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Entrywise lower and upper bounds for the Perron vector
Publikationstyp
Journal Article
Date Issued
2022-11-15
Sprache
English
Author(s)
Institut
Volume
653
Start Page
314
End Page
319
Citation
Linear Algebra and Its Applications 653 : 314-319 (2022-11-15)
Publisher DOI
Scopus ID
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,jxi⁎/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.
Subjects
M-matrix
Perron vector
Perron-Frobenius theory