A note on the resolvent of a nonnegative matrix and its applications
The main result of this paper is the following. Let \(X' = (x'_{ij} )\) be a nonnegative matrix obtained from \(X = (x_{ij} )\) by decreasing a single entry \(x_{hk}\). Then for all indices \(i,j\) and for all \(t\) in \([0,1/\max (r_X ,r_{X'} ))\), where \(r_X ,r_{X'}\) are the spectral radii of \(X\) and respectively \(X'\), the entries of \(Y = (I - tX)^{ - 1}\) and \(Y' = (I - tX')^{ - 1}\) satisfy \(y_{ij} y'_{ik} \leqslant y'_{ij} y_{ik}\) and \(y_{ij} y'_{hj} \leqslant y'_{ij} y_{hj}\). From this result, two corollaries for Google page-rank model and the open Leontief model are derived.
scientific article; zbMATH DE number 5689419