  An exceptional point of degeneracy (EPD) occurs when both the eigenvalues and
the corresponding eigenvectors of a square matrix coincide and the matrix has a
nontrivial Jordan block structure. It is not easy to achieve an EPD exactly. In
our prior studies, we synthesized simple conservative (lossless) circuits with
evolution matrices featuring EPDs by using two LC loops coupled by a gyrator.
In this paper, we advance even a simpler circuit with an EPD consisting of only
two LC loops with one capacitor shared. Consequently, this circuit involves
only four elements and it is perfectly reciprocal. The shared capacitance and
parallel inductance are negative with values determined by explicit formulas
which lead to EPD. This circuit can have the same Jordan canonical form as the
nonreciprocal circuit we introduced before. This implies that the Jordan
canonical form does not necessarily manifest systems' nonreciprocity. It is
natural to ask how nonreciprocity is manifested in the system's spectral data.
Our analysis of this issue shows that nonreciprocity is manifested explicitly
in: (i) the circuit Lagrangian and (ii) the breakdown of certain symmetries in
the set of eigenmodes. All our significant theoretical findings were thoroughly
tested and confirmed by extensive numerical simulations using commercial
circuit simulator software.
