Research On Inertia Of Combinatorial Matrix Theory And Its Application

Matrix theory is an important tool for solving some engineering problems in precision instruments, computer science and other disciplines. As an important branch in matrix theory, the matrix eigenvalue problem has a wide range of applications in engineering and technology problems. In practical application, on the one hand, we need to strengthen the study for the application of eigenvalue theory in engineering technology, and on the other hand, we also need to sduty the basic theory for matrix eigenvalues itself, that is the theoretical foundation for the engineering application. The inertia of matrix is further deepening to the eigenvalue theory, it is a combination research from the numerical research of matrix eigenvalue. So it has important theoretical significance and application value that the matrix theory among the inertia of basic theoretical and applied research.In this paper, we mainly focus on basic issues matrix eigenvalue theory and applied research, and introduce some basic theory of matrix spectral, the perturbation theory of matrix eigenvalue, Gerschgorin Disc theorem and analysis theory of matrix. We introduce sign pattern matrix, zero-nonzero pattern matrix, ray pattern matrix, complex sign pattern, and several common using methods of demonstrating spectrally arbitrary pattern. And we introduce the application of matrix eigenvalue theory for array signal processing. On the basis, we focus on the research of the related problems.The main work and innovations in this dissertation is listed as follows.(1) We discuss the inertias and spectrums of some specific types of sign patterns. Firstly, we show that a zero-nonzero pattern is spectrally arbitrary, and proof that the pattern is also inertially arbitrary. Secondly, we construct a class of almost inertially arbitrary sign pattern matrix by both low-level matrix. Finally, we characterize a minimal inertially arbitrary sign pattern.(2) We consider the spectrum and the number of the nonzero entries of ray pattern matrix. We provide three spectrally arbitrary ray patterns, and we proof that there exist a ray pattern that have exactly 3n ?1 nonzeros.(3) By the extended Nilpotent-Jacobian method, we proof a family of spectrally arbitrary complex sign pattern that have exactly 3n nonzeros.(4) We study the application of matrix eigenvalues theory in array signal processing for vector sensor. Firstly, we proposed a signal number estimation method with eigenvalue based on the traditional method for acoustic pressure array using Gerschgorin Disc theorem. Secondly, based on the feature that the eigenvector is influenced by noise less than the eigenvalue, we proposed a new signal number estimation method with eigenvector using the consistency array steering vector and signal space, Finally, we verify the performance of the proposed algorithm by MATLAB simulation.