Linear and Isometric Automorphism Groups of the PSD Cone - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T02:06:51Z http://mathoverflow.net/feeds/question/76029 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76029/linear-and-isometric-automorphism-groups-of-the-psd-cone Linear and Isometric Automorphism Groups of the PSD Cone Colin Tan 2011-09-21T08:03:27Z 2011-09-21T08:57:13Z <p>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 $\langle A,B\rangle = tr (AB)=tr(BA)$.</p> <p>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 $S_+$?</p> <p>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.</p> http://mathoverflow.net/questions/76029/linear-and-isometric-automorphism-groups-of-the-psd-cone/76039#76039 Answer by Igor Rivin for Linear and Isometric Automorphism Groups of the PSD Cone Igor Rivin 2011-09-21T08:57:13Z 2011-09-21T08:57:13Z <p>The general linear group $GL(n, R)$ acts on $S+$ by $g(x) =$g x g^t,$and this is the full linear automorphism group.</p> <p><a href="http://www.math.umbc.edu/~gowda/tech-reports/trGOW11-03.pdf" rel="nofollow">http://www.math.umbc.edu/~gowda/tech-reports/trGOW11-03.pdf</a></p> <p>and references therein for more details and related results. I am pretty sure the result goes back to at least Minkowski (for$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).</p>