Structure of quasi-primitive multiports

A quasi-primitive n-port is described by an admittance matrix of the form Y(p)=p^-1 A+B+pC in which A, B, C, are real, symmetric, semidefinite n×n matrices. Any m-port, m<n, derived by ignoring some chosen (m-n) ports of the n-port described by Y(·) is a descendant of Y(·). Any such descendant is described by its own admittance matrix Y´(·) called then a descendant of Y(·). This note presents two results, the first a “canonical” form into which any given quasi-primitive Y(·) can be put by a transformation of coordinates. Specifically, given Y(·), quasi-primitive and of size n×n, there exists a real, invertible, n×n matrix Q such that Q*Y(·)Q is of block-diagonal form bldiag{Y_SP(p), Y_ND(p), Y_TI(p)} in which the blocks are quasi-primitive and describe respectively an n₁-port, an n₂-port, a 2n₃ -port, separate networks, each of which, if not of size zero, exhibits distinctive characteristic properties. The second result, based on this structure, is a property of a given m×m positive-real matrix-valued function Y´(·) necessary that Y´(·) be a descendant of some quasi-primitive matrix Y(·)