How to compute the divergence of the function $h(x)=Ax$, where $A$ is a matrix and $x$ is a vector?

Question

I have to compute the divergence of function h (i.e $div(h)$), where $$h(x)=Mx \quad \text{where} \quad M\in R_{nxn} \quad \text{and}\quad x \in R^{n}$$ I know this definiton of divergence $$div(g(z)) = \sum_i^n \frac{d}{dx_i} g_i(z),$$ but I'm not sure how to aplly it here. I tried to write it as $$h(x)= \Bigg[\begin{matrix} m_{11}x_{1} & ... & m_{1n}x_{n} \\ . & . & . \\m_{n1}x_{n} & ... & m_{nn}x_{n}\end{matrix} \Bigg]=\Bigg[\begin{matrix} \sum_i^n m_{1i}x_{i} \\ ... \\ \sum_i^n m_{ni}x_{i} \end{matrix}\Bigg]. $$ But I don't know what to do next. Thank you for the help!

Answer 1

Your work is good, you just need to continue it. The $j$-th row is: $$h_j=\sum_{i}m_{ji}x_i$$ And you need to differentiate it with respect to $x_j$. Notice that $m_{ji}$ is just a constant, so: $$\frac{\partial h_j}{\partial x_j}=\sum_{i}m_{ji}\frac{\partial x_i}{\partial x_j}$$ Can you calculate $\frac{\partial x_i}{\partial x_j}$?