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5 Classification Of First Order Systems     continued...

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.