Entrywise lower and upper bounds for the Perron vector

Publikationstyp
Journal Article
Publikationsdatum
2022-11-15
Sprache
English
Author
Rump, Siegfried M. orcid-logo
Institut
Zuverlässiges Rechnen E-19 (H) 
TORE-URI
http://hdl.handle.net/11420/13570
Enthalten in
Linear algebra and its applications 
Volume
653
Start Page
314
End Page
319
Citation
Linear Algebra and Its Applications 653 : 314-319 (2022-11-15)
Publisher DOI
10.1016/j.laa.2022.08.011
Scopus ID
2-s2.0-85136612736
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.
Schlagworte
M-matrix
Perron vector
Perron-Frobenius theory