Linear operators preserving idempotent matrices over fields

AbstractSuppose F is a field. We show that if the characteristic of the field is not 2, then the semigroup of linear operators on the n × n matrices over F that preserve idempotence is the group G(F) generated by transposition and similarity. Chan, Lim, and Tan have previously established that theorem for the real and complex fields by other methods. We also show that the semigroup L(F) of linear operators on the n × n matrices over F that preserve both idempotence and nonidempotence is G(F) when the characteristic of F is not 2. We determine the structure of L(F) when the characteristic of F is 2, and present some open problems