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4 Equations with n Independent Variables     continued...

Let P denote the number of positive eigenvalues, and Z the number of zero eigenvalues (i.e., the multiplicity of the eigenvalue zero), of the matrix . Then equation (12)is:

hyperbolic
if Z= 0 and P = 1 or Z= 0 and P = n-1
parabolic
if Z> 0 (equivalently, if )
elliptic
if Z= 0 and P = n or Z= 0 and P = 0
ultrahyperbolic
if Z = 0 and 1 < P < n -1
If any of the is nonconstant, the type of equation (12) can vary with position.

For the PDE the matrix is