Solving General Joint Block Diagonalization Problem via Linearly Independent Eigenvectors of a Matrix Polynomial

TL;DR

This paper introduces a novel approach to solve the general joint block diagonalization problem using eigenvectors of a matrix polynomial, providing theoretical insights and a practical three-stage method with numerical validation.

Contribution

It presents a new eigenvector-based solution for the GJBD problem, establishing theoretical foundations and an effective three-stage algorithm for both exact and approximate cases.

Findings

Eigenvectors of a matrix polynomial form the diagonalizer for GJBD.

Theoretical proof links approximate GJBD to the exact case.

Numerical results demonstrate the method's effectiveness.

Abstract

In this paper, we consider the exact/approximate general joint block diagonalization (GJBD) problem of a matrix set $\{A_i\}_{i=0}^p$ ($p\ge 1$), where a nonsingular matrix $W$ (often referred to as diagonalizer) needs to be found such that the matrices $W^{H}A_iW$'s are all exactly/approximately block diagonal matrices with as many diagonal blocks as possible. We show that the diagonalizer of the exact GJBD problem can be given by $W=[x_1, x_2, \dots, x_n]\Pi$, where $\Pi$ is a permutation matrix, $x_i$'s are eigenvectors of the matrix polynomial $P(\lambda)=\sum_{i=0}^p\lambda^i A_i$, satisfying that $[x_1, x_2, \dots, x_n]$ is nonsingular, and the geometric multiplicity of each $\lambda_i$ corresponding with $x_i$ equals one. And the equivalence of all solutions to the exact GJBD problem is established. Moreover, theoretical proof is given to show why the approximate GJBD problem can be solved similarly to the exact GJBD problem. Based on the theoretical results, a three-stage method is proposed and numerical results show the merits of the method.