# Isometry group of positive definite matrices

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1. May 7, 2015

### 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!

2. May 12, 2015