Geometric interpretation for complex eigenvectors of a 2x2 rotation matric.

*Visualizing the Eigenvectors of a Rotation*:

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.

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.*

*Visualizing the Eigenvectors of a Rotation, Craig Gidney, October 15, 2013*