What is intuition behind matrix differentiation?

Question

I understand the intuition behind normal function differentiation, it tells us how function varies when we wiggle the variable. But I don't understand what matrix differentiation tells us.

Answer 1

Imagine a function f(x,y,z) that varies over the 3D spatial dimension and it can either be scalar valued or vector valued. Then the derivative matrix tells you how the value of the function (in the scalar valued case) or each component of the vector changes with respect to the variables x,y,z.

For a scalar valued function f(x,y,z) = f, the derivative matrix looks like: $\begin{bmatrix}\frac{\partial f}{\partial x}\frac{\partial f}{\partial y}\frac{\partial f}{\partial z}\end{bmatrix}$.

For example, if we consider the temperature in the air at position (x,y,z) given by some scalar valued function f(x,y,z), the derivative matrix would tell you how the temperature changes as the position changes in the x, y, or z directions.

For a vector valued function, say $f(x,y,z) = (f_1(x,y,z),f_2(x,y,z),f_3(x,y,z))$, the derivative matrix looks like: $\begin{bmatrix}\frac{\partial f_1}{\partial x}\frac{\partial f_1}{\partial y}\frac{\partial f_1}{\partial z}\\\frac{\partial f_2}{\partial x}\frac{\partial f_2}{\partial y}\frac{\partial f_2}{\partial z}\\\frac{\partial f_3}{\partial x}\frac{\partial f_3}{\partial y}\frac{\partial f_3}{\partial z}\end{bmatrix}$

An example for this would be to consider the velocity vector of molecules in the air (a vector-valued function). The derivative matrix tells you how each component of the velocity vector $(f_1, f_2, f_3)$ changes over each direction.