Publisher DOI: 10.1016/j.laa.2022.08.011
Title: Entrywise lower and upper bounds for the Perron vector
Language: English
Authors: Rump, Siegfried M.  
Keywords: M-matrix; Perron vector; Perron-Frobenius theory
Issue Date: 15-Nov-2022
Source: Linear Algebra and Its Applications 653 : 314-319 (2022-11-15)
Abstract (english): 
Let an irreducible nonnegative matrix A and a positive vector x be given. Assume αx≤Ax≤βx for some 0<α≤β∈R. Then, by Perron-Frobenius theory, α and β are lower and upper bounds for the Perron root of A. As for the Perron vector x⁎, only bounds for the ratio γ:=maxi,j⁡xi⁎/xj⁎ are known, but no error bounds against some given vector x. In this note we close this gap. For a given positive vector x and provided that α and β as above are not too far apart, we prove entrywise lower and upper bounds of the relative error of x to the Perron vector of A.
ISSN: 0024-3795
Journal: Linear algebra and its applications 
Institute: Zuverlässiges Rechnen E-19 (H) 
Document Type: Article
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