Let $S\_+$ be the cone of psd matrices ($n\times n$ real symmetric positive semidefinite matrices). This cone is a metric space induced from the inner product $\langle A,B\rangle = tr (AB)=tr(BA)$.


The cone $S\_+$ seems exceptionally symmetrical, and I am curious to know its symmetry groups. Let me be precise. An $R\_+$-linear map $f:S\_+\to S\_+$ is an additive map such that $f(\alpha A)=\alpha f(A)$ for $\alpha\ge 0$. What is the group of $R\_+$-linear automorphisms of $S\_+$? What is the subgroup of isometric $R\_+$-linear automorphisms of $S\_+$?


Apparently, from my geometric intuition, such automorphisms should permute the extremal rays of rank 1 matrices. Since each rank 1 matrix corresponds to a codim 1 plane in $R^n$, my guess is that these automorphism groups should be related to the Grassmanian.



Linear and Isometric Automorphism Groups of the PSD Cone
The general linear group $GL(n, R)$ acts on $S+$ by $g(x) = g x g^t,$ and this is the full linear automorphism group.


<http://www.math.umbc.edu/~gowda/tech-reports/trGOW11-03.pdf>


and references therein for more details and related results. I am pretty sure the result goes back to at least Minkowski (for $n=2$ this is the hyperboloid model of hyperbolic space, and the isometry group of $H^2$ (which is identified with $x \in S\_+,$ such that $\det x = 1$) is $SL(2, R).$ For higher $n$ the analogous section $\det x = 1$ is a representation of the symmetric space for $SL(n, R)$ (Siegel half-space, I guess; but Siegel defined a metric which makes it into a Riemannian symmetric space).


