Page 138 - Matrices theory and applications

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7.3. Structure of Classical Groups
−k
Observe that, since G is a group, M ∈ G implies (M ) J = JM
for
∗
every k ∈ IN. By linearity, it follows that p(M )J = Jp(M
every polynomial p ∈ IR[X].
Let us now assume that M ∈ G ∩ HPD n .We then have M =
U diag(d 1 ,... ,d n )U,where U is unitary and the d j ’s are positive real
∗
numbers. Let A be the set formed by the numbers d j and 1/d j .There ex-
ists a polynomial p with real entries such that p(a)= √ ∗ k −1 )holds for
a for every a ∈ A.
√ √ −1
Then we have p(M)= M and p(M −1 )= M .Since M = M,we
∗
√ √ −1 √
have also p(M)J = Jp(M −1 ); that is, MJ = J M . Hence M ∈ G.
From Proposition 7.3.1, G is stable under polar decomposition.
The main result of this section is the following:
Theorem 7.3.1 Under the hypotheses of Proposition 7.3.2, the group G
d
is homeomorphic to (G ∩ U n ) × IR , for a suitable integer d.
Of course, if G = O(p, q)or Sp , the subgroup G∩U n can also be written
m
as G ∩ O n .We call G ∩ U n a maximal compact subgroup of G, because one
can prove that it is not a proper subgroup of a compact subgroup of G.
Another deep result, which is beyond the scope of this book, is that every
maximal compact subgroup of G is a conjugate of G ∩ U n .In the sequel,
when speaking about the maximal compact subgroup of G,weshall always
have in mind G ∩ U n .
Proof
The proof amounts to showing that G∩HPD n is homeomorphic to some
d
IR . To do this, we define
G := {N ∈ M n (k)| exp tN ∈ G, ∀t ∈ IR}.
Lemma 7.3.1 The set G defined above satifies
∗
G = {N ∈ M n (k)|N J + JN =0 n}.
Proof
If N J + JN =0 n,let us set M(t)= exp tN.Then M(0) = I n and
∗
d
∗
∗
∗
M(t) JM(t)= M (t)(N J + JN)M(t)=0 n ,
dt
so that M(t) JM(t) ≡ J.Wethus have N ∈G.Conversely,, if M(t):=
∗
∗
exp tN ∈ G for every t, then the derivative at t =0 of M (t)JM(t)= J
∗
gives N J + JN =0 n .
Lemma 7.3.2 The map exp : G∩H n → G∩HPD n is a homeomorphism.
Proof
We must show that exp : G∩ H n → G ∩ HPD n is onto. Let M ∈
G ∩ HPD n and let N be the Hermitian matrix such that exp N = M.
Let p ∈ IR[X] be a polynomial with real entries such that for every λ ∈