A matrix A takes every vector and moves it. Most vectors get rotated and stretched. But a special few come out pointing the same way they went in — only scaled. Those are the eigenvectors, and the scale factor is the eigenvalue λ. Drag the orange probe around the circle: watch Av swing, and light up green exactly when v lands on an eigen-direction.
v and Av — probe & its imageeigenvector 1 (λ₁)eigenvector 2 (λ₂)Eigen-directions solve Av = λv: the image is a pure scaling, no turn. They exist as real lines only when the discriminant (tr A)² − 4·det A ≥ 0; when it's negative the eigenvalues are complex and every real direction gets rotated — that's the Rotate preset, a map with no fixed axis. Eigenvalues run almost all of applied math: |λ| governs whether a dynamical system or control loop is stable (a discrete system is stable when every |λ| < 1); they're the natural frequencies of a vibrating structure, the axes of a covariance ellipse in PCA, and the principal moments of an inertia tensor. Find the invariant directions and a hard problem becomes a set of independent 1-D ones — that is diagonalization.