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## Positive Definite

We reduce it to an equivalent standard HEP. It is done as follows. Choose a decomposition for :
 (90)

Then the generalized eigenvalue problem for is equivalent to the standard HEP for . Both share the same eigenvalues since

which also says that if is an eigenvector for the pair, is an eigenvector for the matrix , and on the other hand if is an eigenvector for , is an eigenvector for the pair. Common choices for are:
1. , the unique positive definite square root of . In this case, . This choice is good enough for theoretical investigations.
2. is the Cholesky factor; optionally with pivoting, i.e., is lower triangular with positive diagonal entries. This choice is preferred for numerical computations.
3. Analogously is upper triangular with positive diagonal entries. It shares the same advantage of the second choice.
In what follows, sometimes it is more convenient to use the inner product induced by a positive definite matrix , the corresponding vector norm , and the two-vector angle function (more precisely, angle between the subspaces spanned by two vectors) . In our case, or . They are defined as follows.

When , all three reduce to the usual definitions. It is rather easy to see that
 (91)

With some extra work, we can relate to the usual angle function, e.g., for , as follows.
 (92)

Subsections

Next: Residual Vector. Up: Stability and Accuracy Assessments Previous: Stability and Accuracy Assessments   Contents   Index
Susan Blackford 2000-11-20