Preserving Problems Between Matrix Spaces

Preserving problems concern the characterization of maps between matrixspaces that preserve some invariants. During the past few decades, one of the veryactive research areas in matrix theory is the study of preserving problems. Firstly, ithas important theoretical values. Secondly, it has wide applications in other areas,such as, quantum mechanics, differential geometry, differential equation, system con-trol, mathematical statistics, and so on. Thirdly, studying preserving problems canhelp us to understand matrix invariants, functions, sets and relations.According to the property of maps, preserving problems can be divided intothree categories, that is, linear preserving problems, additive preserving problemsand general preserving problems. According to the property of invariants, preservingproblems can be divided into four categories, that is, preserving subsets, preservingrelations, preserving functions and preserving transformations.This paper considers some preserving problems on matrix spaces and obtains thefollowing five results:(1) By using the conclusion of linear maps on alternate matrix space which pre-serve rank 2 and rank 4, the structure of linear mapφ: K_n(F)→K_m(F) whichpreserves adjoint matrices is characterized. It can be concluded that the characteriza-tion of alternate matrix spaces of different dimensions can be induced to characterizethe same dimensions.(2) The characterization of linear mapφ: Tn(F)→Tn(F) which pre-serves rank-additivity is given. And then, applications to several related preserv-ing problems are considered. The linear mapsφ: T_n(F)→T_n(F) which pre-serve rank-subtractivity, or satisfy rank(A + B) = |rankA - rankB| which impliesrankφ(A + B) = |rankφ(A) - rankφ(B)|, are characterized respectively.(3) The linear mapsφ: S_n(F)→M_m(F) andφ: S_n(F)→S_m(F) whichpreserve group inverses, are characterized respectively, where the character of thefield F is distinguished into two cases, that is chF = 2 and chF = 2.(4) The structure of additive surjective mapφ: T_n(F)→T_n(F) which preservesrank commutativity is obtained. (5) An additive surjective mapφ: M_n(K)→M_n(K) which preserves anon-trivial multiplicative matrix function is characterized, where K is a divisionring whose character is not 2. As applications, some additive surjective mapsφ: M_n(K)→M_n(K) which preserve the Dieudonn′e determinant, invertible ma-trices, are characterized respectively. Some additive bijective maps which preserverank-additivity are also characterized. And then, the structure of additive surjectivemapφ: M_n(Q)→M_n(Q) which preserves detq is obtained, where Q is a quaternionfield.