"matrix representation of the gradient"

Question

I'm reading a paper and they say "it's convenient to employ a matrix representation for the gradient of f ". Then they simply give the matrix form, but obviously I'm a bit lost. Here are the equations shown in the paper

$ f:(x,t) \mapsto (z,t)=f(x,t) $

$ \frac{\partial f}{\partial(x,t)} = \begin{bmatrix}\frac{\partial z}{\partial x}&\frac{\partial z}{\partial t}\\0^T&1\end{bmatrix}$

I don't know if showing the math helps, but what do they mean by "matrix representation of the gradient"? The bottom row of the matrix they've shown has me confused.

Answer 1

The gradient of $f = (f_1,f_2)$ is given by $$\frac{\partial f}{\partial(x,t)} =\pmatrix{\frac{\partial f_1}{\partial x}&\frac{\partial f_2}{\partial x}\\ \frac{\partial f_1}{\partial t}&\frac{\partial f_2}{\partial t}}.$$ Using $f_1 = z$ and $f_2 = t$, you get the result.

Answer 2

For the vector-valued function $$z=f(x)$$ the Jacobian matrix $J$ is defined such that $$dz=J\,dx$$ In other words $$\eqalign{ J &= \frac{\partial z}{\partial x} = \begin{bmatrix}\frac{\partial z_1}{\partial x_1}&\frac{\partial z_1}{\partial x_2}\\\frac{\partial z_2}{\partial x_1}&\frac{\partial z_2}{\partial x_2}\end{bmatrix} \cr\cr }$$ The paper you're reading refers to $J$ as the gradient, which is true, from a certain perspective.