Uncoupling the Perron eigenvector problem (Q1120636)

A method is given to find the unique normalized Perron vector \(\pi\) satisfying \(A\pi =\rho \pi\) where A is a nonnegative irreducible \(m\times m\) matrix with spectral radius \(\rho\), \(\pi =(\pi_ 1,-\pi_ m)^ T\) and \(\pi_ 1+...+\pi_ m=1\). The matrix is uncoupled into two or more smaller matrices \(P_ 1,P_ 2,...,P_ k\) such that this sequence has the following properties: (1) Each \(P_ i\) is irreducible and nonnegative and has a unique Perron vector \(\pi^{(i)}\). (2) Each \(P_ i\) has the spectral radius \(\rho\). (3) The Perron vectors \(\pi^{(i)}\) for \(P_ i\) can be determined independently. (4) The smaller Perron vectors \(\pi^{(i)}\) can easily be coupled back together to form the Perron vector \(\pi\) for A.