Your questino is a bit confused. Do you want to perform the eigen decomposition of D^-1 ?
If this is the case, for 3x3 matrix you can even derive analytical solutions. You don't even need to invert D. You compute the roots determinant of | D - l*I |, where l is an eigen value and I is the identity matrix.
The 3 roots are the eigenvalues, and then you can compute the eigenvectors as :
U_i = [ d_xy * d_yz - d_xz * (d_yy - l_i) ]
[ d_xy * d_xz - d_yz * (d_xx - l_i) ]
[(d_xx - l_i) * (d_yy - l_i) - d_xy^2 ]
D and D^-1 share the same eigenvectors and their eigenvalues are just invert of each other.