The Random Perturbation Analysis Of Matrix Eigenvalue Problems

Matrix eigenvalue perturbation analysis theory has many important applications in studying the stability of the matrix eigenvalue algorithm etc,and in general,the error in actually are random,so the random perturbation of matrix eigenvalue is a very significative problem.In this paper,we mainly study the random perturbation problem of the eigenvalues of symmetric matrix with the background of spectral clustering algorithm,and we also consider the absolute perturbation bounds of generalized eigenvalue.Firstly,the mathematical problem based on the stability of spectral clustering algorithm had been proposed,we use the determined perturbation bounds of symmetric matrix eigenvalue and the properties of random matrices to refine the permissible range of perturbation of this problem;we also extend this problem and obtain the corresponding results by the step of spectral clustering algorithm.Secondly,for the perturbation problem of generalized eigenvalue of Hermitian positive definite pair,we obtain a new absolute perturbation bound by the first-order analytical expression of generalized eigenvalue;simultaneously,we transform the generalized eigenvalue problem to the standard Hermitian eigenvalue problem,and combine to the first-order analytical expression of Hermitian matrix eigenvalue,then we obtain two new absolute perturbation bounds.Finally,we show the refined permissible range and the new absolute bounds of generalized eigenvalue are sharper than before by the numerical simulation experiments.