Solving Matrix Derivatives: Theorems & Examples

[olds442]

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!

 

[Dick]

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.

 

[olds442]

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...

 

[Dick]

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!