Information Geometry of Covariance Matrix: Cartan-Siegel Homogeneous Bounded Domains, Mostow/Berger Fibration and Fréchet Median - Matrix Information Geometry - page 217

XAX = B ⇒ A 1 / 2 X A 1 / 2 A 1 / 2 XA 1 / 2

= A 1 / 2 BA 1 / 2

A 1 / 2 XA 1 / 2 2

= A 1 / 2 BA 1 / 2 ⇒

A 1 / 2 BA 1 / 2 1 / 2

⇒ X = A − 1 / 2 A 1 / 2 BA 1 / 2 1 / 2 ⇒ A 1 / 2 XA 1 / 2 A − 1 / 2

(9.96)

=

We can observe that X is geodesic center of A − 1 and B for symmetric space of
Hermitian positive definite matrices.

Corollary If M is Hermitian Positive Definite, there exist a unique real symmetric
matrix S such that:

M ∗ = e S M − 1 e S (9.97)

M Positive Definite Hermitian Matrix, M ∗ and M − 1 with same property. From
previous Lemma, there exist a unique hermitian positive definite matrix X such that:

M ∗ = XM − 1 X (9.98)

Exponential providing an homeomorphism between symmetric and positive def-
inite symmetric spaces, it can be proved proof that X is positive definite

M ∗ ) ∗ = X ∗ M ∗− 1 X ∗ ⇒ M ∗− 1 X ∗− 1 MX ∗− 1 M ∗ = X ∗ M − 1 X ∗ M = ( = ⇒

M ∗ = XM − 1 X X ∗ = because ⇒ X

(9.99)

Ue iA e S

If we come back to Mostow Theorem: M =

M + M e S e 2 iA e S ⇒ P = =

e 2 S e − S e − 2 iA e − S e 2 S

P ∗ = e S e − 2 iA e S P ∗ = e 2 S P − 1 e 2 S ⇒ = ⇒ (9.100)

Lemma and corollary will induce:

P 1 / 2 P − 1 / 2 P ∗ P − 1 / 2 1 / 2

P ∗ = e 2 S P − 1 e 2 S e 2 S P 1 / 2 ⇒ = (9.101)

And then:

log P 1 / 2 P − 1 / 2 P ∗ P − 1 / 2 1 / 2 P 1 / 2 with M + M

S = 1 / 2 · P = (9.102)

Based on Exponential injectivity from e 2 iA e − S Pe − S , we can deduce that: =

log e − S Pe − S with

1
2 i

M + M

A = P = (9.103)