The joint eigenvalue decomposition algorithm based on first-order Taylor expansion via the exterior penalty function method

Abstract

The Joint Eigenvalue Decomposition (JEVD) problem has many applications, especially in the field of signal processing. In this paper, we establish two new algorithms to solve the JEVD for the real or complex case. As the JEVD algorithm requires to inverse a matrix, it can be usefully derived by using a first order Taylor expansion under a reasonable condition.In order to possibly enforce the condition, we propose to take advantage of the Penalty Function Method. By adding a quadratic penalty to the regularized criterion, we turn the constrained problem into a single unconstrained problem. Consequently, we show that this regularized criterion leads to a regularized optimal solution. Thus, we propose the Joint Eigenvalue Decomposition Algorithm Based on First-order Taylor Expansion via the Exterior Penalty Function Method (JDTE-EP) and Sweeping JDTE-EP (SJDTE-EP) algorithms based on this idea. And the two algorithms turn out to enjoy low complexity and good accuracy in all numerical experiments.