Linear Algebra
Eigenvectors: The Directions That Survive
Arrows that keep their line
Apply a matrix to most vectors and they get knocked off their direction — rotated, sheared, dragged somewhere new. But almost every transformation has a few special directions that survive: arrows that get stretched or squashed along their own line without being turned.
Those are the eigenvectors, and the stretch factor of each is its eigenvalue. Eigenvalue 3 → the arrow triples along its axis; 0.5 → it halves; negative → it flips, but stays on the line.
Picture a spinning globe: most points trace huge arcs, but points on the axis stay put. The axis is the eigen-direction.
analogy
An eigenvector is the transformation's natural grain — like how wood splits cleanly along the grain and resists across it. Every matrix has preferred directions; eigenvectors are them.