On Chebyshev Polynomials of Matrices

0346245 - ÚI 2011 RIV US eng J - Journal Article
Faber, V. - Liesen, J. - Tichý, Petr
On Chebyshev Polynomials of Matrices.
SIAM Journal on Matrix Analysis and Applications. Roč. 31, č. 4 (2010), s. 2205-2221. ISSN 0895-4798. E-ISSN 1095-7162
R&D Projects: GA AV ČR IAA100300802
Grant - others:GA AV ČR(CZ) M100300901
Institutional research plan: CEZ:AV0Z10300504
Source of funding: I
Keywords : matrix approximation problems * Chebyshev polynomials * complex approximation theory * Krylov subspace methods * Arnoldi's method
Subject RIV: BA - General Mathematics
Impact factor: 1.725, year: 2010

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.
Permanent Link: http://hdl.handle.net/11104/0187315