Common Matrix Theory And Matrix Algorithm In Linear Model

Matrix theory has come into wide use in many branches of the statistics,economics,engineering calculation and has become an indispensable tool in today's scientific computing, With the development of matrix theory, matrix theory also has a wide application in linear model. In this paper, some further problems related linear model are introduced in the matrix theory and get some meaningful results. Those results are generalized based on the original results and also discuss the problem of matrix convergence speed in matrix algorithm.The main results of this paper are listed in the following.1. In chapter three, we mainly get the extension of the two types of matrix inequalities. Firstly, in section II of this chapter,we introduce the extensions of the matrix Cauchy-Schwarz and Marshall Olkin inequalities. Cauchy-Schwarz inequalities play an important role in linear model. Marshall and Olkin generalized it to matrix versions. In this section, we get new extensions of the Cauchy-Schwarz and Marshall Olkin inequalities. Its applying scope is expanded. Secondly, in section III of this chapter, we get the extension of the inequalities of Hermitian matrix trace. The trace inequality of matrix is an important aspect in matrix theory , The trace inequality of matrix has a wide use in many branches.In this section the results have been extended to the general form,Those results are also very useful in the linear model.2. In chapter four, we simply introduce the property of Moore-Penrose generalized inverse matrix. With regard to the properties of the generalized inverse matrix ,[1-2]summarized the research founding before the 1990s, [3]summarized some properties of the plus generalized inverse matrix,in the more stringent conditions, this article proves three new properties of plus generalized inverse matrix.3. In chapter five, we introduce the problem of matrix convergence speed . In this chapter, we present the preconditioned AOR-type iterative method for solving the linear systems Ax = b, where A is a Z-matrix, and prove its convergence. Then we give some comparison theorems to show that the rate of convergence of the preconditioned AOR-type iterative method is faster than the rate of convergence of the AOR-type iterative method. Finally, we give a numerical examples to illustrate our results.