Nonlinear Equations - UFRJ

[SEC. 8.5: CONDITION NUMBERS IN GENERAL 115
Example 8.11. As in the previous section, let M = C n+1 \ {0},
H = C × and F i = H Di . In that case, M \ H = P n , and we set 〈·, ·〉 P n
equal to the Fubini-Study metric. In that case, e i = d i = D i .
Example 8.12. Assume that F 1 , . . . F s are non-degenerate fewspaces
and that M/H is compact. Let
〈·, ·〉 = 〈·, ·〉 1 + · · · + 〈·, ·〉 s .
There we can take d i = 1. Because F i is a non-degenerate fewspace
we know that 〈·, ·〉 i is non-degenerate. By compactness, e i > 0.
In [58], we introduced this mysterious local invariant:
Definition 8.13. Let 〈·, ·〉 be Hermitian inner products in an n-
dimensional complex vector space E. Their mixed dilation is
∆ =
min max max ‖T u‖=1 〈u, u〉 i
.
T ∈L(E,C n ) i min ‖T u‖=1 〈u, u〉 i
Finiteness of ∆ follows from the fact that the fraction in its expression
is always ≥ 1 and finite. The reader can check that the
minimum is attained for some T .
The quotient manifold M/H or a compact subset therein may be
endowed with a ‘minimal dilation metric’, namely
〈u, v〉 x = v ∗ T ∗ T u
where T is a point of minimum of the dilation at that point x. This
metric is arbitrary up to a multiple, so we may scale the metric so
that, for instance,
tr〈·, ·〉 = ∑ 〈·, ·〉 i
Open Problem 8.14. Under what conditions this local metric extends
to a Hermitian metric on all of M/H? It would be nice to find a
uniform bound for the dilation that is polynomially bounded in the
input size.
From now on, we fix a Hermitian metric 〈·, ·〉 on M/H for reference.