Eigenvector Geometry

Differential Equations · Axiom Academy

Some vectors are unshakeable — a matrix can stretch or shrink them, but never turn them. Watch, then drive, what makes a vector special. Most vectors get knocked off course Multiply almost any vector by a matrix A and it rotates — it ends up pointing somewhere new. But a special few vectors don't rotate at all: A only stretches or shrinks them. Those are eigenvectors , and geometry is the fastest way to see why they matter. Watch v sweep a full turn while its image is traced alongside it. Almost everywhere the two arrows splay apart — except at two exact directions, where they snap into a single line. The two moments the arrows line up perfectly are exactly the eigenvector directions of A — that's not a coincidence, it's the definition. Drag v — hunt for the directions that don't rotate Drag the blue handle anywhere around the plane. The red arrow always shows Av . Most places you drop it, the two point in different directions. Find where they overlap. When the banner turns green, you've landed on an eigenvector: for that direction. Change the matrix, the special directions change too Matrix only scales each axis — no mixing between x and y . Drag v around again: where do v and Bv line up this time? For a diagonal matrix, the eigenvectors are always the coordinate axes — the standard basis vectors (1,0) and (0,1) . That's why diagonal matrices are the easiest systems to solve. The straight-line trajectories of a system

This is the written version of the interactive lesson above. See the full Differential Equations course.