Adjacency preserving mappings of symmetric and Hermitian matrices
The space \(\mathcal{H}_n(D)\), where \(n\) is a natural number and \(D\) is a division ring, denotes the graph on the \((n \times n)\)-Hermitian matrices over \(D\) in which two matrices \(A\) and \(B\) are adjacent if the rank of the matrix \(A-B\) is \(1\). The authors prove the following theorem: Let \(n \geq 2\), let \(D\) be a division ring which admits an involution \(\;\bar{}\;\), and let \(F\) be the set of fixed elements of \(\;\bar{}\;\). If \(\;\bar{}\;\) is the identity, assume furthermore that the characteristic of the field \(D = F\) is distinct from \(2\). If \(\;\bar{}\;\) is not the identity, assume that \(F\) is a central subfield of \(D\). If a bijective map \(\phi\) from \(\mathcal{H}_n(D)\) onto itself preserves adjacency in \(\mathcal{H}_n(D)\), then so does its inverse \(\phi^{-1}\).
scientific article; zbMATH DE number 2119230