Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 50, No. 1, pp. 215218 (2009) 

Preservers of the rank of matrices over a fieldJozef KalinowskiUniwersytet Sl\c aski, Instytut Matematyki, ul. Bankowa 14, 40007 Katowice, Poland, email: kalinows@ux2.math.us.edu.plAbstract: In 2001, Li and Pierce characterized the linear operators that preserve the g rank of real matrices. The present paper describes all rank preserving maps $F: M_{m,n}(K) \longrightarrow M_{m,n}(K)$ of the $m\times n$ matrices over an arbitrary field $K$ which are of the form $F(a_{i,j})=(f_{i,j}(a_{i,j}))$. The linearity of $F$ is not a priorily assumed, and it turns out that if $\min \{m,n\}\leq 2$, then nonlinear rank preserving maps indeed exist. Keywords: preservers; rank of matrices Classification (MSC2000): 15A03 Full text of the article:
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