The circulant operator in the banach algebra of matrices

Raymond H. CHAN*, Xiao Qing JIN, Man Chung YEUNG

*Corresponding author for this work

Research output: Journal PublicationsJournal Article (refereed)peer-review

65 Citations (Scopus)


We study an operator c which maps every n-by-n matrix An to a circulant matrix c(An) that minimizes the Frobenius norm {norm of matrix}An - Cn{norm of matrix}F over all n-by-n circulant matrices Cn. The circulant matrix c(An), called the optimal circulant preconditioner, has proved to be a good preconditioner for a general class of Toeplitz systems. In this paper, we give different formulations of the operator, discuss its algebraic and geometric properties, and compute its operator norms in different Banach algebras of matrices. Using these results, we are able to give an efficient algorithm for finding the superoptimal circulant preconditioner which is defined to be the minimizer of {norm of matrix}I - Cn-1An{norm of matrix}F over all nonsingular circulant matrices Cn.

Original languageEnglish
Pages (from-to)41-53
Number of pages13
JournalLinear Algebra and Its Applications
Publication statusPublished - 15 Apr 1991
Externally publishedYes


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