In chapter one , we give the definition of the sub - generalized positive definite matrix , study the inverse problem of matix eqution ax = b on the sub - generalized positive definite matrix 第一章，给出了次广义正定矩阵的定义，进而研究了矩阵方程ax = b在次广义正定矩阵类上的反问题解。
When the covariance matrix formed by securities yields is positive definite , we provide the model with transaction costs , the risk is b index risk , researching the model under short sale and no short sale separately 在证券收益率之间的协方差阵为正定矩阵时，给出了以值风险为风险指标的含有交易费的证券组合投资模型，并分别在允许卖空和不允许卖空两种情形下进行了讨论。
In the first part , we shall prove several inequalities involving symmetric positive semidefinite , general m - matrices and inverse m - matrice which are generalization of the classical oppenheim ' s inequality for symmetric positive semidefinite matrices 第一个部分给出了半正定矩阵，一般的m -矩阵以及逆m -矩阵的一些相关不等式，而这些不等式都是有关半正定矩阵的经典的oppenheim不等式的推广。
It gives a number of sufficient conditions on the subpositive definiteness of the matrix product of two subpositive definite matrices , and also provides some necessary and sufficient conditions on the subpositive definiteness of the matrix product of two special subpositive definite matrices 摘要主要给出了两个亚正定矩阵的乘积仍是亚正定矩阵的几个充分条件以及两个特殊的亚正定矩阵的乘积仍是亚正定矩阵的充要条件。
This paper will put the concept of definiting generalized positive definite matrices in paper [ 1 , 4 ] into the further extensions , discuss the inclusion relation betweenness generalized positive definite matrices in all the different definitions , and give four new definitions on an equality with m - matrices 摘要将文[ 1 , 4 ]中定义广义正定矩阵的概念再作推广，并讨论各种不同定义下的广义正定矩阵间的包含关系，给出m -矩阵等价的四种新定义。
In this dissertation , we systematically discuss the fundamental properties and the necessary and sufficient conditions of 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 . finally , we discuss the relation among these four types of positive definite matrices 本论文比较系统地研究了实对称正定矩阵、 hermite正定矩阵和两类广义正定矩阵? ?亚正定矩阵和复正定矩阵的基本性质尤其是它们的若干充分和必要条件，最后还讨论了这四类正定矩阵之间的关系。
If the proper parameter is chosen , the linear pek method is superior to the linear pe method . ( 2 ) in order to improve the linear pe method and the linear pek method , the quadratic pe method and the quadratic pek method are put forward . then the solvability and the convergence of the two methods are discussed about the hermite positive definite matrix and m - matrix , and the interrelated conclusions are obtained ( 2 )将一次pe方法和一次pe _ k方法改进为二次pe方法和二次pe _ k方法，讨论了当系数矩阵a为hermite正定矩阵和m -矩阵时，二次pe方法和二次pe _ k方法的可解性和收敛性等问题，得到了相关的结论。
The second part applies divide - and - conquer algorithm to calculate the eigenvalues of symmetrical matrices . the eigenvalues problem of symmetrical matrices ax = x can be transformed the eigenvalues problem of symmetric tridiagonal matrices tx = ux through householder transform . we divide t into t1 , t2 and apply symmetrical qr algorithm to compute the eigenvalues of t1 , t2 第二部分是利用分而治之算法计算对称矩阵的特征值，对称矩阵特征值问题ax = x ，通过householder变换，转化为三对角对称正定矩阵的特征值问题ty = y ，再将t分割成两个子矩阵t _ 1 ， t _ 2 ，然后利用对称qr方法分别求t _ 1 ， t _ 2的特征值
In this thesis , we study some open problems and conjectures about the linear complementarity problem . it consists of the next three aspects : firstly , we study murthys " open problem whether the augmented matrix is a q0 - matrix for an arbitary square matrix a , provide an affirmable answer to this problem , obtain the augmented matrix of a sufficient matrix is a sufficient matrix and prove the graves algorithm can be used to solve linear complementarity problem with bisymmetry po - matrices ; secondly , we study murthys " conjecture about positive semidefinite matrices and provide some sufficient conditions such that a matrix is a positive semidefinite matrix , we also study pang ' s conjecture , obtain two conditions when r0 - matrices and q - matrices are equivelent and some properties about e0 q - matrices ; lastly , we give a counterexample to prove danao ' s conjecture that if a is a po - matrix , a e " a p1 * is false , point out some mistakes of murthys in [ 20 ] , obtain when n = 2 or 3 , a e " a p1 * , i . e . the condition of theorem 3 . 2 of [ 25 ] that a p0 can be deleted and obtain a e " a is an almost e - matrix if a is a co - matrix or column sufficient matrix 本文分为三个部分，主要研究了线性互补问题的几个相关的公开问题以及猜想： ( 1 )研究了murthy等在[ 2 ]中提出的公开问题，即对任意的矩阵a ，其扩充矩阵是否为q _ 0 -矩阵，给出了肯定的回答，得到充分矩阵的扩充矩阵是充分矩阵，并讨论了graves算法，证明了若a是双对称的p _ 0 -矩阵时， lcp ( q ， a )可由graves算法给出； ( 2 )研究了murthy等在[ 6 ]中提出关于半正定矩阵的猜想，给出了半正定矩阵的一些充分条件，并研究了pang ~ -猜想，得到了只r _ 0 -矩阵与q -矩阵的二个等价条件，以及e _ 0 q -矩阵的一些性质； ( 3 )研究了danao在[ 25 ]中提出的danao猜想，即，若a为p _ 0 -矩阵，则，我们给出了反例证明了此猜想当n 4时不成立，指出了murthy等在[ 20 ]中的一些错误，得到n = 2 ， 3时，即[ 25 ]中定理3 . 2中a p _ 0的条件可以去掉。
It firstly utilizes the characteristic of layout geometric information to partition the global network into some sub networks . after these sub networks are compressed by cholesky factorization , the sub networks edge nodes are combined with other nodes . calculating the final linear equations group , the algorithm can be very effective 算法首先利用版图几何特征，按最小割原理将电源地网划分成子网，利用对称正定矩阵cholesky分解压缩子网内点；将子网等效成与网络其它部分连接的结点集的等效网络，最后通过计算压缩后的网络，获得线性方程组求解答案。
