eigs(): Eigenvalues and Eigenvectors

eigs() finds eigenvalues and associated eigenvectors of a square matrix. An eigenvector keeps its direction under a transformation, changing only by its eigenvalue. These quantities describe stability, principal directions, vibration modes, and repeated transformations.

Matrix requirements

Eigenvalue problems require a square matrix. Check its dimensions with size(). Results can be complex even when all matrix entries are real, so do not discard imaginary parts without a mathematical reason; inspect them with re() and im().

For symmetric matrices, eigenvectors have especially useful orthogonality properties. Related decompositions include svd() and qr(). Numerical results may vary slightly in ordering or precision, so validate by checking A×v against λ×v using multiply().