# Matrix derivative and product rule

by hamster on wheels   Last Updated May 16, 2019 04:20 AM

$$A$$ is a square matrix with non-negative elements. $$n$$ is a positive integer.

How to evaluate the following for all possible $$n$$?

$$f_{n}(A, i, j) =\frac{\partial }{\partial A_{ij}}\left(\vec{1}A^{n}\vec{1}^{\intercal}\right)$$

where $$\vec{1}$$ is a row vector and $$\vec{1}^{\intercal}$$ is transpose of $$\vec{1}$$.

Attempt to solve

I thought

$$f_{3}(A, i, j) =\vec{1}(BAA + ABA +AAB)\vec{1}^{\intercal}$$

where

$$B_{i'j'} = \begin{cases} 1, & i = i', j=j' \\ 0, &\text{otherwise}\end{cases}$$

But the problem is that the expression is always non-negative.

Unless the function is monotonic, I would not expect the first derivative to always have the same sign.

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