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!

 

[jvicens]

Thank you for your interesting question regarding the isometry group on the space of positive definite matrices, P. It is indeed true that the metric tensor given by <X,Y>_p = trace(p^-1 X p^-1 Y) makes P into a Riemannian manifold. As you mentioned, 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 action is transitive, meaning that for any two points in P, there exists a g in G that maps one point to the other. Furthermore, this action is by isometries, meaning that it preserves the metric tensor, <X,Y>_p.

However, it is important to note that this action does not necessarily constitute the entire isometry group on P. The isometry group of a Riemannian manifold is the set of all transformations that preserve the metric tensor, not just a specific action. In other words, there may be other transformations besides those in the form of phi(g,p) that preserve the metric tensor on P.

To determine whether the action of G on P constitutes the entire isometry group, we need to consider the group of all isometries on P, which is denoted by Iso(P). This group includes all possible transformations that preserve the metric tensor on P, not just the ones in the form of phi(g,p).

In general, it is not easy to determine the entire isometry group of a given Riemannian manifold. However, in the case of P, it is known that the isometry group is actually larger than the action of G. In fact, the isometry group of P is the group of all affine transformations on P, which includes not only the action of G, but also translations and dilations.

To summarize, while the action of G on P is certainly important and useful, it is not the entire isometry group on P. The full isometry group, Iso(P), includes other transformations such as translations and dilations. I hope this helps clarify your question.