If you set aside for a moment the constraint on the vector x, then the problem of maximizing
x_{i}D_{ij}x_{j}
(a sum over repeated indices is implied) translates, upon taking the derivative with respect to x_{i},
into the equation
D_{ij}x_{j} = 0.
This is a special form of the
eigenvalue equation for the matrix D, with the
eigenvalue being 0. In order for nontrivial 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_{i}x_{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.
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