My Math Forum Optimization of a Matrix with one constraint.
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 December 16th, 2007, 07:03 AM #1 Newbie   Joined: Dec 2007 Posts: 3 Thanks: 0 Optimization of a Matrix with one constraint. Hello! I have a minor problem, that wont get solved... Its been posted in an old competition at our university, but I didnt manage to solve it in time. Maybe someone in here could help me? Where D is a symmetric n x n matrix, and x is an unknown n x 1 vector. I can only figure out how to test for a maximum later on, but is there something im missing with the charataristic roots? Regards
 December 16th, 2007, 05:56 PM #2 Global Moderator   Joined: May 2007 Posts: 6,661 Thanks: 648 I must confess I'm a little rusty here. However by taking first derivatives with respect to each of the xi, you will end with a matrix equation with (x1,....,xn) any eigenvector, as a solution. Normalize the eigenvectors to unit length and see which one gives you the maximum.
 December 16th, 2007, 11:37 PM #3 Newbie   Joined: Dec 2007 Posts: 3 Thanks: 0 You dont sound rusty at all in this, compared to me that is. We tried just to solve it for all x's and came to the following: But we can't eliminate lamda. Do we need to diagonalise D before doing so?
 December 17th, 2007, 02:11 PM #4 Global Moderator   Joined: May 2007 Posts: 6,661 Thanks: 648 If you look at the line which says "D is symmetric" and you make the equations into one big vector, you will have Dx=Lx (L=lambda), which means the solution is an eigenvector and L is an eigenvalue. In general there will be n solutions.

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