Bounds for the Positive Eigenvectors of Nonnegative Matrices and for their Approximations by Decomposition

Reviewer: Robert James Plemmons

The authors use the theory of nonnegative matrices to obtain upper and lower bounds on the Perron vector of irreducible nonnegative matrices, such as those arising in the study of discrete ergodic Markov chains. The interest here is in bounding the stationary distribution vector for such chains. The authors indicate that the results given in this paper extend and simplify those given by Stewart [1]. The computation of these bounds can itself be regarded as a new approximation technique, called here bounded aggregation. Simple LU factorization methods are also effective in computing stationary distribution vectors, as indicated in [2]. However, for nearly completely decomposable chains with a large number of states, the methods of Courtois and Semal provide very valuable analytical tools for error analysis.