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Maximizing quantity which is a product of matrices/vectors

  1. Apr 17, 2013 #1
    I am trying to follow the following reasoning:

    Given a known matrix A, we want to find w that maximizes the quantity

    w'Aw​

    (where w' denotes the transpose of w) subject to the constraint w'w = 1.

    To do so, use a lagrange multiplier, L:

    w'Aw + L(w'w - 1)
    and differentiate to obtain

    Aw = Lw.​

    Thus, we seek the eigenvector of A with the largest eigenvalue. ​


    I do not understand how they differentiated w'Aw + L(w'w-1) to get Aw = Lw. Can someone explain to me what is going on at that step?
     
  2. jcsd
  3. Apr 17, 2013 #2

    fzero

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    It would probably help to write things down explicitly in terms of components,

    $$ m = w'Aw + L(w'w - 1) = \sum_{ij} A_{ij} w_i w_j + L \left( \sum_i w_i^2 -1 \right).$$

    This is a function of the ##n## variables ##w_i##. At an extremum, ##\partial m /\partial w_k =0##. If you actually work out this set of equations, you'll see they are the components of the eigenvalue equation that you quoted. You'll need to use the fact that ##A## can be taken to be a symmetric matrix.
     
  4. Apr 17, 2013 #3

    Bacle2

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  5. Apr 18, 2013 #4
    Both posts were helpful. Thanks.
     
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