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Solving the general joint block diagonalization problem via linearly independent eigenvectors of a matrix polynomial
Author(s) -
Cai Yunfeng,
Cheng Guanghui,
Shi Decai
Publication year - 2019
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2238
Subject(s) - mathematics , invertible matrix , eigenvalues and eigenvectors , block matrix , combinatorics , matrix polynomial , matrix (chemical analysis) , polynomial matrix , diagonal matrix , diagonal , permutation matrix , exact solutions in general relativity , multiplicity (mathematics) , diagonalizable matrix , symmetric matrix , polynomial , discrete mathematics , pure mathematics , mathematical analysis , circulant matrix , geometry , physics , materials science , quantum mechanics , composite material
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 pλ iA 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.

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