# Matrix Vector Product, as a Derivative?

Let $$p\in\mathbb{R}^d$$ and $$A\in \mathbb{R}^{d\times d}$$, can the product $$Ap$$ be the derivative of some function?

In particular given a matrix $$A$$, how can I construct $$H:\mathbb{R}^d\to \mathbb{R}$$, such that $$\nabla H(p)=Ap$$. Note below I can do this for specific $$A$$ but I want it for general $$A$$ (say positive semi-definite, but possibly singular).

$$\textbf{My Work :}$$

• If $$A$$ is the identity matrix then we have $$\nabla H(p)=\nabla \frac{1}{2}\|p\|^2=p=Ap$$.

• If $$A$$ can be written as $$B^TB$$ for some matrix $$B\in\mathbb{R}^{d\times d}$$, then take $$\nabla H(p)=\nabla \frac{1}{2}\|Bp\|^2=B^TBp=Ap$$.

What about more general $$A$$?

If $$A$$ is symetric you can set $$H(p) = \frac{1}{2}p^TAp$$.

If you then calculate the derivative using the chain rule you get $$\nabla H(p) = Ap$$.

If $$A$$ is not symetric you can't find such a function $$H$$.

For example let $$a_{12} \neq a_{21}$$. You can write down the product of $$Ap$$ and start integrating the first element of the vector with respect to $$p_1$$:

Then

$$H(p) = \frac{1}{2}a_{11}p_1^2 + p_1a_{12}p_2 + p_1a_{13}p_3 +...$$

Similarly when you integrate the second element with respect to $$p_2$$ you get:

$$H(p) = p_1a_{21}p_2 + \frac{1}{2}a_{22}p_1^2 +...$$

So we get a contradiction. You can do a similar argument for any non symetrical matrix.