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Main Title: On Chebyshev polynomials of matrices
Author(s): Liesen, Jörg
Faber, Vance
Tichý, Petr
Type: Article
Language Code: en
Abstract: The mth Chebyshev polynomial of a square matrix A is the monic polynomial that minimizes the matrix 2-norm of $p(A)$ over all monic polynomials $p(z)$ of degree m. This polynomial is uniquely defined if m is less than the degree of the minimal polynomial of A. We study general properties of Chebyshev polynomials of matrices, which in some cases turn out to be generalizations of well-known properties of Chebyshev polynomials of compact sets in the complex plane. We also derive explicit formulas of the Chebyshev polynomials of certain classes of matrices, and explore the relation between Chebyshev polynomials of one of these matrix classes and Chebyshev polynomials of lemniscatic regions in the complex plane.
Issue Date: 24-Jun-2010
Date Available: 14-Dec-2017
DDC Class: 512 Algebra
Subject(s): matrix approximation problems
Chebyshev polynomials
complex approximation theory
Krylov subspace methods
Arnoldi's method
Journal Title: SIAM Journal on Matrix Analysis and Applications
Publisher: Society for Industrial and Applied Mathematics
Publisher Place: Philadelphia, Pa
Volume: 31
Issue: 4
Publisher DOI: 10.1137/090779486
Page Start: 2205
Page End: 2221
EISSN: 1095-7162
ISSN: 0895-4798
Appears in Collections:FG Numerische Lineare Algebra » Publications

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