• Support PF! Buy your school textbooks, materials and every day products Here!

Matrix Derivatives

  • Thread starter olds442
  • Start date
6
0
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!
 

Answers and Replies

Dick
Science Advisor
Homework Helper
26,258
618
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.
 
6
0
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
Science Advisor
Homework Helper
26,258
618
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!
 

Related Threads for: Matrix Derivatives

  • Last Post
Replies
5
Views
2K
Replies
0
Views
2K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
3
Views
2K
Replies
3
Views
2K
  • Last Post
Replies
8
Views
4K
Replies
4
Views
4K
Top