I am currently working with some matrix functions $\left(\mathbb{C}^2\right)^2\to\left(\mathbb{C}^2\right)^2$, and I would like to make some sort of graphic.
I had an idea to split the output into two 4-dimensional vectors, one for the real part and one for the imaginary part, and create two animated 3-dimensional vector field plots with each frame corresponding to a value in the fourth coordinate.
This isn't sufficient to depict the function over the entire domain, though, since the imaginary part of the input can effect the real part of the output and vice-versa unless $f(\mathbf{Z})=f(\Re[\mathbf{Z}])+if(\Im[\mathbf{Z}])$ for all $\mathbf{Z}\in\left(\mathbb{C}^2\right)^2$.
What could I use as a graphical representation for these functions?
Also, it might help to know if there's a standard way to visualize octonions, given that the real dimension is the same (8).
Note: $\left(\mathbb{C}^2\right)^2$ is the space of $2\times2$ complex matrices, it is the same thing as $\mathbb{C}^2\otimes\mathbb{C}^2$, $M_2(\mathbb{C})$, $\mathbb{C}^{2\times2}$, etc.
Tagged "several complex variables" because $\left(\mathbb{C}^2\right)^2$ is isomorphic to $\mathbb{C}^4$, and this is probably a problem dealt with in functions of several complex variables.
The issue is that $\mathbb{C}^4$ is isomorphic to $\mathbb{R}^8$. You can split your function $f:\mathbb{C}^4\rightarrow\mathbb{C}^4$ into eight fuctions $g_{i\in\left[1;8\right]}:\mathbb{R}^8\rightarrow\mathbb{R}$. I think that this step might not be that easy depending on the function.
If your function $f$ is a linear map then you can try to represent it as a matrix. It is equivalent to represent each $g_i$ as a vector of $\mathbb{R}^8$.
You can graphically represent those eight vectors in a barchart still it will not represent your function values (each bar will be the image of the canonical base of $\mathbb{R}^8$ by a $g_i$). This plot would be $\left(g_i\left(e_j\right)\right)_{(i,j)\in\left[1;8\right]^2}$. This would consist in $8\times8=64$ bars.
It is impossible to represent more than three dimensions in the same time (two axes plus time) ...