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The necessary and sufficient conditions are given that a matrix with arbitrary complex number elements be the impedance, admittance or scattering matrix of a physical linear reciprocal or non-reciprocal network. A canonical form for non-reciprocal network synthesis is presented which applies to any linear n-terminal pair (n-port) system at any fixed frequency. If the network is passive the only circuit element required in addition to lossless inductors, capacitors, transformers and positive resistors is the gyrator. If the network is active, negative resistors and gyrators must be used in addition to conventional passive elements. Some discussion of matrixes with frequency variable elements is also given.