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## Related Eigenproblems

1. If and are Hermitian, is not positive definite, but is positive definite for some choice of real numbers and , one can solve the generalized Hermitian eigenproblem instead. Let ; then the eigenvectors of and are identical. The eigenvalues of and the eigenvalues of are related by .

2. If and are non-Hermitian, but and are Hermitian, with positive definite, for easily determined , and nonsingular and , then one can compute the eigenvalues and eigenvectors of . One can convert these to eigenvalues and eigenvectors of via and . For example, if is Hermitian positive definite but is skew-Hermitian (i.e., ), then is Hermitian, so we may choose , , and . See §2.5 for further discussion.

3. If one has the GHEP , where and are Hermitian and is positive definite, then it can be converted to a HEP as follows. First, factor , where is any nonsingular matrix (this is typically done using Cholesky factorization). Then solve the HEP for . The eigenvalues of and are identical, and if is an eigenvector of , then satisfies . Indeed, this is a standard algorithm for .

4. If and are positive definite with and for some rectangular matrices and , then the eigenproblem for is equivalent to the quotient singular value decomposition (QSVD) of and , discussed in §2.4. The state of algorithms is such that it is probably better to try solving the eigenproblem for than computing the QSVD of and .

Next: Example Up: Generalized Hermitian Eigenproblems   Previous: Specifying an Eigenproblem   Contents   Index
Susan Blackford 2000-11-20