Play: Eigenvalue Dynamics
Test the rules yourself. Set the two eigenvalues and the starting point, and watch the neural activity trace its path. Predict before you drag: explode, decay, or something in between?
The white path is the neurons' activity over time. It gets pulled toward the eigenvector with the larger eigenvalue. Both λ > 1 → explode; both < 1 → decay; one each → a saddle. Just two numbers (the eigenvalues) tell you the fate of the whole system — the core trick behind analyzing stability in neuroscience and deep learning.
Try each case:
1. Both λ > 1 (e.g. 1.4 and 1.2) → the path explodes outward, curving toward the larger eigenvalue's line.
2. Both λ < 1 (e.g. 0.8 and 0.4) → everything spirals in to zero.
3. One above, one below 1 → a saddle: it explodes unless you start exactly on the small-eigenvalue line, where it decays toward the origin first.
Notice how the trajectory always bends toward the eigenvector with the bigger eigenvalue — because its λⁱ outgrows the other as the steps pile up. You're watching two numbers decide the fate of a whole dynamical system in real time.