To express the transmission line equations in the matrix notation of equation (17), introduce the notation
In most applications the matrix 
is nonsingular. In all
that follows we assume this to be the case; therefore, we
take 
.
Associated with the system (11) is a characteristic
polynomial defined by
Since 
and 
are 
matrices and 
, the polynomial F
has degree n.
If 
has n
distinct real zeros, we classify the first order system
(17) as hyperbolic. The system is also called hyperbolic if
has n real zeros and the generalized eigenvalue problem
has n linearly
independent solutions. If 
has no real zeros, then (17) is called elliptic. If 
has n real zeros, but 
does not have n linearly
independent solutions, then the system (11) is classified
as parabolic. An exhaustive classification cannot be carried
out when 
has both real and complex zeros.
