Page 132 - Matrices theory and applications

Theorem 7.1.1 For every M ∈ GL n (CC), there exists a unique pair
(H, Q) ∈ HPD n × U n
such that M = HQ.If M ∈ GL n (IR),then (H, Q) ∈ SPD n × O n .
The map M → (H, Q), called the polar decomposition of M,is a
homeomorphism between GL n (CC) and HPD n × U n (respectively between
GL n (IR) and SPD n × O n ). 7.1. The Polar Decomposition 115
Theorem 7.1.2 Let H be a positive definite Hermitian matrix. There ex-
2
ists a unique positive definite Hermitian matrix h such that h = H.If H
is real-valued, then so is h.The matrix h is called the square root of H,
√
and is denoted by h = H.
Proof
We prove Theorem 7.1.1 and obtain Theorem 7.1.2 as a by-product.
Existence.Since MM ∗ ∈ HPD n , we can diagonalize MM ∗ by a
unitary matrix
∗
∗
MM = U DU, D =diag(d 1 ,... ,d n ),
√ √
where d j ∈ (0, +∞). The matrix H := U diag( d 1 ,... , d n )U is Hermi-
∗
2
tian positive definite and satisfies H = HH = MM .Then Q := H −1 M
∗
∗
∗ −1
satisfies Q Q = M H −2 M = M (MM ) M = I n , hence Q ∈ U n .If
∗
∗
∗
M ∈ M n(IR), then clearly MM is real symmetric. In fact, U is orthogo-
∗
nal and H is real symmetric. Hence Q is real orthogonal. Note: H is called
the square root of MM .
∗
Uniqueness.Let M = H Q be another suitable decomposition. Then

−1
1

N := H −1 H = Q(Q ) is unitary, so that Sp(N) ⊂ S .Let S ∈ HPD n
be a positive definite Hermitian square root of H (we shall prove below

that it is unique). Then N is similar to N := SH −1 S. However, N ∈

HPD n . Hence N is diagonalizable, with real positive eigenvalues. Hence
Sp(N)= {1},and N is therefore similar, and thus equal, to I n .
This proves that the positive definite Hermitian square root of a matrix
of HPD n is unique in HPD n , since otherwise, our construction would
provide several polar decompositions. We have thus proved Theorem 7.1.2
in passing.
Smoothness.The map (H, Q) → HQ is polynomial, hence continuous.
Conversely,, it is enough to prove that M → (H, Q) is sequentially con-
tinuous, since GL n (CC) is a metric space. Let (M k ) k∈IN be a convergent
sequence in GL n (CC)and let M be its limit. Let us denote by M k = H k Q k
and M = HQ their respective polar decompositions. Let R be a cluster
point of the sequence (Q k ) k∈IN , that is, the limit of some subsequence
) Q ∗ converges to S := MR .
∗
(Q k l l∈IN ,with k l → +∞.Then H k l = M k l
k l
The matrix S is Hermitian positive semidefinite (because it is the limit
’s) and invertible (because it is the product of M and R ). It
∗
of the H k l
is thus positive definite. Hence, SR is a polar decomposition of M.The
uniqueness part ensures that R = Q and S = H. The sequence (Q k ) k∈IN ,