Next: Invariant Subspaces Up: Hermitian Eigenproblems   J. Previous: Hermitian Eigenproblems   J.   Contents   Index

## Eigenvalues and Eigenvectors

The polynomial is called the characteristic polynomial of . The roots of are called the 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 . Since , left and right eigenvectors are identical.

All eigenvalues of a Hermitian matrix 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 eigenvalue has an eigenvector . We may assume without loss of generality that . Each is real if is real. Though the may not be unique (e.g., any vector is an eigenvector of the identity matrix), they may be chosen to all be orthogonal to one another: if . When an eigenvalue is distinct from all the other eigenvalues, its eigenvector is unique (up to multiplication by scalars).

Next: Invariant Subspaces Up: Hermitian Eigenproblems   J. Previous: Hermitian Eigenproblems   J.   Contents   Index
Susan Blackford 2000-11-20