An Alternating Projection Algorithm for Approximate Simultaneous Diagonalization

Abstract

In this paper, we present a novel formulation of Approximate Simultaneous Diagonalization (ASD) with a nonconvex feasibility problem to find a structured low rank matrix whose building blocks are the Kronecker sums of given multiple matrices. To tackle this feasibility problem, we propose an alternating projection algorithm that can generate a matrix sequence approaching monotonically to a solution. By this algorithm, simultaneously diagonalizable matrices are obtained in the neighborhood of the given matrices which are not necessarily diagonalizable simultaneously. By using further the Diagonalize-One-then-Diagonalize-the-Other (DODO) method, we can obtain finally a common similarity transformation which diagonalizes the simultaneously diagonalizable matrices. Numerical experiments show that, compared with a Jacobi-like method, the proposed algorithm achieves a better approximation to the desired common similarity transformation.