On the Convergence of Discrete-Time Linear Systems: A Linear Time-Varying Mann Iteration Converges IFF Its 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 socalled 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.