Linear, time-invariant, multi-channel, general proper systems with direct control feedthrough are considered. A polynomial matrix method is presented to derive explicit conditions characterizing the systems which can be made controllable or stabilizable through a single channel by applying non-dynamic decentralized feedback to all other channels. These conditions are then applied to the decen- tralized pole-assignment and stabilization problems for the systems, yielding a complete criterion for determining when the closed-loop spectrum can be freely assigned or stabilized with decentralized control. It can be expected that the results obtained will have direct application to periodic discrete-time control systems.