TAILIEUCHUNG - Báo cáo toán học: "The Bounds on Components of the Solution for Consistent Linear Systems"

Đối với một hệ thống phù hợp tuyến tính Ax = b, trong đó A là Z-ma trận đường chéo chi phối, chúng tôi trình bày các ràng buộc trên các thành phần của giải pháp cho hệ thống tuyến tính, khái quát các kết quả tương ứng thu được bằng cách Milaszewicz et al. [3]. | Vietnam Journal of Mathematics 33 1 2005 91-95 V Í e It ini ai m J o mt r im ai I of MATHEMATICS VAST 2005 The Bounds on Components of the Solution for Consistent Linear Systems Wen Li Department of Mathematics South China Normal University Guangzhou 510631 P. R. China Received February 27 2004 Abstract. For a consistent linear system where A is a diagonally dominant -matrix we present the bound on components of solutions for this linear system which generalizes the corresponding result obtained by Milaszewicz et al. 3 . 1. Introduction and Definitions In 2 3 the authors consider the following consistent linear system . A 1 where A is an A A A A -matrix A is an A dimension vector in . The study of the solution of the linear system 1 is very important in Leontief model of input-output analysis and in finite Markov chain see 1 2 . In this article we will discuss a special A -matrix linear system when the matrix A in linear system 1 is a diagonally dominant -matrix this matrix class often appears in input-output model and finite Markov chain . see 1 . In order to give our main result we first introduce some definitions and notations. Let A be a directed graph. Two vertices A and A are called strongly connected if there are paths from A to A and from A to A. A vertex is regarded as trivially strongly connected to itself. It is easy to see that strong connectivity defines an equivalence relation on vertices of A and yields a partition Thls work was supported by the Natural Science Foundation of Guangdong Province 31496 Natural Science Foundation of Universities of Guangdong Province 0119 and Excellent Talent Foundation of Guangdong Province Q02084 . 92 Wen Li of the vertices of The directed subgraph G Vi with the vertex set i of G is called a strongly connected component of G G G be an associated directed graph of A nonempty subset of G is said to be a nucleus if it is a strongly connected component of G see 3 . For a nucleus G K denotes the set of indices

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