  This study first introduces the frequency-wise phases of n-port linear
time-invariant networks based on recently defined phases of complex matrices.
Such a phase characterization can be used to quantify the well-known notion of
passivity for networks. Further, a class of matrix operations induced by fairly
common n-port network connections is examined. The intrinsic phase properties
of networks under such connections are preserved. Concretely, a scalable
phase-preserving criterion is proposed, which involves only the phase
properties of individual subnetworks, under the matrix operations featured by
connections. This criterion ensures that the phase range of the integrated
network can be verified effectively and that the scalability of the analyses
can be maintained. In addition, the inverse operations of the considered
connections, that is, network subtractions with correspondences are examined.
With the known phase ranges of the integrated network and one of its
subnetworks, the maximal allowable phase range of the remaining subnetwork can
also be determined explicitly in a unified form for all types of subtractions.
Finally, we extend the phase-preserving properties from the aforementioned
connections to more general matrix operations defined using a certain
indefinite inner product.
