Apologies if this is not at an appropriate level for this site or if it's too broad/scrambled of a question, but I was wondering how best to visualize a matrix-valued function of a matrix variable?
For some context: in my single variable calculus class, we were taught that the derivative of $f(x)$ at a point $x = a$ is $f'(a)\in\mathbb{R}$. This represents the slope of the tangent line to the graph of $f(x)\vert_{x = a}$. The equation of this tangent line is $y = f'(a)(x-a) + f(a)$, and this is the best linear approximation of $f(x)$ near $x = a$.
In my analysis class, we were taught that the derivative of a map $f:U\to F$, where $U\subseteq E$ is open, $x\in U$, and $E,F$ are complete normed linear spaces, is a continuous linear map $\lambda=f'(x):E\to F$ satisfying $$f(x+h)-f(x)=f'(x)(h)+|h|\psi(h) $$ with $\lim_{h\to 0}\psi(h)=0$ and $\psi(0)=0.$ There is no confusion here. This definition is in accordance with our definition from single variable calculus, it's a special case as expected, where the standard matrix of this linear transformation (in the one dimensional case) is the $1\times 1$ matrix $[f'(a)]$. If we consider the special case of our scenario where $E$ and $F$ are Euclidean spaces rather than arbitrary complete normed linear spaces, i.e. for $f : \mathbb{R}^n \to \mathbb{R}^m$ at a point $a \in \mathbb{R}^n$ then the derivative $Df(a)$ is an $m\times n$ matrix $Df(a) = \left[\frac{\partial f_i}{\partial x_j}( a)\right]$, or the Jacobian of $f$ at $a$.
On our exam, we were asked to compute the derivative of $f:M_{n\times n}(\mathbb{R})\to M_{n\times n}(\mathbb{R}) $ given by $f(x)=x^3$. This was a fairly straightforward computational task, however I have been stuck about how to visualize/understand my answer. I know you could just say that $M_{n\times n}(\mathbb{R})$ and $\mathbb{R}^{n^2}$ are isomorphic as complete normed linear spaces, so think of it in terms of the framework that has already been established.
My question is: I know how to visualize/conceptualize this exam question when $f$ is a real-valued function of a real variable, but in the case from the exam is there a way to understand it in terms of what a linear map actually does (some combination of rotation and scaling) or where the vectors that comprise the matrix live with respect to the function (in some tangent subspace would be my intuition, but I'm not sure since I don't have that level of background in such a subject)? Is there a meaningful way to answer/subject to address my question or is the whole matter moot since even for a $3\times 3$ matrix, that would correspond to a function of $9$ variables, which we cannot visualize?