Very cool, but as a layman I was a very confused by the description of eigenspaces and the S1/S2 lines. I'm just guessing here (reasoning below) but I'd like to suggest phrasing like:

"Eigenspaces are special lines, where any starting-point along them yields an eigenvalue that lands back on the same line. In these examples two exist, labeled, S1 and S2."

"Eigenspaces show where there is 'stability' from repeated applications of the eigenvector. Some act like 'troughs' which attract nearby series of points (S1) while others are like hills (S2) where any point even slightly outside the stable peak yields eigenvalues further away.

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Original post / detailed-reaction:

First, every point on the same line as an eigenvector is another eigenvector. That line is an eigenspace.

At first I though this statement-of-fact meant that the whole tweakable quadrant of the X/Y plot (at a minimum) is an unbroken 2D Eigenspace, because every point within it can be "covered" by a dashed line (a 2D "vector") if I pick the appropriate start-point.

However, the last sentence also says eigenspaces are (despite the "space" in their name) lines, which throws the earlier interpretation into doubt.

As you can see below, eigenspaces attract this sequence

S1 and S2 were displayed earlier, but not explained, now this section implies that those lines are the Eigenspaces? If so, what is the difference between S1 and S2? Playing with the chart, I assume they are the "forward" and "reverse" for repeat-applications of the transformation.