Adjacency preserving mappings on real symmetric matrices (Q2897921)

The author characterizes adjacency preserving maps \(\varphi\) between spaces of real symmetric matrices \(S_n({\mathbb R})\) and \(S_m({\mathbb R})\) under the additional assumption that \(\varphi(0)=0\). It is shown that either \(\varphi\) maps the whole space \(S_n({\mathbb R})\) into \({\mathbb R}B\), where \(B\) is a symmetric matrix of rank 1, or there exist \(c\in\{\pm 1\}\) and an invertible matrix \(R\in M_m({\mathbb R})\) such that for all \(A\in S_n({\mathbb R})\) it holds that \(\varphi (A)=cR (A \oplus 0_{m-n}) R^t\), here \(m\geq n\), and if \(m=n\) then there are no zero blocks on the righthand side.