How to visualize matrix functions?

Question

I’m a middle school (6th grade) student who is self-learning these topics, including linear algebra, and this is another question I have (teacher didn’t know again):

I recently came across exponentiating matrices here, which we basically define by plugging in a matrix $A$ into the series; $\exp(A) = e^A = \sum_{n = 0}^{\infty} \frac{1}{n!}A^n$.

This is when I learned about matrix functions. Matrix functions take in matrices as inputs and output matrices: $f: A_x \rightarrow A_y$. The function $\exp(A)$ is probably the most noteworthy example of a matrix function (do correct me if I’m wrong).

One of the beauties of functions is their visualizations; graphs, vector fields, linear transformations, and so on. Now, I think that a valid question to ask is

How do we visualize matrix functions? It would require, for the bare minimum of $2 \times 2$ matrices; 8 dimensions…?

However, we already have two main methods for visualizing $4$-dimensional functions; transformations, and the one I prefer, vector fields. Is there something analogous to transformations or vector fields to visualize them?

I would also (really) like to know if matrix functions are linear, that is, for some matrix function $M$, if

$$M(A + B) = M(A) + M(B)$$ $$\text{and}$$ $$M(kA) = kM(A)\text{.}$$

I have already tested it for one simple function; $f(A) = 2A$, and the result I got is that $M(A + B)$ is indeed equal to $M(A) + M(B)$ and that $M(kA)$ is equal to $kM(A)$, but maybe this is a coincidence as I selected a simple function; $M$ is just a scaling function in the first place.

Related in some sense: Does the exponential of a function converge? What can we do with it?

Answer 1

Matrix functions are just functions on multiple variables. You can visualize them in the same way (or can't, as usually matrix functions are at least 8 dimensional, which is pretty hard to visualize as a graph). But you can view Matirces as just another vector and apply higher dimensional functions to them.
Matrix functions, in general, are not linear, but there are linear ones of course, which behave the same way you'd expect them, just on other dimensions.
So a linear Matrix function taking in a mxn Matrix is just a linear transformation of an nm dimensional space. (E.g. a linear Matrix function on $\mathbb{R}^{2\times2}$, can be represented as a 4x4 Matrix!) How would you transform a Matrix in $\mathbb{R}^{2\times 2}$ to a vector in $\mathbb{R}^{4}$ so you can apply Matrix vector multiplication on it? How would the Matrix which is your function look like (So f(A)=2A)? A Matrix function which wouldn't be linear, for example, would be the function which maps $M$ to $I_n$ (the identity Matrix) (Why is it not linear?)