Special Directions

Linear Algebra (Matrices) · Axiom Academy

A matrix turns almost every vector off its line. A precious few it leaves pointing exactly where they were — those are the eigenvectors, and finding them cracks the whole transformation open. Most vectors get knocked off their line. A few refuse. Picture a matrix as a machine that grabs the whole plane and reshapes it — stretching, squashing, shearing. Feed it a vector and you usually get back a vector pointing somewhere new : the machine rotated it off its original line. But hidden in every such transformation are a handful of special directions the machine can't turn. A vector pointing along one of those comes back pointing the exact same way — only longer or shorter. Before any formula, just watch it happen. Watch a fan of arrows pointing in every direction. Press play and the matrix A transforms each one. Most swing off their faded starting line — their direction changed. But two arrows lock dead onto their line: the matrix only restretches them. Those two are the special directions. The two arrows that never leave their line are the eigenvectors of A. Everything else turns. Hunt for a direction the matrix can't turn Drag the blue arrow around the circle. The matrix instantly shows you in orange. Watch the gap between their directions: for most arrows the orange one points somewhere else, so A turned . Sweep slowly and feel for the spots where the gap snaps to 0° — there, lands right on top of 's own line. You found an eigenvector.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.