# Linear and Isometric Automorphism Groups of the PSD Cone

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.

• I am sorry. I have redone the latex. All Rs really should be in blackboard font.
– user2529
Sep 21, 2011 at 8:17

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.
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).