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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 ${\mathbb{R}}+$-linear R_+$-linear map$f:S+\to f:S_+\to S_+$is an additive map such that$f(\alpha A)=\alpha f(A)$for$\alpha\ge 0$. What is the group of${\mathbb{R}}+$-linear R_+$-linear automorphisms of $S+$? S_+$? What is the subgroup of isometric${\mathbb{R}}+$-linear R_+$-linear automorphisms of $S+$?S_+$? Apparently, from my geometric intuition, such automorphisms should permute the extremal rays of rank 1 matrices. Since each rank 1 matrix corresponds to a codim 1 plane in${\mathbb{R}}^n$, R^n$, my guess is that these automorphism groups should be related to the Grassmanian.

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# 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 ${\mathbb{R}}+$-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 ${\mathbb{R}}+$-linear automorphisms of $S+$? What is the subgroup of isometric ${\mathbb{R}}+$-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 ${\mathbb{R}}^n$, my guess is that these automorphism groups should be related to the Grassmanian.