Nonnegative iterations with asymptotically constant coefficients (Q732071)

Perron's theory yields a unique (normalized) nonnegative eigenvector for nonnegative matrices. The eigenvalue equals the spectral radius. Now let \(A\) be a matrix that is not necessarily nonnegative, but has the properties above. If the sequence \(A_k\) of matrices converges to \(A\), and \(x_{k+1}=A_k x_k\) is a sequence of nonnegative vectors, and some technical assumptions hold, then \(x_k/\|x_k\|\) converges to the nonnegative eigenvector of \(A\).