How to visualize $\mathbb{C}^2$?

Question

In a homework question I had to do, the rotational matrix $A = \begin{pmatrix} 0&-1\\1&0 \end{pmatrix}$ was given. Its eigenvalues in $\mathbb{C}$ are $i$ and $-i$. The set of all eigenvectors with eigenvalue $i$ is $\{(a,ia)|a \in \mathbb{C} \}$; the set of all eigenvectors with eigenvalue $-i$ is $\{(ia,a)|a \in \mathbb{C} \}$. Just calculating this stuff was pretty much the substance of the homework question, but I'm trying to visualize this better.

Intuitively, I can see why $A$ does not have any real eigenvalues: rotating a point $(a,b)$ in $\mathbb{R}^2$ about the origin by $\pi/2$ (counterclockwise) would result in the point $(b,-a)$. The dot product of the corresponding vectors is $\langle a,b \rangle \cdot \langle b,-a \rangle = 0$, so they are perpendicular and thus not parallel. Since they are not parallel, there are no real eigenvalues.

However, I have no idea where to start when thinking about $\mathbb{C}^2$. Should I be thinking of something similar-looking to $\mathbb{R}^4$, similar to how the complex plane is visualized with a real and imaginary axis, or something completely different? What does it mean that for all $a \in \mathbb{C}$, $(a + 0i, 0 + ai) * A = i\cdot (a+0i, 0+ai)$? Multiplying some $z \in \mathbb{C}$ by $i$ in the complex plane also rotates $z$ by $\pi/2$ about the origin (counterclockwise), is this something I could use to help gain intuition here?

Any help in understanding this would be greatly appreciated.

Answer 1

I can't visualize $\Bbb R^4$ well enough for these kinds of things, so here is what I would do: imagine two copies of $\Bbb C$ sitting next to each other, with independent operations. Meaning, given two pairs of complex numbers, you sum the complex numbers in the first copy of $\Bbb C$ and also sum the complex numbers in the second copy of $\Bbb C$. If you can visualize adding complex numbers in $\Bbb C$, all you need to do is duplicate your mental picture. And multiplying by a scalar does the usual thing to complex numbers, but now it does it to both numbers.

What the application of $A$ (from the right, blegh) does is swap the two complex numbers and then negates the second one (or, first it negates the first one and then swaps them, either way yields the same result). We should be able to picture swapping two numbers between our two mental copies of the complex plane, and that negating a number flips it across the origin. Try to picture how this operation acting on $(1,i)$ yields the same result as simply rotating both $1$ and $i$ an angle of $90^\circ$ counterclockwise, which is what happens when you multiply by $i$! Same idea for $(i,1)$ and $-i$.

Here's a picture:

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Answer 2

Something that I do to visualize higher dimensional spaces is consider nice subspaces, and try to piece them together, so to speak. For example, in $\mathbb R^3$, we can think of this as trying to assemble the space out of a bunch of two dimensional slices. This is usually how I go about visualizing the shapes of graphs and other objects that sit in $\mathbb R^3$. Likewise, if I wanted to visualize the graph in $\mathbb R^4$, I might try to consider two dimensional slices, and try to build a picture out of it.