Nonlinear Equations - UFRJ

118 [CH. 8: CONDITION NUMBER THEORY
8.6 Inequalities about the condition number
The following is easy:
Lemma 8.19. Assume that ‖f‖ = ‖g‖ = 1. Then
µ(f, x) −1 − ‖f − g‖ ≤ µ(g, x) −1 ≤ µ(f, x) −1 + ‖f − g‖
Definition 8.20. A symmetry group G is a Lie group acting on
M/H and leaving ω, ω 1 , . . . , ω n invariant. It acts transitively iff for
all x, y ∈ M/H there is Q ∈ G such that Gx = y. The action is
smooth if Q, x ↦→ Qx is smooth.
The action of G in M/H induces an action on each F i , by
f i
Q
fi ◦ Q −1 .
When each f ↦→ f ◦ Q is an isometry, we say that G acts on F i
by isometries. In this later case, µ and ¯µ are G-invariants.
Example 8.21. The group U(n + 1) is a symmetry group acting
smoothly and transitively on P n . It acts on each H di by isometries.
Proposition 8.22. Let G be a compact, connected symmetry group
acting smoothly and transitively on M/H, such that the induced action
into the F i is by isometries.
Then, there is D such that for all f ∈ F and Q ∈ G, ‖f‖ = 1,
‖f − f ◦ Q −1 ‖ ≤ Dd(x, Qx)
where d denotes Riemannian distance. In the particular case F = H d
and G = U(n + 1), D = max d i .
Proof. The existence of D is easy: take Q(t) so that Q(t)x is a minimizing
geodesic between x and Qx. Since the action is smooth,
is also smooth. Hence
f i ◦ Q ∗ t : x ↦→ 〈f i (·), K i (·, Q ∗ t x)〉
D =
sup ‖DK i (·, ˙Qx)‖
i, ˙Q∈T I G