Adjacency preserving mappings on real symmetric matrices

TL;DR

This paper characterizes maps between real symmetric matrices that preserve adjacency, specifically those that maintain rank-one differences, for dimensions two and higher.

Contribution

It provides a complete characterization of adjacency-preserving maps on real symmetric matrices for all dimensions n ≥ 2.

Findings

Maps preserving adjacency are characterized for n=2 and n>2.

The structure of such maps is explicitly described.

Results extend understanding of matrix transformations preserving rank properties.

Abstract

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.