Page 131 - Matrices theory and applications

7
Exponential of a Matrix, Polar
Decomposition, and Classical Groups

7.1 The Polar Decomposition

The polar decomposition of matrices is defined by analogy with that in the
∗
complex plane: If z ∈ CC , there exists a unique pair (r, q) ∈ (0, +∞) × S 1
1
(S denotes the unit circle, the set of complex numbers of modulus 1) such
that z = rq.If z acts on CC (or on CC ) by multiplication, this action can
∗
be decomposed as the product of a rotation of angle θ (where q =exp(iθ))
with a homothety of ratio r> 0. The fact that these two actions commute
∗
is a consequence of the commutativity of the multiplicative group CC ;this
property does not hold for the polar decomposition in GL n (k), k = IR or
CC, because the general linear group is not commutative.
Let us recall that HPD n denotes the (open) cone of matrices of M n (CC)
that are Hermitian positive definite, while U n denotes the group of unitary
matrices. In M n(IR), SPD n is the set of symmetric positive definite ma-
trices, and O n is the orthogonal group. The group U n is compact, since it
is closed and bounded in M n (CC). Indeed, the columns of unitary matrices
are unit vectors, so that U n is bounded. On the other hand, U n is defined
∗
∗
by an equation U U = I n ,where themap U → U U is continuous; hence
U n is closed. By the same arguments, O n is compact.
Polar decomposition is a fundamental tool in the theory of finite-
dimensional Lie groups and Lie algebras. For this reason, it is intimately
related to the exponential map. We shall not consider these two notions
here in their full generality, but we shall restrict attention to their matricial
aspects.