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Preservers of the rank of matrices over a field. (English) Zbl 1171.15002

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. 6mm
(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 \(x\).
(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 \(x\).

MSC:

15A04 Linear transformations, semilinear transformations
15A03 Vector spaces, linear dependence, rank, lineability
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