Derivative of matrix B with respect to matrix B

[Guillaume F.]

Homework Statement

Homework Equations

Create an algorithm to calculate

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

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

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

 

[mighty2000]

Dear Guillaume,

Thank you for your question. you are correct in your approach to solving this problem using the chain rule. However, I would like to offer some suggestions to improve your algorithm.

Firstly, instead of creating a four-dimensional array, it would be more efficient to create a k x k matrix of zeros and then fill in the appropriate elements with ones using a for loop. This way, you will save memory and time in your computation.

Additionally, you can use the Kronecker delta function to simplify your algorithm. The Kronecker delta function, denoted by δ_ij, is defined as 1 when i = j and 0 otherwise. Hence, your algorithm can be written as:

1. Create a k x k matrix of zeros, let's call it D.
2. Use a for loop to iterate through the elements of D.
3. For each element (i,j) of D, use the Kronecker delta function to determine whether to assign a 1 or 0 to that element.
4. Once D is filled with ones and zeros, you can simply multiply it by A and A' to get the final result.

I hope this helps you in solving the problem. Let me know if you have any further questions.