If you set aside for a moment the constraint on the vector x, then the problem of maximizing x_iD_ijx_j (a sum over repeated indices is implied) translates, upon taking the derivative with respect to x_i, into the equation D_ijx_j = 0. This is a special form of the eigenvalue equation for the matrix D, with the eigenvalue being 0. In order for non-trivial solutions, the matrix D cannot have an inverse, so it's determinant must vanish. Finding the eigenvectors can be done with PDL; see Mastering Algorithms with Perl for a discussion. The Math::Cephes::Matrix module can also do this for real symmetric matrices. Generally, the components of the eigenvectors found in this way are not completely determined; by convention, the normalization x_ix_i = 1 is imposed.

Update: The preceding needs to be augmented by a test on the Hessian of the matrix to determine what type of critical point is present.