A Novel Unified and Self-Stabilizing Algorithm for Generalized Eigenpairs Extraction

Generalized eigendecomposition problem has been widely employed in many signal processing applications. In this paper, we propose a unified and self-stabilizing algorithm, which is able to extract the first principal and minor generalized eigenvectors of a matrix pencil of two vector sequences adaptively. Furthermore, we extend the proposed algorithm to extract multiple generalized eigenvectors. The performance analysis shows that only the desired equilibrium point of the proposed algorithm is stable and all others are (unstable) repellers or saddle points. Convergence analysis based on the deterministic discrete-time approach shows that, for a step size within a certain range, the norm of the principal/minor state vector converges to a fixed value that relates to the corresponding principal/minor generalized eigenvalue. Thus, the proposed algorithm is a generalized eigenpairs (eigenvectors and eigenvalues) extraction algorithm. Finally, the simulation experiments are carried to further demonstrate the efficiency of the proposed algorithm.