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December 16th, 2007, 06:03 AM   #1
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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
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December 16th, 2007, 04:56 PM   #2
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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.
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December 16th, 2007, 10:37 PM   #3
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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?
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December 17th, 2007, 01:11 PM   #4
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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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