主理想整环上保对称矩阵群逆的线性算子

Linear operator on group inverses of symmetric matrices over principal ideal domain

广义逆在数值分析、数理统计、测量学和最优化等领域具有广泛重要的应用,尤其是在最小二乘问题,病态线性、非线性问题,不适定问题,回归、分布估计、马尔可夫链等统计问题,随机规划问题,控制论和系统识别问题等等研究中,广义逆更是发挥着重要的作用.线性保持问题不仅在数学理论研究中有重要应用,而且在量子力学、微分几何、系统控制、数理统计等领域有着广泛的实际应用背景.随着对广义逆和线性保持问题的深入研究,使得广义逆的保持问题有着广泛的实际应用前景.在文中,R是一个特征为2的可交换的主理想整环,至少有4个单位.利用刻画基底的形式证明了主理想整环上保持对称矩阵群逆的可逆线性算子的形式.

Generalized inverses are widely applied in the fields of numerical analysis, mathematical statislics, surveying and optimization. They play a particularly important role in statistical problems such as least square, morbidity linearity, non-linearity, ill-posed property, regression, estimation of distribution, Markov chains, stochastic programming, cybernetics, and system identification. The linear preserver problem not only has important applications in theoretical mathematics, but also has extensive applications in the fields of quantum mechanics, differential geometry, system control, mathematical statistics and so on. Along with further research on linear preservers and generalized inverses, the linear preserver of the generalized inverse has many potential applications. In this paper, let R be a PID（principal ideal domain）, ch= 2, has at least four units. Using the formal method of characterizing the images about bases of space, it is proved that f is the invertible linear operator from Sn （R） into Sn （R） preserving group inverses of symmetric matrices, if and only if there exists.