Eigenvectors: the Directions a Matrix Does Not Turn

A three by three matrix A acts on a unit vector x, in blue, and sends it to A x, in red. In general A x points somewhere else entirely.

Sweep x over the sphere, and A x swings around with it, always tilted away. The grey arc measures the angle between them.

But for a few special directions the angle closes: A x lies on the same line as x. Those directions are the eigenvectors, and the stretch factor is the eigenvalue: A x equals lambda x.

This matrix has three of them, with eigenvalues two and a half, one, and one half. Along the first, A stretches; along the last, it shrinks.

Lambda is an eigenvalue exactly when A minus lambda I is singular, the characteristic equation; then x is a vector in its null space.

Your turn. Turn x with the pad. Find a direction that A does not turn, so that A x is parallel to x.

An eigenvector is a direction the matrix only stretches. The eigenvalues say by how much.