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Some Combination of and is Positive Definite

This is the general case for the definite matrix pair, and now may be singular. To be able to handle infinite eigenvalues, it is standard practice [425] to introduce a homogeneous representation of an eigenvalue by a nonzero pair of numbers :

When , such pairs represent eigenvalue , and this occurs when is singular. Such representations are clearly not unique since represents the same ratio for any , and consequently the same eigenvalue. So really a pair is a representative from a class of pairs that give the same ratio. The difference of two eigenvalues is measured by the chordal metric: for and ,
 (99)

An equivalent definition for a Hermitian matrix pair being a definite pair is that the Crawford number

It can be proved [425] that if is a definite pair, then
• there is a such that is positive definite and , the smallest eigenvalue of , where
 (100)

The reader is referred to [86,87] for an algorithm that realizes such a , which in turn yields the lower bound on needed for estimating error bounds below.
• there is an nonsingular matrix such that
 (101)

where
 (102)

The decompositions (5.37) and (5.38) give a complete picture of the underlying eigenvalue problems. In fact, all eigenvalues are given by pairs with corresponding eigenvectors . If, in addition, in (5.37) and (5.38) for all , then [423]

 (103)

Subsections

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