This paper introduces the connection matrix as a generalization of the transmittance matrix approach used in signal flow-graph analysis. The connection matrix is shown to be a simple and computationally efficient way to describe composite systems formed from an interconnection of single-variable and multi-variable subsystems. The approach allows system descriptions in state-space form, transfer-function form, and system-matrix form, and allows the transmissions between arbitrarily nominated sets of inputs and outputs to be readily determined.