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**

For the PDE the matrix is