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## Related Eigenproblems

Since the HEP is one of the best understood eigenproblems, it is helpful to recognize when other eigenproblems can be converted to it.

1. If is non-Hermitian, but is Hermitian for easily determined and , it may be advisable to compute the eigenvalues and eigenvectors of . One can convert these to eigenvalues and eigenvectors of via and . For example, multiplying a skew-Hermitian matrix (i.e., ) by the constant makes it Hermitian. See §2.5 for further discussion.

2. If for some rectangular matrix , then the eigenproblem for is equivalent to the SVD of , discussed in §2.4. Suppose is by , so is by . Generally speaking, if is about as small or smaller than (, or just a little bigger), the eigenproblem for is usually cheaper than the SVD of . But it may be less accurate to compute the small eigenvalues of than the small singular values of . See §2.4 for further discussion.

3. If one has the generalized HEP , where and are Hermitian and is positive definite, it can be converted to a Hermitian eigenproblem 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 . See §2.3 for further discussion.

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