The Eigenvalue Distribution Of The Schur Complement For Some Kinds Of Matrix With Diagonally Dominant Substructure

The matrix theory is widely used in areas of control theory, computationmathematic, statistics and so on. The Schur complement theory, as one of im-portant branches of matrix theory, plays an vital role in the reducing order oflarge scale matrices and is one of important topics in the felds of numericalalgebra and matrix analysis.In this paper, we generalize some kinds of criterions for H-matrix and blockH-matrix; and present some eigenvalue distributions of the Schur complementfor some kinds of matrices with diagonally dominant substructure by the entriesof the original matrix, which generalize and improve some existing results.In chapter one, we present some background knowledge and recent worksfor the theory of the Schur complement and generalized Schur complement; andintroduce some basic symbols and defnitions in this paper.In chapter two, by using the equivalent relations of H-matrix with gener-alized γ-diagonally dominant matrix and generalized product γ-diagonally dom-inant matrix, and according to some inequality scaling techniques, we obtainsome criterions of H-matrix and block H-matrix, which generalize some existingresults.In chapter three, by applying the criterions of H-matrix and block H-matrixwhich obtained in chapter two, and according to some properties of set opera-tions, some properties of Schur complement determinant and the nonsingularityof H-matrix and block H-matrix, we obtain some eigenvalue distributions of theSchur complement for some kinds of matrix with the diagonally dominant sub-structure by the entries of original matrix, which generalize and improve someexisting results; furthermore, by using matrix singularity decomposition and someproperties of unitary matrices and spectral norm, we obtain an distribution forthe eigenvalue of generalized Schur complement.