Diameter preserving surjection on alternate matrices
Let \(F^{m\times n}\) be the set of all \(m\times n\) matrices over a field \(F\). A matrix \(A\in F^{m\times m}\) is called an alternate matrix if \(^tA=-A\) with zero diagonal entries, where \(^tA\) is the transpose of \(A\). The set of all \(m\times m\) alternate matrices over \(F\) is denoted by \( \mathcal {K}_m(F)\). Note that a matrix \(A\in F^{m\times m}\) is an alternate matrix if and only if \(^txAx=0\) for all \(x\in F^{m\times 1}\). If the characteristic of \(F\) is not 2, \(A\) is alternate if and only if \(A\) is skew-symmetric. The arithmetic distance between \(A, B\in \mathcal {K}_m(F)\) is \(d(A, B)= \text{rank}(A-B)\). \(A\) and \(B\) are said to be adjacent if \(d(A, B)=2\) and denoted by \(A\sim B\). The diameter of \(\mathcal {K}_m(F)\) is defined as \(d_{\max}=\max \{ d(A, B): A, B\in \mathcal {K}_m(F)\} \). Let \(\varphi : \mathcal {K}_m(F)\to \mathcal {K}_m(F)\) be a map. Then \(\varphi \) is said to be (1) an adjacency preserving map if \[ A \sim B \Rightarrow \varphi (A) \sim \varphi (B); \] (2) an adjacency preserving map in both directions if \[ A\sim B \Leftrightarrow \varphi (A) \sim \varphi (B); \] (3) an arithmetic distance preserving map if \[ d(\varphi (A), \varphi (B))=d(A, B); \] (4) a diameter preserving map if \[ d(A, B)=d_{\max } \Rightarrow d(\varphi (A), \varphi (B))=d_{\max }; \] (5) a diameter preserving map in both directions if \[ d(A, B)=d_{\max } \Leftrightarrow d(\varphi (A), \varphi (B))=d_{\max }; \] (6) a preserver of distance-additivity if \[ d(A, B)=d(A, C)+d(C, B) \Rightarrow d(\varphi (A), \varphi (B))=d(\varphi (A), \varphi (C))+d(\varphi (C), \varphi (B)), \] for all \(A, B, C\in \mathcal {K}_m(F)\). The aim of this paper is to investigate these preservers on alternate matrices. The following result is obtained: Let \(F\) be a field with \(|F|\geq 3\) and \(m\geq 4\) be an integer. If \(\varphi \) is a map from \(\mathcal {K}_m(F)\) into itself, then the following statements are equivalent: (i) \(\varphi \) is a diameter preserving surjective map in both directions; (ii) \(\varphi \) is both an adjacency preserving surjective map and a diameter preserving map; (iii) \(\varphi \) is a bijective map preserving arithmetic distance; (iv) \(\varphi \) is an adjacency preserving surjective map in both directions; (v) \(\varphi \) is a bijective preserver of distance-additivity.
scientific article; zbMATH DE number 5645776
scientific article