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Re: Help with Matrix math!by randyk (Parson) 
on Dec 16, 2007 at 21:33 UTC ( #657319=note: print w/ replies, xml )  Need Help?? 
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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