Exceptional-Point Eigenvalue Monodromy
Follow an eigenvalue through two parameter loops and expose a defective point's singular projectors.
Free theory articleFollow an eigenvalue through two parameter loops and expose a defective point's singular projectors.
Concept & interpretation
An exceptional point occurs when both eigenvalues and their corresponding eigenvectors coalesce, making the matrix defective (non-diagonalizable). This module tracks an eigenvalue and its eigenvector around a closed parameter loop in the complex plane, revealing the monodromy: a single 360-degree loop swaps the two eigenvalues, and two full loops (720 degrees) are required to restore the original state.
Experiment: Observe the monodromy and exceptional point
Open the Exceptional-Point Eigenvalue Monodromy module to interactively track the eigenvalues.
- Set Parameter radius to 0.25 and Branch to "Positive start". Observe the two distinct eigenvalues at ±0.5.
- Change Unwrapped degrees from 0 to 360. Watch the tracked eigenvalue move along the upper half of the Riemann surface to arrive at -0.5. The eigenvalues have swapped!
- Change Unwrapped degrees to 720. The tracked eigenvalue completes the second half of the loop and returns to 0.5.
- Set Parameter radius to 0. Watch the two eigenvalue sheets merge at the origin. The matrix becomes a nilpotent Jordan block, and the eigenvectors coalesce.