The Technology And Extension Of Minimal Gerschgorin Disc-theorem

In the study of matrix theory, eigenvalue is an important concept and it has been studied by many authors. For the large matrices, it is very difficult to calculate the exact value of their eigenvalues. However, it is unnecessary to find the exact eigenvalues for many problems in practical applications, but only need to estimate the domain where they are. In such circumstances, Gerschgorin disc-theorem and its some improvements are proposed. Nowdays, this theory has gradually becomed perfect. This paper mainly research the minimal Gerschgorin disc-theorem. Firstly, in this work we will simply improve the Gerschgorin disc-theorem and reduce the disc region theoretically. But in practical applications, it is very difficult to obtain the dominant expression of these Gerschgorin sets. Therefore, approximate sets of the minimal Gerschgorin of the general matrix and irreducible matrix are investigated. Secondly, we simply introduce and research the Gerschgorin disc-theorem of generalized eigenvalues in this article.Chapterl is the introduction, we introduce the development of Gerschgorin disc-theorem and minimal Gerschgorin disc-theorem, and describe the basic concepts and notes.Chapter2, by using the Gerschgorin disc-theorem, is to present the minimal Gerschgorin area accurately. Moreover by using the given special collections, we will obtain approximate disc area of the general matrices. While for the irreducible matrix, its particular properties and appropriate methods are used to get the corresponding approximate circular region.Chapter3 will apply Gerschgorin disc-theorem to the generalized eigenvalue and classify them according to whether the matrix B is a strictly diagonally dominant. Finally, the disc-area and the minimum disk theorem of generalized eigenvalues are obtained.