Preservers On Group Inverses Of Symmetric Matrices Problems Over PID

Generalized inverse is widely applied in the fields of numerical analysis, mathematical statistics, surveying and optimization. It especially plays important role in the problems of statistics such as the least square, morbidity linearity, non-linear, ill-posed, regression, estimation of distribution, Markov Chain, the problems of stochastic programming, cybernetics, system identification. The linear preserving not only has important applications in the theoretical research of mathematics, but also has extensively applied backgrounds in the fields of quantum mechanics, differential geometry, system control, mathematical statistics and so on. Along with the further research of linear preserving and generalized inverse, the linear preserving of generalized inverse has extensively applied foregrounds in the future.This paper studies the invariant, which is the linear preserving problem based on generalized inverse of matrices. The generalized inverse of matrices and the research status of preserving problems of generalized inverse are outlined. In the basis of deeply understanding the basic knowledge of linear maps, the definition, characteristics of generalized inverse and decomposition of matrix, the author analyzes the decomposition form of preserving idempotence and preserving tripotence in the PID, then studies on the linear maps form of preserving group inverses of symmetric matrices. The studies on field and PID, which characters are 2, are rarely researched. Because of hard difficulty, the study has no results on the addition maps and pluses more conditions on the base fields and PID when the characters of field and PID are all 2. In this paper, R is a PID with at least four units, M_n(R) is the n×n full matrix algebra over R,S_n(R) isthe n×n symmetric matrix algebra over R and f denotes the linear operator over S_n(R). Using the formal method of characterizing theimages about bases of space, the author gives the forms of linear (?)ijection from S_n(R) to itself preserving group inverses of matriceswhen the characteristic of the PID is 2.