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Existence of a low rank or ℋ︁‐matrix approximant to the solution of a Sylvester equation
Author(s) -
Grasedyck L.
Publication year - 2004
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.366
Subject(s) - rank (graph theory) , mathematics , generalization , matrix (chemical analysis) , sylvester equation , combinatorics , singular value , sylvester matrix , complex matrix , pure mathematics , mathematical analysis , eigenvalues and eigenvectors , physics , quantum mechanics , chemistry , chromatography , polynomial matrix , matrix polynomial , polynomial
We consider the Sylvester equation AX − XB + C =0 where the matrix C ∈ℂ n × m is of low rank and the spectra of A ∈ℂ n × n and B ∈ℂ m × m are separated by a line. We prove that the singular values of the solution X decay exponentially, that means for any ε ∈(0,1) there exists a matrix X̃ of rank k = O (log(1/ ε )) such that ∥ X − X̃ ∥ 2 ⩽ ε ∥ X ∥ 2 . As a generalization we prove that if A,B,C are hierarchical matrices then the solution X can be approximated by the hierarchical matrix format described in Hackbusch ( Computing 2000; 62 : 89–108). The blockwise rank of the approximation is again proportional to log(1/ ε ). Copyright © 2004 John Wiley & Sons, Ltd.