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Math Help - Differentiate with respect to vector

  1. #1
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    Smile Differentiate with respect to vector

    I have this problem:

    L=\frac{\mathbf{a}^T\mathbf{w}\mathbf{w}^T\mathbf{  a}}{\mathbf{w}^T(\mathbf{B}\mathbf{w}+\mathbf{C}\m  athbf{w})}

    Here, \mathbf{a} is a known d-by-1 vector, \mathbf{w} is a d-by-1 vector to be identified, both \mathbf{B} and \mathbf{C} are known d-by-d matrix.

    And I want to find the maximum value of L by taking the first order derivative with respect to \mathbf{w}.

    Can anyone teach me how to take the first order derivative of this scalar with respect to a vector here?

    Thanks a lot!!
    Last edited by mr fantastic; November 10th 2010 at 06:29 PM. Reason: Deleted begging from title.
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  2. #2
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    Note that both the denominator and the numerator are of the form of w'Aw, where w' is the transpose of w. Let f(w) = w'Aw, calcuate the differential like this:
    Let t be a real number and x be a vector, then
    f(w+tx) = (w'+tx')A(w+tx) = f(w) + tw'(A+A')x + o(t)
    Thus the linear map df(x) = w'(A+A')x is the differental of f at the point w.

    For your case, L = f(w)/g(w), where f(w) = w'(a*a')w, g(w) = w'(B+C)w
    dL(x) = [df(x)g(w) - f(w)dg(x)] / [g(w)]^2
    let dL = 0, we get df(x)g(w) = f(w)dg(x) for any vector x.
    Solve this you'll get the answer.
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