Abstract
Suppose π½ is an arbitrary field of characteristic not 2 and π½ββ βπ½3. Let M n (π½) be the space of all nβΓβn full matrices over π½ and P n (π½) the subset of M n (π½) consisting of all nβΓβn idempotent matrices and GL n (π½) the subset of M n (π½) consisting of all nβΓβn invertible matrices. Let Ξ¦π½(n,βm) denote the set of all maps from M n (π½) to M m (π½) satisfying AβββΞ»BβββP n (π½)βββΟ(A)βββΞ»Ο(B)βββP m (π½) for every A,βBβββM n (π½) and Ξ»βββπ½, where m and n are integers with 3ββ€βnββ€βm. It is shown that if ΟβββΞ¦π½(n,βm), then there exists TβββGL m (π½) such that Ο(A)β=βTβ[AβββI p βββA t βββI q βββ0]Tββ1 for every AβββM n (π½), where I 0β=β0. This improves the results of some related references.
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Acknowledgements
This work is supported by the Postdoctoral Foundation of China (No. 520-415099) and the NSF of P.R. China (No.10871056). The authors would like to thank the referee for his valuable comments and suggestions to the earlier version of this article.