Eigenvalues & Eigenvectors
Push a plane of arrows through the same matrix and almost every one swings to a new heading, yet a stubborn few, like the arrow reaching to `[2, 1]`, come out pointing exactly the way they went in, only longer.
- ▸Push every arrow through one matrix and only a couple come out still aimed along their own line.
- ▸One number tells the whole story: above one stretches, below one shrinks, negative flips the arrow, zero crushes it.
- ▸Stretch that surviving arrow ten times longer and its stretch number never moves, the line was special, not the arrow.
Two arrows sit on the same grid, both starting at the origin. One is the arrow you steer; the other is where a fixed matrix sends it. Its four numbers reach into your arrow, pair each row against it, multiply and sum, and hand back a brand-new arrow. That answer is the first surprise: for almost every way you aim the arrow you steer, the matrix points its answer somewhere else entirely, swung off to a different heading like a compass needle knocked by a passing magnet.
So steer the arrow slowly all the way around and watch the answer chase it. Through most of the circle the two disagree, the answer leaning off at its own stubborn angle. But twice on the way around something snaps into place: the answer lands right on top of the line your arrow is already on. Same heading, or dead opposite, never off to the side. On those two headings the matrix has stopped turning your arrow and is only sliding it along the line it was already on, longer, shorter, maybe reversed, but never rotated.
Those two special headings are the whole point of this topic. Every other direction is bookkeeping, knocked sideways in a tangle you would need the full grid of numbers to track. Find the directions a matrix refuses to turn and you have found the skeleton the entire transformation hangs on. The next part reads the one number waiting on each of them.