Matrix Spectral Estimation And The Inverse N <sub> 0 </ Sub> - The Nature Of The Matrix

For matrices of high orders,it is very difficult to obtain their exact eigenvalues and singular values,so it is particularly important to local the eigenvalues by rows,columns or minors of matrices or local the eigenvalues of AA^* in a similar way.But this method requires intensive computational errorts.On the other side,nonnegative matrices have useful properties,moreover inverse M-matrices are important nonnegative matrices and have valuable application,many scholars are interested in the general properties of inverse M-matrices.For inverse M-matrices,a large number of properties have been studied.However,some beautiful properties of inverse N₀-matrices need to be studied furtherly.The main results are as follows:1.The background of this paper and some current state of eigenvalue and relevant research performance are summarized,and our purposes are presented.2.The general methods to local the eigenvalues of matrices and the relation of eigenvalues and traces are mainly discussed,and the current research achievement of the Perron root are introduced.We also obtain a decreasing sequence of upper bounds for the Perron root of a nonnegative matrix with at most r+1 nonzero eigenvalues using the traces.These upper bounds are new and readily computable.3.The classical results and development trend about the smallest singular value are discussed.We apply the concept of G-function introduced by Nowosad and Hoffman to obtain the lower bound for the smallest singular value of matrices.4.Some beautiful properties of inverse No-matrices are discussed on the basis of perrron complement and generalized Perron complement of inverse M-matrices.The eigenvalues of Hadamard Product of inverse M-matrice and N₀-matrices are estimated on the basis of the eigenvalues estimation of Hadamard Product of inverse M-matrice and M-matrices in order to obtain the bound for the smallest eigenvalues of N₀-matrices in module.