The Diagonally Dominant Degree For The Schur Complement Of Diagonally Dominant Matrices And Its Applications

Matrix theories has been in a signifcant position when we comes to scienceresearch and specifc practice. This paper is to study the theory for the Schur com-plement of matrices which is one of the important parts of matrix theories. Andthe Schur complement plays an important role in many mathematical problems,particularly in matrix analysis, statistics and numerical analysis.Many domestic and foreign scholars have made a great of research on the Schurcomplement of matrices. In this paper, applying the properties of Schur comple-ment, utilizing some inequality techniques, some new estimates of the diagonallydominant degree on the Schur complement of diagonally dominant matrices areobtained, which improve some relative results. Further, as applications of thesederived results, we present some distribution for the eigenvalues of the Schur com-plement and new bounds of the determinant of diagonally dominant matrices.In chapter one, we frst introduce some background knowledge and recent worksfor the Schur complement. Then we give some basic symbols and defnitions usedin our paper.In chapter two, by combining with the criterion of H matrix and constructingthe suitable H matrix, applying some techniques of inequalities, we get several newestimations of diagonally dominant degree, γ diagonally dominant degree and prod-uct γ diagonally dominant degree, respectively. Moreover, we use the estimationsof the determinants of the diagonally dominant matrices and employ inequalitiestechniques to extend some existing results. And some numerical examples are givento verify their efectiveness.In chapter three, we obtain some new estimations of bounds for the eigenvaluesof the diagonally dominant matrices by using the Ostrowski T heorem. Accordingto many numerical examples, we further explain the results in our paper is superiorto the former.