H- Matrix (tensor) Identification And Schur Complement Research

The theory of H-matrices is one of the important research topics in numerical algebra and matrix analysis, and has a wide range of applications in many fields such as computational mathematics, cybernetics, optimization theories, mechan-ics, management science and engineering and so on. But in practice, it is difficult to determine a matrix, especially for a large scale matrix, is an H-matrix or not, so to study the criteria for H-matrices has important theoretical and practical significance. The Schur complements of matrix play an important role in many areas such linear control theories, matrix theories, numerical analysis and statis-tics etc.. The class of higher order tensors is a generalization of matrices, and has a wide range of practical applications, such as data mining, data analysis, and signal processing etc..In this dissertation, we research the criteria for H-matrices, the Schur com-plement problems, algorithm for indentifying H-tensors. The main results are as follows.In chapter one, we introduced some background knowledge, significance and the main works of this dissertation.In chapter two, we obtain some new criteria for H-matrices by constructing positively diagonal matrix factors progressively. Meanwhile, by improving iter-ation factor and applying interleaved iterative method, we present an iterative algorithm for identifying H-matrices.In chapter three, we focus on the Schur complement problems of special H-matrices. By constructing low dimension matrix which keep original matrix’s properties to estimate the diagonally dominant degree and the a-diagonally dom-inant degree of the Schur complements of special H-matrices. We obtain that the diagonally dominant degree of each row of the Schur complement is superior to the diagonally dominant degree of relevant row of its original matrix. Fur-thermore, using the Gersgorin disc theorem, the Ostrowski disc theorem and the Brauer Ovals theorem, we obtain some bounds for the eigenvalues of the Schur complements of special H-matrices, which are expressed by the original matrix’s entries. In chapter four, we research the criteria for H-tensors as a generalization of H-matrices. We obtain some new sufficient conditions of H-tensors, which provide an algorithm to identify H-tensors. As an application, we construct an algorithm to identify the positive definiteness of an even-degree homogeneous polynomial form.