Study On The Numerical Characteristics Of Matrices And The Trace Dominant Matrices

Matrix analysis and matrix computations have wide applications in mathematic physics, computational mathematics, control theory, image analysis and economics etc. The purpose of this thesis is to discuss the numerical characteristics of matrices and the properties and iterative algorithm of trace dominant matrices. The main results and innovations are as follows:1. The estimations for lower bounds for the rank of non-normal matrix are investigated. Three lower bounds for the rank are obtained. In many cases, the estimations are more accurate than Ky Fan-Hoffman inequality.2. For h ( A) > 0 such that , we prove that all the eigenvalues are located in the following disk Based on matrix partition, similarity transformation and mean inequality the disk theorems which conclude all the eigenvalues are shown.3. Some improvements of Brauer theorem and Ky Fan theorem are presented, the eigenvalues inclusion regions are given as follows and The distributions for the eigenvalues of tensor product of matrices are obtained, several estimations for the real part and imaginary part of eigenvalues are shown.4. Based on singular value decomposition and inequality theories, the upper bound and lower bound for the singular value are obtained, the following inequality holds5. Several properties of trace dominant matrices and generalized trace dominant matrices are obtained based on the concepts of trace dominant matrices and generalized trace dominant matrices. Combining with the optimization theory and the properties of generalized trace dominant matrices, an algorithm is presented to determine whether a matrix is a generalized trace dominant matrix or not. Some numerical examples show the effectiveness of our algorithm.