Study On The Estimation And The Location Of The Matrix Eigenvalues And The Algebraic Polynomial Roots In Complex Domain

Matrix theory and matrix analysis have a wide application in computationalmathematics, physics, economics, image processing and engineering science. This papermainly studies the estimation and location of numerical characteristics of the matrix andthe estimation areas of characteristic roots of the polynomial, the main contents andinnovations are as follows:1. Using a new way, we prove all eigenvalues of arbitrarily given n×ncomplexmatrix with its characteristic polynomial having real coefficients can be located by thefollowing elliptic region:2. With the properties of singular matrix and further study, we present a ellipticalregion contains all eigenvalues of a singular matrix, and we get some more accurateinferences.3. Based on some basic conclusions of the linear algebra and the particularity ofthe normal matrix, we get the estimation of all eigenvalues of the normal matrix,compared to the previous results to be accurate.4. Based on the particularity of block matrix and further study, we found that alleigenvalues of the complex matrix are located in a new disk which is more accuratethan the disks before.5. From the point of the view of algebra, we inversely think the calculation of thematrix algebra roots of the polynomial and associate the estimation and location of thepolynomial roots with the estimation and location of the matrix eigenvalues to discussthe estimation area of characteristic roots of the polynomial.