Taylor expansions of eigenvalues of perturbed matrices with applications to spectral radii of nonnegative matrices

Moshe Haviv*, Ya'acov Ritov, Uriel G. Rothblum

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

Abstract

Let A and B be two n×n complex matrices, and let λ be an eigenvalue of A. The purpose of this paper is to derive, under certain conditions, Taylor power series expansions of the form λ+σk=1λkεk and σ∞k=0v{script}kεk, respectively, for eigenvalues and corresponding eigenvectors of the perturbed matrices A+εB for ε that has sufficiently small absolute value. Our results apply to the case where λ is a simple eigenvalue of A, e.g., when A is nonnegative and irreducible and λ is the spectral radius of A. In particular, if A+εB is nonnegative for sufficiently small nonnegative ε and A is irreducible, we obtain power series expansions for the spectral radii of the perturbed matrices A+εB and for corresponding eigenvectors. The coefficients of the expansions yield explicit expressions for the regular and mixed derivatives of the spectral radius and of a corresponding eigenvector of a nonnegative irreducible matrix when viewed as a function of the elements of the matrix. Our approach is constructive, and we present a recursive algorithm that will compute the coefficients of the above series.

Original languageEnglish
Pages (from-to)159-188
Number of pages30
JournalLinear Algebra and Its Applications
Volume168
Issue numberC
DOIs
StatePublished - 15 Apr 1992

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