The scattering matrices for lossless physical waveguide junctions give an apparently unexplored class of lossless FDN prototypes. However, this is just a subset of all possible lossless feedback matrices. We are therefore interested in the most general conditions for losslessness of an FDN feedback matrix. The results below are adapted from [463,385].
Consider the general case in which
is allowed to be any
scattering matrix, i.e., it is associated with a
not-necessarily-physical junction of physical waveguides.
Following the definition of losslessness in classical network theory,
we may say that a waveguide scattering matrix
is said to be
lossless if the total complex power
 at the junction is scattering invariant, i.e.,
where is any Hermitian, positive-definite matrix (which has an interpretation as a generalized junction admittance). The form is by definition the square of the elliptic norm of induced by , or . Setting , we obtain that must be unitary. This is the case commonly used in current FDN practice.
The following theorem gives a general characterization of lossless scattering:
Proof: Since is positive definite, it can be factored (by the Cholesky factorization) into the form , where is an upper triangular matrix, and denotes the Hermitian transpose of , i.e., . Since is positive definite, is nonsingular and can be used as a similarity transformation matrix. Applying the Cholesky decomposition in Eq.(C.125) yields
where , and
for each eigenvalue of
that there exists a matrix
of linearly independent
. The matrix
diag. Taking the Hermitian transform of
this equation gives
. Multiplying, we
. Thus, (C.125) is satisfied for
which is Hermitian and positive
Thus, lossless scattering matrices may be fully parametrized as , where is any unit-modulus diagonal matrix, and is any invertible matrix. In the real case, we have diag and .
Note that not all lossless scattering matrices have a simple physical interpretation as a scattering matrix for an intersection of lossless reflectively terminated waveguides. In addition to these cases (generated by all non-negative branch impedances), there are additional cases corresponding to sign flips and branch permutations at the junction. In terms of classical network theory , such additional cases can be seen as arising from the use of ``gyrators'' and/or ``circulators'' at the scattering junction ).