Adjacency preserving mappings on real symmetric matrices

Let $S_{n}$ denote the space of all $n \times n$ real symmetric matrices. For n=2 or n>2 we characterize maps F from $S_{n}$ to $S_{m}$ which preserve adjacency, i.e. if rank(A-B)=1, then rank(F(A)-F(B))=1.

💡 Research Summary

The paper investigates mappings between spaces of real symmetric matrices that preserve the adjacency relation defined by a rank‑one difference. Let (S_n) denote the vector space of all (n\times n) real symmetric matrices. Two matrices (A,B\in S_n) are called adjacent if (\operatorname{rank}(A-B)=1). A map (F:S_n\rightarrow S_m) is said to preserve adjacency when (A\sim B) implies (F(A)\sim F(B)). The main goal is to give a complete description of all such maps for the two essentially different regimes (n=2) and (n\ge 3).

Main Results

1. General Form (n≥3). If (F:S_n\to S_m) preserves adjacency, then either
   a) (F) is constant, i.e. there exists a fixed matrix (C\in S_m) such that (F(A)=C) for every (A); or
   b) there exist a non‑zero scalar (\lambda\in\mathbb{R}), an invertible real matrix (P\in GL_m(\mathbb{R})) (not necessarily orthogonal), and a constant matrix (C\in S_m) such that for all (A\in S_n)
   \