Convergence of products of matrices in projective spaces

AbstractLet Ak, k∈N be a sequence of n×n complex valued matrices which converge to a matrix A. If A and each Ak is positive then the product AkAk-1⋯A2A1‖AkAk-1⋯A2A1‖ converges to a rank one matrix positive matrix uwT, where u is a positive column eigenvector of A. If each Ak is nonsingular and A has exactly one simple eigenvalue λ of the maximal modulus with the corresponding eigenvector u, then e-1θkAkAk-1⋯A2A1‖AkAk-1⋯A2A1‖, θk∈R converges to a rank one matrix uwT