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Structured eigenvalue condition numbers and linearizations for matrix polynomials

by B. Adhikari and R. Alam and D. Kressner

(Report number 2009-01)

Abstract
This work is concerned with eigenvalue problems for structured matrix polynomials, including complex symmetric, Hermitian, even, odd, palindromic, and anti-palindromic matrix polynomials. Most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size. Recently, linearizations have been classified for which the pencil reflects the structure of the original polynomial. A question of practical importance is whether this process of linearization increases the sensitivity of the eigenvalue with respect to structured perturbations. For all structures under consideration, we show that this is not the case: there is always a linearization for which the structured condition number of an eigenvalue does not differ significantly. This implies, for example, that a structure-preserving algorithm applied to the linearization fully benefits from a potentially low structured eigenvalue condition number of the original matrix polynomial.

Keywords: Eigenvalue problem, matrix polynomial, linearization, structured condition number

@Techreport{AAK09_394,
  author = {B. Adhikari and R. Alam and D. Kressner},
  title = {Structured eigenvalue condition numbers and linearizations for matrix polynomials},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2009-01},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-01.pdf },
  year = {2009}
}

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