Minimum Singular Values ​​and Eigenvalues ​​and Iterative Matrix Spectral Radius Is Estimated

This paper investigates the smallest singular value of matrices, the spectral radius of the iterative matrix, the minimal eigenvalue of irreducible M-matrices and some spectral properties of tensor product of matrices. The thesis consists of four chapters.In chapter 1, based on the technology of block matrices and the properties of block diagonal dominances, new results are obtained for the lower bounds of the smallest singular value. Furthermore, we apply the concept of G-function introduced by Nowsoad and Hoffman to obtain the lower bounds for the smallest singular values of matrices.In chapter 2, for the iterative matrix M^-1N in solving linear system, upper bounds of the spectral radius of some iterative matrix are discussed. Particularly, for some generalized diagonally dominant matrix M, upper bounds of the spectral radius of the iterative matrix M^-1Nare presented.In chapter 3, based on the relationship between the minimal eigenvalue and the spectral radius of non-negative matrices, we derive some others upper and lower bounds on irreducible M-matrices.In chapter 4, the incorrectness of two results in the paper "Disk theory for tensor product of matrix" and the book "spectral theory of matrix" is shown and the cause of these mistakes is pointed out.