Solving the general joint block diagonalization problem via linearly independent eigenvectors of a matrix polynomial

Abstract

SummaryIn this paper, we consider the exact/approximate general joint block diagonalization (GJBD) problem of a matrix set ( p ≥ 1), where a nonsingular matrix W (often referred to as a diagonalizer) needs to be found such that the matrices W HAiW '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 = [x1,x2,…,xn]Π, where Π is a permutation matrix and xi's are eigenvectors of the matrix polynomial , satisfying that [x1,x2,…,xn] is nonsingular and where the geometric multiplicity of each λi corresponding with xi is equal to 1. In addition, the equivalence of all solutions to the exact GJBD problem is established. Moreover, a 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.