Some Generalizations And Characterizations For Nekrasov Matrices

Generalized strictly diagonally dominant matrix is a class of matrix whichplays an important role in matrix theory and real application. It has numerousapplications in many areas such as numerical algebra, control theory, electricalsystem theory, economical mathematics and statistic. Many scolars pay atten-tions to it. In this paper, we research mainly a kind of special matrices, i.e.,Nekrasov matrix. Nekrasov matrix is a special case of generalized strictly diago-nally dominant matrix, many scholars have researched many aspects of Nekrasovmatrix such as the nonsingularity, determinant and spectral radius for iterationmatrix and they have obtained lots of important results.In this paper, based on some recent documents, we research the proper-ties about Nekrasov matrix, improving and generalizing some conclusions aboutNekerasov matrix.In chapter one, we introduce the importance of generalized strictly diago-nally dominant matrix in theory and applied background. We detail the recentlyimportant results about Nekarasov matrix and recall some defines and conclusionsused in this paper.In chapter two,we first point out some scarceness of some results in recentresearch about Nekrasov matrix. Further, by constructing special matrices andspecial vectores, using the properties of the submatrix of Nekrasov matrix andsome inequalities techniques, we revise this improper conclusions and obtain someequivalent conditions of the generalized Nekrasov matrix. Finally, advantages ofour results are illustrated by a numerical example.In chapter three, we extend the Nekrasov matrix by giving some new kinds ofmatrix, which are the k-layer Nekrasov matrix, the generalized k-layer Nekrasovmatrix and the k-layer weak Nekrasov matrix. Further, by constructing specialpositive diagonal matrices, using the properties and definitions of matrix and someinequalities techniques, we obtain the relationship between these kinds of newmatrix and generalized strictly diaganally dominant matrix. Finally, advantagesof these results are illustrated by a numerical example.In chapter four, we summary all of this paper, and give the author's expec-tation.