【作者】 陈涛；

【导师】 游宏；

【作者基本信息】 哈尔滨工业大学 ， 基础数学， 2013， 硕士

【摘要】 线性保持问题是矩阵理论研究领域中的一个重要课题,主要考虑保持矩阵的某些特殊性质和不变量的映射和算子,它在微分方程,系统控制等领域都有着广泛的应用。近十几年,人们又将限制在映射上的条件削弱,如,将“线性”改为仅保“加法运算”等。本文在介绍保持问题的背景和发展概况之后,讨论了交换局部环上全矩阵模上的保持矩阵逆的加法单射。主要结果如下：设R为含1交换局部环,2,3∈R*,则f是Mn(R)到Mn(R)的保持矩阵逆的加法单射当且仅当f为如下两种形式之一i)存在P∈GLn(R)使得f(A)=εPAδJP-1对任意A∈Mn(R)成立,其中ε=±1；ii)存在P∈GLn(R)使得f(A)=εP(AT)σP-1对任意A∈Mn(R)成立,其中ε=±1,AT是矩阵A的转置。上两式中,δ为R到自身的单自同态。更多还原

【Abstract】 Linear preserver problem(LPP) is a very active topic in the field of matrix theory, which concerns the maps and operators preserving some invariant of matrices. It has been widely applied in the differential equations, systems control and other fields. In the recent decades, the limits on LPP has been weakened, such as the linear condition has been substituted by additive or some other ones.After introducing the background and the development of preserver problem, we study the problem of additive maps preserving inverses of matrices from matrix modules onto matrix modules over commutative local rings. The main results obtained in this thesis are as follows:Let R be a commutative local ring with identity and2,3∈R*. Then f is an additive injective map from Mn(R) to Mn(R) that preserves inverses of matrices, if and only if f is one of the following two formsi) there exits P∈GLn(R) such that for all A∈Mn(R), f(A)=εPAδP-1, where ε=±1.ⅱ) there exits P∈GLn (R) such that for all A∈Mn (R), f(A)=εP(AT)δP-1, where ε=±1, AT is the transpose of matrix A. In ⅰ) and ⅱ),δ is an injective endomorphism of R.更多还原