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If you set aside for a moment the constraint on the vector x, then the problem of maximizing xiDijxj (a sum over repeated indices is implied) translates, upon taking the derivative with respect to xi, into the equation Dijxj = 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 xixi = 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.

In reply to Re: Help with Matrix math! by randyk

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