A New Algorithm for Complex Non-Orthogonal Joint Diagonalization Based on Shear and Givens Rotations

Abstract

This paper introduces a new algorithm to approximate non orthogonal joint diagonalization (NOJD) of a set of complex matrices. This algorithm is based on the Frobenius norm formulation of the JD problem and takes advantage from combining Givens and Shear rotations to attempt the approximate joint diagonalization (JD). It represents a non trivial generalization of the JDi (Joint Diagonalization) algorithm (Souloumiac 2009) to the complex case. The JDi is first slightly modified then generalized to the CJDi (i.e. Complex JDi) using complex to real matrix transformation. Also, since several methods exist already in the literature, we propose herein a brief overview of existing NOJD algorithms then we provide an extensive comparative study to illustrate the effectiveness and stability of the CJDi w.r.t. various system parameters and application contexts.Index Terms-Non orthogonal joint diagonalization, Performance comparison of NOJD algorithm, Givens and Shear rotations.

Separation of Dependent Autoregressive Sources Using Joint Matrix Diagonalization

“…Note that this problem is different from the one considered in [14] which is the joint diagonalization by congruence. JUST estimates the diagonalizer matrix W as a product of generalized Givens rotations 2 which minimize iteratively the off-diagonal elements of transformed matrices M ′ k :…”

Blind modal identification of non-classically damped structures under non-stationary excitations

“…Based on Eq. , we can employ complex‐valued non‐unitary joint approximate diagonalization methods (e.g., ) on all matrices to identify the complex‐valued mode shape matrix. It may be even more preferable to work with real‐valued data instead of complex‐valued ones.…”

Hybrid Joint Diagonalization Algorithms

“…We propose to minimize (3) using real Givens rotations G p,q (θ). However, to preserve the special structure of A, one needs to simultaneously apply Givens rotations G p,q (θ) and G p+n,q+n (θ) and also simultaneously apply G p,q+n (θ ) and G q,p+n (θ ) (for more details refer to [12]). The unmixing matrix is defined as The solutions are the unit-norm principal eigenvectors of matrices Q, Q and Q , respectively.…”

“…, which has the advantage of closed-form solutions for the optimal pairs of parameters (θ, y) and (θ , y ). Motivated by the effectiveness of the CJDi algorithm especially in the adverse scenarios (see [12] for details), we propose to generalize it for solving our hybrid JD problem. As in [12], we proceed by first transforming the…”