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Derivative of matrix B with respect to matrix B

  1. Jul 18, 2009 #1
    1. The problem statement, all variables and given/known data
    2. Relevant equations

    Create an algorithm to calculate

    [tex] \frac{\partial\mathbf{ABA'}} {\partial\mathbf{B}} [/tex]

    where [tex]\mathbf{B}[/tex] is a k x k symmetrical matrix.

    3. The attempt at a solution

    We know that

    [tex] \frac{\partial\mathbf{ABA'}} {\partial\mathbf{B}} =
    \mathbf{A} \frac{\partial\mathbf{B}} {\partial\mathbf{B}} \mathbf{A'}.

    Hence, we need to calculate [tex] \frac{\partial\mathbf{B}} {\partial\mathbf{B}} [/tex].

    This will result in a four dimensional k x k x k x k array, with each element (:,:,i,j) corresponding to the matrix of partial derivative [tex] \frac{\partial\mathbf{B}} {\partial {B_{i,j}}} [/tex].

    Furthermore, we know that

    [tex] \frac{\partial{B_{g,h}}}
    {\partial {B_{i,j}}} [/tex] equals 1 if g = i and h = j, and 0 otherwise. Hence,

    [tex] (\frac{\partial\mathbf{B}} {\partial\mathbf{B}})_{g,h,i,j}[/tex] will equal 1 when g = i and h = j, and 0 otherwise.

    However, I am still not capable of finding a good way to build [tex] \frac{\partial\mathbf{B}} {\partial\mathbf{B}} [/tex]. Could you help me on this one, or point me in a good direction?

  2. jcsd
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