Entrywise lower and upper bounds for the Perron vector

Abstract

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.