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) ...

  • $\begingroup$ Five dimensions, actually - three axes plus time plus color (six, if you add sound), but yes this still falls short of the required 8. I was hoping there might be a way around this. Not necessarily a perfect representation, mind you, just something to give a little more visual meaning to some otherwise abstract math. $\endgroup$ – R. Burton Mar 13 at 21:48
  • $\begingroup$ @R.Burton No, color and sound will not allow you to add dimensions. Color and sound provide a scale as the axis does but for every value of your axis you have to plot a n-1 dimensional space, if you have three variables, one is color, for each tone you have to plot a 2D surface... $\endgroup$ – H.C. Lefevre Mar 13 at 22:07
  • $\begingroup$ I was thinking more along the lines of an arrow at every lattice point in 3d space (3I->3O) and having the arrows change direction and length as a function of time (4I->3O), with a slider to change the point on the color axis using the input color for the tail of each arrow and output color for the tip, with a nice color gradient for aesthetics (5I->4O). $\endgroup$ – R. Burton Mar 14 at 0:21
  • $\begingroup$ Then the strength of each harmonic of a note could represent the input value on the sound axis, the strength of each harmonic three or four octaves higher could represent the output on the sound axis (6I->5O) Still, you're right that there are limitations to how many dimensions I can reasonably represent. $\endgroup$ – R. Burton Mar 14 at 0:22
  • $\begingroup$ @R.Burton why not, sill it depends on your goal while attempting to represent this function. I do not think that a regular brain will be able to interpret such a representation. Still do you have more details on the application (Does it have some properties ? Linearity ? Is it quadratic ?) $\endgroup$ – H.C. Lefevre Mar 14 at 7:20

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