Error Bounds on an Approximation to the Dominant Eigenvector of a Nonnegative Matrix

Let [formula omitted] be an (unknown) irreducible nonnegative matrix. Suppose only the following information on A is given: (1) its spectral radius ρ(A) (an example for such an information is in knowing that A is stochastic); (2) lower and upper bounds on each of the entries of A, namely two nonnegative matrices B and E such that B ⩽ A ⩽ B + E are given. The purpose of this paper is to bound the error of the dominant eigenvector of A. The technique used is as follows: the error vector is shown to satisfy a set of linear constraints. Then, a set of linear programming problems is solved to obtain bounds on the values for the entries of the error vector.