Preservers of the rank of matrices over a field
Let \(M_{m,n}(K)\) denote the set of \(m\times n\) matrices over a field \(K\). Let \(A=(a_{ij})\in M_{m,n}(K)\). Let \(F:M_{m,n}(K)\to M_{m,n}(K)\) be a map such that \(F(A)=(f_{ij}(a_{ij}))\), where \(f_{ij}: K\to K\). The author verifies which \(F\) preserves rank of all matrices. The following theorem is proved. {\parindent6mm \begin{itemize}\item[(a)] \(\min\{m,n\}=2\). Then \(F\) preserves rank of all matrices if and only if there exist nonzero \(u_1, \dots,u_m,v_1,\dots,v_n\in K\) and a nonzero homomorphism \(g\) of the multiplicative group \(K^\times\) such that \(g(0)=0\) and \(f_{ij}(x)=u_iv_j g(x)\) for every \(x\). \item[(b)] \(\min\{m,n\}\geq 3\). Then \(F\) preserves rank of all matrices if and only if there exist nonzero \(u_1,\dots,u_m, v_1,\dots,v_n\in K\) and an embedding \(g\) of \(K\) into \(K\) such that \(f_{ij}(x)=u_iv_jg(x)\) for every \(x\). \end{itemize}}
scientific article; zbMATH DE number 5529364