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Therefore, in order to offer reference to readers, the paper systematically expound and prove the eigenvalue of special matrix that base on idempotent matrix, antiidempotent matrix, involutory matrix, anntiinvolutory matrix, nilpotent matrix, orthogonal matrix, polynomial matrix, the shape of , matrix, diagonal matrix, invertidle matrix, adjoint matrix, similar matrix, transposed matrix, numerical matrix, companion matrix, and practicality and superiority of the achievement was showed by some examples.

为此本文系统地阐述幂等矩阵，反幂等矩阵，对合矩阵，反对合矩阵，幂零矩阵，正交矩阵，多项式矩阵，形为：，矩阵，对角矩阵，可逆矩阵，伴随矩阵，相似矩阵，转置矩阵，友矩阵一系列特殊矩阵的特征值问题并加以证明，并通过一些具体例子展示所得成果的实用性和优越性。

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In this paper, firstly, not only the incidence matrix ,adjacent matrix, cycle matrix, cut-set matrix of an undirected graph are summarized, but also the close contact between a graph and its corresponding matrix are discussed ; secondly, many problems of a graph which are solved by analysing its matrix are listed as follows:1、The co-tree set of a graph is obtained by using its cycle-matrix ; 2、The branches of its spanning tree are given by using its cut-set matrix ; 3、By making use of the incidence matrix of a graph ,not only its vertex cut 、cut vertex 、isolated point and spanning tree can be obtained ,but also the two sides which are whether parallel or not can be judged ;4、By using their adjacent matrix ,the two graphes which are whether isomorphous or not can be judged; once more, there is a detailed introduction in view of special graph (for example: bigaritite graph ,regular graph and so on);last but not least, a graph method of calculating the N power of a matrix is given and the practical applications of the theorem for degree is indicated.

本文首先综述了无向图的关联矩阵，邻接矩阵，圈矩阵，割集矩阵以及图和它对应矩阵之间的关系；其次总结出了利用上述各类矩阵可以解决的图的若干问题：1、利用图的圈矩阵可以求其连枝集；2、利用图的割集矩阵可以求其生成树的树枝；3、利用图的关联矩阵不仅可以求其割点、点割集、连通度、孤立点和生成树，而且可以判断两条边是否平行；4、利用图的邻接矩阵可以判断两个图是否同构；再次，针对特殊图（例如：二分图、正则图等等）的邻接矩阵作了详细介绍；最后，得到了利用图计算矩阵的N次幂的方法，指出度数定理的实际应用。

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.

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。

Summary: The concept of matrix and its determinant computing, matrix determinant, matrix sub-block with the elementary transformation, invertible matrix, rank of matrix; vector and its computation, the linear relationship between vector, vector group of rank; linear equations of the nature and structure of linear equations; matrix eigenvalue and eigenvector, similar to matrix and matrix diagonalization conditions, the standard quadratic form with the normal forms, quadratic and symmetric matrix There are qualitative.

内容提要：行列式矩阵的概念及其运算，方阵的行列式，矩阵的分块与初等变换，可逆矩阵，矩阵的秩；向量及其运算，向量间的线性关系，向量组的秩；线性方程组的性质与结构，线性方程组的求解；矩阵的特征值与特征向量，相似矩阵与矩阵可对角化条件，二次型的标准形与规范形，二次型和对称阵的有定性。

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Therefore, in order to offer reference to Readers, based on idempotent matrix, involutory matrix, nilpotent matrix, diagonal matrix, the main character of special matrix are proved in this paper after the Defined and algorithm of eigenvalue of matrix .for example , some problems of the eigenvalues of matrix are solved in a special method based on the eigenvalues of matrix .

为此， 本文除了介绍矩阵特征值的定义和算法外，还围绕幂等矩阵、幂零矩阵、对角矩阵、等特殊矩阵给出了其主要性质并加以证明，同时还介绍了一些特殊矩阵的特征值的算法，例如：本文利用矩阵的特征值，对与矩阵的特征值相关的一些典型问题给出了较好的处理方法。

On the basis of the definition of matrix traces , this paper discusses their characteristics at first and then according to the norm of the F of square matrix and Cauchy-Schwarz inequality gives how to prove the zero matrix, unsimilar matrix, number cloth matrix, column matrix idempotent matrix and non-equality matrix.

根据矩阵迹的定义，首先给出了矩阵迹的性质，然后依据方阵的F—范数定义Cauchy—Schwarz不等式，给出了零矩阵，不相似矩阵，数幂矩阵，列矩阵，幂等矩阵及矩阵不等式的证法。

This dissertation mainly investigated two frameworks of H-matrix, such as SPB framework consisting of integer Subscript matrix, Permutation matrix and Bidiagonal matrix and MSPT framework consisting of Masking matrix, sparse Subscript matrix, Permutation matrix and approximately lower Triangular array matrix.

本文主要研究了两种H矩阵的类随机框架结构模型，一是SPB框架，由整数下标矩阵、置换矩阵、双对角线矩阵构成；二是MSPT框架，由稀疏下标矩阵、模板矩阵、置换矩阵和近似下三角阵列矩阵构成。

Selfconjugate matrix, skewselfconjugate matrix, perselfconjugate matrix, skewperselfconjugate matrix, centrosymmetric matrix, skewcentrosymmetric matrix, bisymmetric matrix, and skewbisymmetric matrix over a ring with an involutorial antiautomorphism are defined. Significant criteria for matrices to be bisymmetric and skewbisymmetric are obtained.

在具有对合反自同构的环上定义了自共轭矩阵，斜自共轭矩阵，广自共轭矩阵，斜广自共轭矩阵，中心对称矩阵，斜中心对称矩阵，双对称矩阵和斜双对称矩阵，建立了双对称矩阵和斜双对称矩阵的重要判定定理。

"Yes, now you can give yourself airs," she said, you have got what you wanted.

"对了，您现在高兴了，"她说道，这是您所期待的。

Then the LORD said to me: Rebel Israel is inwardly more just than traitorous Judah.

上主于是对我说：＂失节的以色列比失信的犹大，更显得正义。