A new algorithm for complex non orthogonal joint diagonalization based on Shear and Givens rotations

TL;DR

This paper presents a novel algorithm for complex non orthogonal joint diagonalization using Givens and Shear rotations, extending the JDi algorithm to complex matrices and demonstrating its effectiveness through extensive comparisons.

Contribution

The paper introduces the CJDi algorithm, a generalization of JDi for complex matrices, combining Givens and Shear rotations for improved joint diagonalization.

Findings

CJDi outperforms existing NOJD algorithms in stability and effectiveness.

The algorithm is effective across various system parameters and application contexts.

Extensive comparative study validates the proposed method's advantages.

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.