Solving the general joint block diagonalization problem via linearly independent eigenvectors of a matrix polynomial. Issue 4 (27th March 2019)

• Record Type: Journal Article
• Authors: Cai, Yunfeng; Cheng, Guanghui; Shi, Decai
• Is Part Of: Numerical linear algebra with applications. Volume 26:Issue 4(2019)
• Page Start: n/a
• Page End: n/a
• Publication Date: 2019-03-27
• Subjects: general joint block diagonalization -- matrix polynomial -- tensor decomposition; Algebras, Linear -- Periodicals; 512.5
• DOI: 10.1002/nla.2238
• Languages: English
• ISSNs: 1070-5325

Abstract

Summary: In this paper, we consider the exact/approximate general joint block diagonalization (GJBD) problem of a matrix set { A i } i = 0 p ( p ≥ 1), where a nonsingular matrix W (often referred to as a diagonalizer) needs to be found such that the matrices W H A i W '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, …, x n ]Π, where Π is a permutation matrix and x i 's are eigenvectors of the matrix polynomial P ( λ ) = ∑ i = 0 p λ i A i, satisfying that [ x 1, x 2, …, x n ] is nonsingular and where the geometric multiplicity of each λ i corresponding with x i 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.