Page 145 - Matrices theory and applications

7. Exponential of a Matrix, Polar Decomposition, and Classical Groups
128
T
∈ Sp ,we also have
But since M
n
T
T
T
T
T
T
Let us combine these equations:
T
T
T
T
B = B(A A+C C)= AB A+(AD −I n )C = A(B A+D C)−C = −C.
Similarly, T AB = BA , T AD − BC = I n , CD = DC .
T
T
T
T
T
T
D = D(A A+C C)=(I n +CB )A+CD C = A+C(B A+D C)= A.
Hence

A B
M = .
−B A
The remaining conditions are
T
T
T
T
A A + B B = I n , A B = B A.
This amounts to saying that A + iB is unitary. One immediately checks
that the map M → A + iB is an isomorphism from Sp onto U n .
n
Finally, if

A B
N = T
B D
is symmetric and NJ + JN =0 2n, we have, in fact,

A B
N = ,
B −A
where A and B are symmetric. Hence G∩ Sym is homeomorphic to
2n
Sym × Sym ,that is, to IR n(n+1) .
n n
Proposition 7.6.1 The symplectic group Sp is homeomorphic to U n ×
n
n(n+1)
IR .
Corollary 7.6.1 In particular, every symplectic matrix has determinant
+1.
Indeed, Proposition 7.6.1 shows that Sp is connected. Since the de-
n
terminant is continuous, with values in {−1, 1}, it is constant, equal to
+1.
7.7 Singular Value Decomposition
As we shall see in Exercise 8 (see also Exercise 12 in Chapter 4), the
eigenvalues of the matrix H in the polar decomposition of a given matrix
M are of some importance. They are called the singular values of M.Since
∗
these are the square roots of the eigenvalues of M M,one may evenspeak