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Matrix Derivatives

  1. Apr 2, 2008 #1
    I have encountered some problems that have to do with the derivatives of matrices... I have NO experience with these and had little luck finding any theorems... I looked on wikipedia for some help and found a few definitions, but I am still unclear about how this is proven or attained... here is an example from wikipedia:

    d tr(AXB)/ dX = A^T B^T

    my question is... how are they getting that?!? I seem to be having a big mind block with this..

    Any theorems about how to take derivatives of vectors or matrices would be great!

    any help would be appreciated!
     
  2. jcsd
  3. Apr 2, 2008 #2

    Dick

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    Think indices. tr(AXB)=A_ij*X_jk*B_ki. The lm component of the derivative matrix is the derivative of that with respect to X_lm. The only terms that contribute to that are terms where j=l and k=m. Removing the X_lm since it is differentiated leaves A_il*B_mi. That's (A^T)_li*(B^T)_im=(A^T*B^T)_lm. So the lm component of d(tr(AXB))/dX is the same as (A^T*B^T). So they are equal as matrices.
     
  4. Apr 2, 2008 #3
    hmmm.. if they are defined as square matrices, the tr(AXB) would be given by A_ii*X_ii*B_ii so that tr(AXB) is a square matrix whose diagonal elements are all AXB correct? If not, there is definitely something here that I am missing...
     
  5. Apr 2, 2008 #4

    Dick

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    AXB_ij=A_ik*X_kl*B_lj. Repeated indices are summed over (I don't think I emphasized that). To get the trace, just set i=j and sum over it. Leave the summed dummy indices alone!
     
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