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Multiple Eigenvalues.

Multiple eigenvalues can be defective (having only a single eigenvector and a chain of principal vectors) or derogatory (having several linearly independent eigenvectors) or both. A normal (or more generally, a diagonalizable) matrix has only derogatory multiple eigenvalues. The eigenvalue problem for defective matrices is ill posed. As mentioned above in the discussion of convergence properties, perturbation (read approximation) scatters the defective eigenvalue into a cluster of poorly conditioned eigenvalues.

The Lanczos algorithm has the theoretical advantage that the characteristic polynomials of the tridiagonal matrices $\{ T_j\}$ approximate the minimal polynomial of $A$ [234]. This means that in exact arithmetic the Lanczos algorithm computes a complete chain of principal vectors for a defective multiple eigenvalue if run for at least as many steps as the length of the chain.

Susan Blackford 2000-11-20