Uncoupling the Perron eigenvector problem

AbstractFor a nonnegative irreducible matrix A with spectral radius ϱ, this paper is concerned with the determination of the unique normalized Perron vector π which satisfies Aπ = ϱπ, π #&62; 0, Σjπj = 1. It is explained how to uncouple a large matrix A into two or more smaller matrices—say P11,P22,…,Pkk—such that this sequence of smaller matrices has the following properties: (1) Each Pii is also nonnegative and irreducible, so that each Pii has a unique Perron vector π(i). (2) Each Pii has the same spectral radius ϱ as A. (3) It is possible to determine the π(i)'s completely independently of each other, so that one can execute the computation of the π(i)'s parallel. (4) It is easy to couple the smaller Perron vectors π(i) back together in order to produce the Perron vector π for the original matrix A