Weighted non‐linear criterion‐based adaptive generalised eigendecomposition

Generalised eigendecomposition problem for a symmetric matrix pencil is reinterpreted as an unconstrained minimisation problem with a weighted non‐linear criterion. The analytical results show that the proposed criterion has a unique global minimum which corresponds to the principal generalised eigenvectors, thus guaranteeing the global convergence via iterative methods to search the minimum. A gradient‐based adaptive algorithm and a fixed point iteration‐based adaptive algorithm are derived for the generalised eigendecomposition, which both work in parallel and avoid the error propagation effect of sequential‐type algorithms. By applying the stochastic approximation theory, the global convergence of the proposed adaptive algorithm is proved. The performance of the proposed method is evaluated by simulations in terms of convergence rate, estimation accuracy as well as tracking capability.