( 2 ) on the results of new quadratic pek method , the quadratic epek method is discussed . the convergence of the quadratic epek method is proved in hermite positive definite matrix . then the condition of hermite positive definite matrix is extended to the condition of the positive - definable matrix ( 2 )在新型二次pe _ k方法的基础上，讨论了新型二次pe _ k方法的外插迭代二次epe _ k方法的收敛性，证明了当系数矩阵为对称正定矩阵时二次epe _ k方法的收敛性，并将在正定矩阵条件下收敛的情形推广到了可正定化矩阵的情形。
The relation between some special matrices and its squares are discussed , in the sense of the star partial ordering , the minus partial ordering and lower partial ordering , the related results of the definite positive matrices are extended , some related results on the general matrices are obtained 研究矩阵及其平方矩阵偏序之间的关系，解决了j . k . baksalary和f . pukesheim提出的公开问题，并推广到更一般的情形，对于一般矩阵同时得到了若干新结果，推广了关于(半)正定矩阵的相应结果。
It is to be discussed as follows : ( 1 ) the new quadratic pek method is derived on the base of incompletely lu triangular decomposition . based on the results , the convergence of the new quadratic pek method is proved about the hermite positive definite matrix , m - matrix and h - matrix . in the end , some examples are given to illustrate that the convergence of the new quadratic pek method is better than pe method and quadratic pe method to proper parameter k 本文对这种类型系数矩阵的方程组给出了一种迭代解法，主要讨论了如下几个问题： ( 1 )在对系数矩阵进行不完全lu分解的基础上，导出了新型二次pe _ k方法，对系数矩阵a为hermite正定矩阵， m -矩阵及h -矩阵，证明了新型二次pe _ k方法的收敛性，并且在最后通过实例计算说明了对适当的参数k ，新型二次pe _ k方法比pe方法及二次pe方法的收敛性都好。
In chapter 2 , we give some important properties of metapositive definite matrices , the expression of real and imaginary part about eigenvalues of real square matrices as well as the range of eigenvalues of real square matrices ( especially the range of their imaginary part ) . on the other hand , we sum up , establish and extend some necessary and sufficient conditions of metapositive definite matrices roundly . in chapter 3 , we give some important properties of complex positive definite matrices , the expression of real and imaginary part about eigenvalues of complex square matrices as well as the range of eigenvalues of complex square matrices . on the other hand , we sum up , establish and extend some necessary and sufficient conditions of complex positive definite matrices roundly . in summary , we discuss in terms of set theory the relation among real symmetric positive definite matrices , hermitian positive definite matrices and two types of generalized positive definite matrices - metapositive definite matrices and complex positive definite matrices 第一章：除引入hermite矩阵、 hermite正定矩阵和实对称正定矩阵较重要的性质外，还建立了它们的其它一些重要性质，尤其较全面地总结和建立了它们的若干充分和必要条件；第二章：建立了亚正定阵的一些重要性质，给出了实方阵的特征值实部和虚部的表达式，并导出了实方阵的特征值范围尤其是导出了其虚部范围，同时着重较全面地总结、建立和推广了亚正定阵的若干充分和必要条件；第三章：建立了复正定矩阵的一些重要性质，给出了复方阵的特征值实部和虚部的表达式，并导出了复方阵的特征值范围，同时着重较全面地总结、建立和推广了复正定矩阵的若干充分和必要条件；全文小结：从集合论的观点出发，对已讨论过的四类正定矩阵? ?实对称正定矩阵、 hermite正定矩阵、亚正定矩阵和复正定矩阵之间的关系作了比较细致的论证与探讨。
