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Related Eigenproblems

1. If is nonsingular, then the NHEP has the same eigenvalues and corresponding right eigenvectors as . Similarly, has the same eigenvalues as and right eigenvectors . If is nonsingular, has the same right eigenvectors as , and its eigenvalues are reciprocals . Finally, if is nonsingular, has reciprocal eigenvalues and right eigenvectors . Analogous statements can be made about left eigenvectors.

2. More generally, suppose has eigenvalues and corresponding right eigenvectors . Let , , , and be scalars such that . Then has the same eigenvectors as and eigenvalues . If one or both of and are nonsingular, then the method in item 1 above can be applied.

3. Let be an -by- matrix polynomial, where is not identically zero. An eigenpair of satisfies . Define the by block companion pencil of as where all entries are by blocks and all entries not explicitly shown are 0. Then is a regular generalized eigenvalue problem, and the eigenvalues of are the eigenvalues of . Note that there are eigenvalues. If is an eigenpair of , then is a right eigenvector of .     Next: Example Up: Generalized Non-Hermitian Eigenproblems   Previous: Specifying an Eigenproblem   Contents   Index
Susan Blackford 2000-11-20