Hybrid Description and the Spectrum of Linear Multiports

We introduce the spectrum as the set of the possible numbers of nullator-norator pairs for embedding a linear time-invariant resistive multiport into a uniquely solvable network, and as a generalization of an earlier result of Iri and Tomizawa we find the minimum of the spectrum using the matroid intersection algorithm (this number is zero if and only if the multiport has at least one hybrid description). We show that the maximum value can also be determined in polynomial time, using the linear matroid parity algorithm. We also prove that the spectrum is an invariant, that is, a multiport has the same spectrum as its negative, inverse, dual and adjoint. The proof clearly shows the difference between the two types of voltage-current symmetries for nonreciprocal networks. Most of the results remain valid for generalized multiports where the number of linearly independent equations need not be equal to the number of the ports.