Convexity and Star-shapedness of Real Linear Images of Special Orthogonal Orbits

Abstract

Let $A\in {\mathbb{R}}^{N\times N}$ and ${\mathrm{SO}}_n:=\{U\in {\mathbb{R}}^{N\times N}:U{U}^t=I_n,\det U>0\}$ be the set of $n\times n$ special orthogonal matrices. Define the (real) special orthogonal orbit of $A$ by \[ O(A):=\{UAV:U,V\in\mathrm{SO}_n\}. \] In this paper, we show that the linear image of $O(A)$ is star-shaped with respect to the origin for arbitrary linear maps $L:{\mathbb{R}}^{N\times N}\to {\mathbb{R}}^{\ell }$ if $n\ge 2^{\ell -1}$. In particular, for linear maps $L:{\mathbb{R}}^{N\times N}\to {\mathbb{R}}^2$ and when $A$ has distinct singular values, we study $B\in O(A)$ such that $L(B)$ is a boundary point of $L(O(A))$. This gives an alternative proof of a result by Li and Tam on the convexity of $L(O(A))$ for linear maps $L:{\mathbb{R}}^{N\times N}\to {\mathbb{R}}^2$.