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## Eigenvalues and Eigenvectors

The polynomial is called the characteristic polynomial of . The roots of are eigenvalues of . Since the degree of is , it has roots, and so has eigenvalues.

A nonzero vector satisfying is a (right) eigenvector for the eigenvalue . The eigenpair also satisfies , so we can also call a left eigenvector.

All eigenvalues of the definite pencil are real. This lets us write them in sorted order . If all , then is called positive definite, and if all , then is called positive semidefinite. Negative definite and negative semidefinite are defined analogously. If there are both positive and negative eigenvalues, is called indefinite.

Each is real if and are real. Though the may not be unique, they may be chosen to all be orthogonal to one another: if . This is also called orthogonality with respect to the inner product induced by the Hermitian positive definite matrix . When an eigenvalue is distinct from all the other eigenvalues, its eigenvector is unique (up to multiplication by scalars).

Next: Eigenspaces Up: Generalized Hermitian Eigenproblems   Previous: Generalized Hermitian Eigenproblems     Contents   Index
Susan Blackford 2000-11-20