Eigenvector Hunt

Each challenge checks itself. Work on this keeps it in view above the controls.

Rotate a unit vector x and watch Ax; find the directions where Ax is parallel to x, read off lambda, then iterate x -> Ax (power method) to see the dominant eigenvector emerge.

How to use

• An eigenvector of A is a nonzero x with Ax = λx: A only stretches, shrinks or flips it, without turning it off its line.
• Turn x with the angle slider and watch Ax. The status box reports the angle between x and Ax. When it is 0° (same direction) or 180° (opposite direction) you have found an eigen-direction, and λ is read off as the stretch factor (negative if Ax points backwards).
• Found directions are drawn as dash-dot lines and listed under the picture. Close to one? Press Snap to the nearest eigen-direction.
• Power iteration starts from the current x and repeatedly replaces x by Ax/‖Ax‖. The numbered dots on the circle show x₀, x₁, x₂, … For most matrices they approach the direction of the eigenvalue with the largest size |λ| (the dominant one).
• Some matrices (rotations, for example) have complex eigenvalues: then no real direction is ever parallel to its image, and the app says so.
• Keyboard: every control is a standard form field; the status box is read aloud by screen readers. Arrows differ by pattern and label: x is solid, Ax is dashed.

Accessibility: every control is a standard form control you can reach with Tab and change with the arrow keys; the status box is announced by screen readers; the picture has a text description (press Describe) that updates as you work; animations are skipped when your system asks for reduced motion.