System behavior in quotient systems (Q1354250)

Let \(A\) be an \(n\times n\) matrix. We consider the discrete matrix systems \(x_{k+1}= A\cdot x_k\) and \(x_{k+1}= A\cdot x_k+b\), where \(b\) and \(x_k\), for all \(k\), are \(n\times 1\) vectors. The author is interested in system behavior when \(|\rho|> 1\), where \(\rho\) is the largest (in modulus) eigenvalue of \(A\). Namely, he studies the case when the iterates do not converge to a vector but do tend to a subspace of vectors and describes such convergence in a quotient space setting. The author gives criteria in terms of norms which determine this type of convergence and the behavior of systems in quotient spaces.