We consider first *diagonal* impedance matrices, since they have
an intuitive physical counterpart in lossless acoustic tubes. Under this assumption,
we have the impedance matrix

(28) |

(29) |

(30) |

(31) |

As a side note, this equation is analogous to the relation between power-wave scattering matrices and voltage-wave scattering matrices as found in the WDF literature [27] for lumped circuit elements.

In the more general case, when the (complex) admittance
matrix is not necessarily diagonal, but remains positive
semidefinite as required for lossless propagation, we have that
admits a Cholesky factorization

(34) |

By an explicit computation of the matrix product in
(36) we can show that the terms of any column of
are *power complementary*, i.e.,

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