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Additive map preserving rank 2 on alternate matrices. (English) Zbl 1104.15002
An $n$-by-$n$ matrix $A$ is called alternate if $x^tAx=0$ for each vector $x$. These matrices are of considerable importance in the theory of quadratic forms and classical groups. The authors classify additive surjections on alternate matrices that preserve the minimal rank. It is shown that such a mapping is automatically injective and it preserves any rank. Rewiever’s remark: A similar classification without the surjectivity assumption was obtained by {\it X. Zhang} [Linear Algebra Appl. 392, 25--38 (2004; Zbl 1059.15010)] (under some constraint on the underlying field), and the reviewer [Linear Multilinear Algebra 53, No. 4, 231--241 (2005; Zbl 1080.15001)].

15A04Linear transformations, semilinear transformations (linear algebra)
15A03Vector spaces, linear dependence, rank