Geometric interpretation for complex eigenvectors of a 2×2 rotation matrix

The rotation matrix

has complex eigenvalues $\{e^{\pm i\theta}\}$ corresponding to eigenvectors $\pmatrix{1 \\i}$ and $\pmatrix{1 \\ -i}$. The real eigenvector of a 3d rotation matrix has a natural interpretation as the axis of rotation. Is there a nice geometric interpretation of the eigenvectors of the $2 \times 2$ matrix?

Tom Oldfield’s answer is great, but you asked for a geometric interpretation so I made some pictures.

The pictures will use what I called a “phased bar chart”, which shows complex values as bars that have been rotated. Each bar corresponds to a vector component, with length showing magnitude and direction showing phase. An example:

The important property we care about is that scaling a vector corresponds to the chart scaling or rotating. Other transformations cause it to distort, so we can use it to recognize eigenvectors based on the lack of distortions. (I go into more depth in this blog post.)

So here’s what it looks like when we rotate <0, 1> and <i, 0>:

Those diagram are not just scaling/rotating. So <0, 1> and <i, 0> are not eigenvectors.

However, they do incorporate horizontal and vertical sinusoidal motion. Any guesses what happens when we put them together?

Trying <1, i> and <1, -i>:

There you have it. The phased bar charts of the rotated eigenvectors are being rotated (corresponding to the components being phased) as the vector is turned. Other vectors get distorting charts when you turn them, so they aren’t eigenvectors.