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We adopt an operator-theoretic perspective to study convergence of linearfixed-point iterations and discrete- time linear systems. We mainly focus onthe so-called Krasnoselskij-Mann iteration x(k+1) = ( 1 - \alpha(k) ) x(k) +\alpha(k) A x(k), which is relevant for distributed computation in optimizationand game theory, when A is not available in a centralized way. We show thatconvergence to a vector in the kernel of (I-A) is equivalent to strictpseudocontractiveness of the linear operator x -> Ax. We also characterize somerelevant operator-theoretic properties of linear operators via eigenvaluelocation and linear matrix inequalities. We apply the convergence conditions tomulti-agent linear systems with vanishing step sizes, in particular, to linearconsensus dynamics and equilibrium seeking in monotone linear-quadratic games.

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