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