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## Eigenvalues and Eigenvectors

The polynomial is the characteristic polynomial of . The degree of is at most . The roots of are called the finite eigenvalues of . If the degree of is , we say that has infinite eigenvalues too. For example, has characteristic polynomial , and so has eigenvalues , , and .

If is a finite eigenvalue, a nonzero vector satisfying is a (right) eigenvector for the eigenvalue . A nonzero vector satisfying is a left eigenvector.

If is an eigenvalue, nonzero vectors and satisfying and are called right and left eigenvectors, respectively.

An by pencil need not have independent eigenvectors. The simplest example is , which is defined in equation (2.3) and discussed in §2.5.1. The fact that independent eigenvectors may not exist (though there is at least one for each distinct eigenvalue) will necessarily complicate both theory and algorithms for the GNHEP.

Since the eigenvalues may be complex or infinite, there is no fixed way to order them. Nonetheless, it is convenient to number them as , with corresponding right eigenvectors and left eigenvectors (if they exist).     Next: Deflating Subspaces Up: Generalized Non-Hermitian Eigenproblems   Previous: Generalized Non-Hermitian Eigenproblems     Contents   Index
Susan Blackford 2000-11-20