Derivative of matrix B with respect to matrix B

[Guillaume F.]

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

Create an algorithm to calculate

\frac{\partial\mathbf{ABA'}} {\partial\mathbf{B}}

where \mathbf{B} is a k x k symmetrical matrix.

3. The attempt at a solution

We know that

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


Hence, we need to calculate \frac{\partial\mathbf{B}} {\partial\mathbf{B}} .

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 \frac{\partial\mathbf{B}} {\partial {B_{i,j}}} .

Furthermore, we know that

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

(\frac{\partial\mathbf{B}} {\partial\mathbf{B}})_{g,h,i,j} 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 \frac{\partial\mathbf{B}} {\partial\mathbf{B}} . Could you help me on this one, or point me in a good direction?

Thanks,
Guillaume