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

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

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

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

where

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]