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Perturbation properties of the generalized spectral radius
Citation Link: https://doi.org/10.15480/882.14972
Publikationstyp
Journal Article
Date Issued
2025
Sprache
English
Author(s)
TORE-DOI
Journal
Volume
73
Issue
4
Start Page
633
End Page
648
Article Number
2366951
Citation
Linear and Multilinear Algebra 73 (4): 633-648 (2025)
Publisher DOI
Scopus ID
Publisher
Taylor & Francis
Let n∈N, K∈{R,C} and Mn(K) be the set of all n-by-n matrices with entries in K. We investigate the sensitivity of the generalized spectral radius ρK(A):=max{|λ|:λ∈Kand|Ax|=|λx|foranx∈Kn∖{0}}of A∈Mn(K) under perturbations, i.e. we ask how ρK(A) is related to ρK(A+E) for perturbations E∈Mn(K). For example and somewhat surprisingly, Elsner's famous bound on the spectral variation of two complex square matrices fully translates to |ρC(A)−ρC(B)|⩽∥A+B∥1−1n2∥A−B∥1n2for all A,B∈Mn(C), where ∥⋅∥2 is the matrix 2-norm. For ρR this result holds true locally.
Subjects
Generalized spectral radius
Hölder continuity
perturbation of eigenvalues
sign-complex spectral radius
sign-real spectral radius
DDC Class
510: Mathematics
Publication version
publishedVersion
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Perturbation properties of the generalized spectral radius.pdf
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