eigshow — a scope for linear maps

eigshow is a free, interactive visualizer for eigenvectors, eigenvalues, and the singular value decomposition (SVD) of 2×2 matrices — steer a probe vector on the unit circle and watch its eigenvalues, singular values, and condition number update live.

x — the probe you steer (on the unit circle) Ax — the response locus of Ax (image of the circle)

Hunt for an eigenvector. Drag x around the unit circle and watch Ax. Almost everywhere the two arrows point in different directions — the map rotates as well as stretches. An eigenvector is a direction where Ax is perfectly parallel (or anti-parallel) to x: the arrows line up, no rotation, pure scaling. That scale factor is the eigenvalue — the Rayleigh quotient x·Ax / x·x, shown the instant you lock on. The green ellipse is the image of the whole circle; its two "flattest/fattest" directions are where eigen-alignment tends to live for symmetric A.