A system criterion is given for a real rationaln×nmatrixS(p) to beJlossless for a givenJ, i.e.J-S∼*(p)JS(p)≥ 0 in Rep> 0 andJ-S∼(-p)JS(p) = 0 whereJ=J∼andJ2=In. This algebraic property is then applied to the synthesis of lossless-chain and chain-scattering matrices and to stable transfer-matrix functions by cascade load and cascade networks using the minimum number of reactive elements.