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## Related Eigenproblems

1. Consider the GNHEP , where and are square and is nonsingular. The matrix has the same eigenvalues and right eigenvectors as . If is a left eigenvector, i.e., , is a left eigenvector of . Analogous statements are true about . If is a factorization of (from Gaussian elimination, QR decomposition, or anything else), has the same eigenvalues as , right eigenvector , and left eigenvector . If is well-conditioned, or , , or can be accurately computed, this is an effective way to solve . If is ill-conditioned, it is preferable to treat it as the GNHEP, see §2.6.

2. Let be a monic polynomial. Define the by companion matrix of as where all entries not explicitly shown are 0. Then the eigenvalues of are the roots of , and the right eigenvectors are . is not diagonalizable if has multiple roots. A reliable, but not optimally efficient, algorithm for finding roots of a polynomial is to find all the eigenvalues of .

3. Let be a monic matrix polynomial, where each is an by matrix. An eigenpair of satisfies . Define the by block companion matrix of as where all entries are by blocks and all entries not explicitly shown are 0. Then 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: Non-Hermitian Eigenproblems  J. Demmel Previous: Specifying an Eigenproblem   Contents   Index
Susan Blackford 2000-11-20