Estimation Of Matrix Eigenvalues And Bounds Of The Spread Of Matrices

Matrix theory is the core of linear algebra and the matrix eigenvalues is the main research in matrix theory. Matrix eigenvalues in quantum mechanics, factor analysis, image processing, mechanics, solid mechanics, pedigree chart have important meanings, such as the frequency in dynamics, the ultimate load in stability analysis, the principal stress in stress analysis and so on. This paper studies the distribution and estimation of matrix eigenvalues、 the spread of matrices and the stability of dynamical systems, which include:1. The distribution of matrix eigenvalues are studied and the elliptical distribution area of matrix eigenvalues are obtained, which is more precise compared with circle domain in the previous articles. Also, the distribution and estimation of interval matrix eigenvalues are studied.2. By constantly improving the upper bounds of squares sum of the module of matrix eigenvalues, three estimation results on the spread of a matrix are obtained. One of the most precise estimation is the following result:At the same time, the estimation of matrix spectral radius are proposed. In addition, by the idea of optimization, the upper bounds of the spread of matrices is presented. Furthermore, the estimation of upper bounds of real spread and imaginary spread are given, respectively.3. By making use of block matrices, the estimation on upper bounds of the spread of matrices are obtained: Also, by the elliptical distribution area of block matrix, the estimation of upper bounds of the spread of matrices and the real spread and imaginary spread of block matrices are presented, respectively.4. Based on the distribution of interval matrix eigenvalues, the spread of interval matrices are discussed and two results on estimation of upper bound of the spread of interval matrices are obtained. Especially, for real symmetric interval matrices, the following estimation are given:5. Based on requirements of the equilibrium point’s stability of multi-dimensional nonlinear dynamical systems, a series of judgment methods on stability and degree of stability of dynamical system and the stability of interval dynamical systems are presented.