Please use this identifier to cite or link to this item: http://dx.doi.org/10.14279/depositonce-6563
Main Title: When is the adjoint of a matrix a low degree rational function in the matrix?
Author(s): Liesen, Jörg
Type: Article
Language Code: en
Abstract: We show that the adjoint $A^+$ of a matrix A with respect to a given inner product is a rational function in A, if and only if A is normal with respect to the inner product. We consider such matrices and analyze the McMillan degrees of the rational functions r such that $A^+=r(A)$. We introduce the McMillan degree of A as the smallest among these degrees, characterize this degree in terms of the number and distribution of the eigenvalues of A, and compare the McMillan degree with the normal degree of A, which is defined as the smallest degree of a polynomial p for which $A^+=p(A)$. We show that unless the eigenvalues of A lie on a single circle in the complex plane, the ratio of the normal degree and the McMillan degree of A is bounded by a small constant that depends neither on the number nor on the distribution of the eigenvalues of A. Our analysis is motivated by applications in the area of short recurrence Krylov subspace methods.
URI: https://depositonce.tu-berlin.de//handle/11303/7290
http://dx.doi.org/10.14279/depositonce-6563
Issue Date: 2007
Date Available: 19-Dec-2017
DDC Class: DDC::500 Naturwissenschaften und Mathematik::510 Mathematik::518 Numerische Analysis
DDC::500 Naturwissenschaften und Mathematik::510 Mathematik::512 Algebra
Subject(s): normal matrices
representation of matrix adjoints
rational interpolation
Krylov subspace methods
short recurrences
Usage rights: Terms of German Copyright Law
Journal Title: SIAM Journal on Matrix Analysis and Applications
Publisher: Society for Industrial and Applied Mathematics
Publisher Place: Philadelphia, Pa
Volume: 29
Issue: 4
Publisher DOI: 10.1137/060675538
Page Start: 1171
Page End: 1180
EISSN: 1095-7162
ISSN: 0895-4798
Appears in Collections:Fachgebiet Numerische Lineare Algebra » Publications

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