On the convergence of discrete-time linear systems: A linear time-varying Mann iteration converges IFF the operator is strictly pseudocontractive

We adopt an operator-theoretic perspective to study convergence of linear fixed-point iterations and discrete-time linear systems. We mainly focus on the so-called Krasnoselskij-Mann iteration, x ( k + 1) = (1-α k) x ( k) + α k A x (k ), which is relevant for distributed computation in optimization and game theory, when A is not available in a centralized way. We show that strict pseudocontractiveness of the linear operator x ↠ Ax is not only sufficient (as known) but also necessary for the convergence to a vector in the kernel of I-A. We also characterize some relevant operator-theoretic properties of linear operators via eigenvalue location and linear matrix inequalities. We apply the convergence conditions to multi-agent linear systems with vanishing step sizes, in particular, to linear consensus dynamics and equilibrium seeking in monotone linear-quadratic games.