A matrix turns most needles: point v one way and its image Av comes out rotated. Sweep the whole rose and you can feel Av lag and lead the needle.
At a few special headings the turning stops: Av lands right on the line of v. That heading is an eigenvector — the map only stretches it.
The stretch factor read off there is the eigenvalue λ, the number in Av = λv. A negative λ flips the needle to point backward.
Not every map has one. Sweep the rotation chip forever and nothing ever aligns — its eigenvalues are complex, so it has no real eigenvectors at all.
Press strike to apply the map again and again: every needle funnels onto the biggest-λ direction. That is power iteration — the seed of PCA.
The compass jammed mid-sweep. The lesson continues without it — everything you have already done is safe.