Claiming "up to a quadratic speedup" in one sentence and then
saying "similar to the FFT speedup or QuickSort" is potentially
confusing --- those both give you speedsup from n^2 to n log n,
and I would not describe them as "quadratic speedup."

"This gift, it seems, doesn't to come at a price. A beautiful free lunch [7] indeed." s/to come at a price/come at a price. Also I disagree with the argument: in general, keeping all the iterates comes at a huge cost in memory, and if you don't keep them, knowing when to stop is far from easy.

I think there is a mistake in the interactive graph right above "Choosing A
Step-Size." It says 2 is the "optimal step size", but it looks
like 2 is way too big, causing the biggest eigenvalue \lambda_3
to not converge at all.

The article introduces a convex quadratic, but doesn't explicitly mentions the connection between that and nonnegative eigenvalues. This could be clarified.