See the Eigenvectors
Watch it happen. The dashed lines are the eigenvector directions. Flip between 'Before' and 'After' — the eigenvectors stay glued to their own line while everything else swings around them.
The dashed lines are the eigenvector directions — the special lines that the transformation only stretches or squashes, never knocks off their own line. Each eigenvalue λ is how much its eigenvector gets scaled. Everything else in space gets rotated or sheared away from its direction; eigenvectors don't.
Read the transformation from its eigenvectors:
• [[3,0],[0,1]] → one eigenvalue is 3 (big horizontal stretch), the other is 1 (vertical unchanged) → a horizontal expansion.
• [[0,1],[1,0]] → eigenvalues +1 and −1 → a reflection across the diagonal.
• [[2,0],[0,0.5]] → stretch one way, squash the other.
The rotation surprise: try [[0,−1],[1,0]]. It has no real eigenvectors — because a pure rotation knocks every direction off its line, so none can stay put. Its eigenvalues are complex, and complex eigenvalues always mean rotation. (You'll use that fact to spot rotating neural dynamics in the next quest.)