roots in SPD n .
One has then
homeo
G ∼ (G ∩ O n) × (G ∩ SPD n ).
Proof
Let M ∈ G be given and let HQ be its polar decomposition. Since
2
MM ∈ G,we have H ∈ G,thatis, H ∈ G, by assumption. Finally, we
∗
have Q = H −1 M ∈ G. An application of Theorem 7.1.1 finishes the proof.
We apply this general result to the classical groups U(p, q), O(p, q)
(where n = p+q)and Sp m (where n =2m). These are respectively the uni-
2
2
2
2
tary group of the Hermitian form |z 1 | +···+|z p| −|z p+1 | −···−|z n| ,the
2
2
2
orthogonal group of the quadratic form x + ···+ x − x 2 p+1 − ···−x ,and
p
n
1
the symplectic group. They are defined by G = {M ∈ M n (k)|M JM = J},
∗
with k = CC for U(p, q), k = IR otherwise. The matrix J equals

I p 0 p×q
,
0 q×p −I q
for U(p, q)and O(p, q), and

0 m I m
,
−I m 0 m
2
for Sp . Ineachcase, J = ±I n .
m
2
Proposition 7.3.2 Let J be a complex n × n matrix satisfying J = ±I n .
The subgroup G of M n (CC) defined by the equation M JM = J is invariant
∗
under polar decomposition. If M ∈ G,then | det M| =1.
Proof
The fact that G is a group is immediate. Let M ∈ G.Then det J =
2
det M det M det J;that is, | det M| =1. Furthermore, M JM(JM )=
∗
∗
∗
2
∗ 2
∗
∗
J M = ±M = M J . Simplifying by M J on the left, there remains
∗
MJM = J,that is, M ∈ G.
∗
∗