An Algebraic Approach to Non-Orthogonal General Joint Block Diagonalization

TL;DR

This paper introduces an algebraic method for solving the non-orthogonal general joint block diagonalization problem by transforming it into a system of linear equations and then applying similarity transformations.

Contribution

It establishes a novel algebraic framework linking joint block diagonalization to linear systems and proposes two numerical methods for solving the problem.

Findings

The proposed methods effectively solve the nogjbd problem.

A necessary and sufficient condition for solution equivalence is provided.

Numerical examples demonstrate the methods' advantages.

Abstract

The exact/approximate non-orthogonal general joint block diagonalization ({\sc nogjbd}) problem of a given real matrix set $\mathcal{A}=\{A_i\}_{i=1}^m$ is to find a nonsingular matrix $W\in\mathbb{R}^{n\times n}$ (diagonalizer) such that $W^T A_i W$ for $i=1,2,\dots, m$ are all exactly/approximately block diagonal matrices with the same diagonal block structure and with as many diagonal blocks as possible. In this paper, we show that a solution to the exact/approximate {\sc nogjbd} problem can be obtained by finding the exact/approximate solutions to the system of linear equations $A_iZ=Z^TA_i$ for $i=1,\dots, m$, followed by a block diagonalization of $Z$ via similarity transformation. A necessary and sufficient condition for the equivalence of the solutions to the exact {\sc nogjbd} problem is established. Two numerical methods are proposed to solve the {\sc nogjbd} problem, and numerical examples are presented to show the merits of the proposed methods.