Isometry group of positive definite matrices

[MichaelL.]

Hi,

Consider P the space of n by n positive definite matrices.

Let <X,Y>_p = trace(p^-1 X p^-1 Y) where p in P be the metric tensor on P so that it is a Riemannian manifold.

The general linear group G acts on P by phi: G X P -> P, phi(g,p) = phi_g(p)=gpg' (g' means g transpose).

This a transitive action by isometries. Does this constitute the isometry group on P however, that is, can every isometry on P be written in this fashion?

Thanks!

 

[Greg Bernhardt]

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?