Page 139 - Matrices theory and applications

7. Exponential of a Matrix, Polar Decomposition, and Classical Groups
122
−1
Sp M ∪ Sp M
,wehave p(λ)=log λ. Such a polynomial exists, since the
numbers λ are real and positive.
∗
Let N = U DU be a unitary diagonalization of N.Then M =exp N =
−1
∗
U (exp D)U and M
=exp(−N)= U exp(−D)U. Hence, p(M)= N
∗
−1
−1
, and therefore
and p(M
)= −N. However, M ∈ G implies MJ = JM
−1
q(M)J = Jq(M
) for every q ∈ IR[X]. With q = p,we obtain NJ = −JN.
These two lemmas complete the proof of the theorem, since G∩H n is an
IR-vector space. The integer d mentionned in the theorem is its dimension.
We wish to warn the reader that neither G,nor H n is a CC-vector space.
We shall see examples in the next section that show that G∩ H n can be
naturally IR-isomorphic to a CC-vector space, which is a source of confusion.
One therefore must be cautious when computing d.
The reader eager to learn more about the theory of classical groups is
e
advised to have a look at the book of R. Mneimn´ and F. Testard [28] or
the one by A. W. Knapp [24].
7.4 The Groups U(p, q)
Let us begin with the study of the maximal compact subgroup of U(p, q).
If M ∈ U(p, q) ∩ U n ,let us write M blockwise:

A B
M = ,
C D
where A ∈ M p (CC), etc. The following equations express that M belongs
to U n :
∗
∗
∗
∗
A A + C C = I p , B B + D D = I q , A B + C D =0 pq .
∗
∗
Similarly, writing that M ∈ U(p, q),
∗
∗
∗
∗
∗
∗
A A − C C = I p , D D − B B = I q , A B − C D =0 pq .
Combining these equations, we obtain first C C =0 p and B B =0 q .For
∗
∗
n
2
every vector X ∈ CC ,we have CX = X C CX = 0; hence CX =0.
∗
∗
2
Finally, C = 0 and similarly B = 0. There remains A ∈ U p and D ∈ U q .
The maximal compact subgroup of U(p, q) is thus isomorphic (not only
homeomorphic) to U p × U q .
Furthermore, G∩ H n is the set of matrices

A B
N = ,
B ∗ D
where A ∈ H p , D ∈ H q , which satisfy NJ + JN =0 n;thatis, A =0 p ,
D =0 q . Hence G∩ H n is isomorphic to M p×q (CC). One therefore has
d =2pq.