MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

I saw a statement in [Murakami, On automorphisms on Siegel domains] that every linear automorphism $\phi$ on the set of positive definite matrices can be represented as conjugation: i.e. there is a matrix $B\in GL(n,\mathbb{R})$ such that $\phi(A)=B^t A B$. It seems an easy statement but I couldn't prove it. Can somebody help me?