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.