The Addition Of The Equation. Ensure Matrix Rank Surjective

In recent years,many mathematicians studied the linear maps or additive maps preserving certain transformations,certain quantitative character,certain relations on matrix space.And they constantly presented new ideas to solve these problems. The study of problem preserving rank equations is an important work.At first,this paper outlines the research of preserver problems and the status of the study of rank equations in the domestic and overseas.In the chapter 2,the characterization of additive operators on Mn(F) over a field of characteristic zero that preserve the equation r(AB) = r(BA) for all A, B￡Mn(F) is given, and additive maps on Mn(F) over an arbitrary field that preserve the equation r(A+B) = r(A) + r(B) have been shown in the [3].This paper is inspired by the research of preservers of rank permutability and rank additivity.In the chapter 3,let F be a field of characteristic zero.We describe additive surjective maps <p preserving following rank equations: ?r{AB) = r(A) + r{B) - n ?r(AB) = r{A) ?r(ABA) = r(A) ?r(ABA) = r(B),for A, Be Mn(F) .This paper makes use of the results of additive surjective maps preserving invertibility on Mn(F) and obtains that <p is either of the following two forms:(i)(?), for all (?); (ii)(?), for all (?).Where PeGLn(F), aeF N.a^O, a is an automorphism of the field F.