Neurodynamic Approach for Generalized Eigenvalue Problems

This paper presents a novel neurodynamic approach for solving generalized eigenvalue problems. A series of neurodynamic systems are proposed for finding all eigenvectors to a given pair (A, B) of matrices. Dynamical analysis shows that each system is globally convergent to an exact eigenvector of the pair (A, B) and hence all the eigenvectors can be found inductively by the proposed neurodynamic systems. It also demonstrates that all the neurodynamic systems are primal in the sense that the system´s neural trajectories will never escape from the feasible region when starting at it. By constructing a series of energy functions, all the system´s stable point sets are guaranteed to be the eigenvector sets of the matrix pair. Compared with the existing neural network models for the generalized eigenvalue problems, the new approach has two robust features: 1) all the eigenvectors of the given pair of matrices (A, B) can be approached by the proposed neurodynamic systems while the existing neural network models can not; 2) the proposed systems can globally converge to the problem´s exact eigenvectors while the existing neural network models do not behave proper stabilities