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Lagrangian gradient question

  1. Mar 23, 2014 #1
    From the attached image problem:
    When deriving the third term in the Lagrangian:
    [itex]\lambda[/itex][itex]_{2}[/itex](w[itex]^{T}[/itex]∑w - [itex]\sigma[/itex][itex]^{2}_{\rho}[/itex]) with respect to w, are w[itex]^{T}[/itex] and w used like a w[itex]^{2}[/itex] to arrive at the gradient or am I oversimplifying and it just happens to work out on certain problems like this?

    (∑ is an n x n symmetric covariance matrix and w is n x 1 vector)
     

    Attached Files:

  2. jcsd
  3. Mar 24, 2014 #2

    Ray Vickson

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    Homework Helper

    The best way of avoiding errors, at least when you are starting, is to write it out in full:
    [tex] w^T \Sigma w = \sum_{i=1}^n \sum_{j=1}^n \sigma_{ij} w_i w_j
    = \sum_{i=1}^n \sigma_{ii} w_i^2 + 2 \sum_{i < j} \sigma_{ij} w_i w_j[/tex]

    BTW: you don't need all those '[ itex]' tags: just enclose the whole expression between '[ itex]' and ' [/itex]' (although I prefer an opening '# #' (no space) followed by a closing '# #' (again, no space). Compare this: ##\lambda_2 (w^T \Sigma w - \sigma_{\rho}^2)## with what you wrote. (Press the quote button, as though you were composing a reply; that will show you the code. You can then just abandon the reply and not post it.)
     
    Last edited: Mar 24, 2014
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