Nonlinear Equations - UFRJ

[SEC. 8.4: CONDITION NUMBERS FOR HOMOGENEOUS SYSTEMS 113
8.4 Condition numbers for homogeneous
systems
We consider now a possibly unmixed situation. Let f ∈ H d1 × · · · ×
H dn , where each f i is homogeneous in n + 1 variables. Let M =
C n+1 \ {0}, H = C × and thus M/H = P n .
Projective space is endowed with the Fubini-Study metric 〈·, ·〉.
Each of the H di has reproducing kernel K i (x, y) = (x 0 ȳ 0 + · · · +
x n ȳ n ) di and therefore (Exercise 5.5) induces a metric 〈·, ·〉 P n ,i =
d i 〈·, ·〉.
Lemma 8.8. Let L = L ix : H di → Tx(P ∗ n ) be defined by
〉
L ix (f) : u ↦→ √
〈f(·),
1
1
√ P x D¯x K(·, x)ū
di K(x, x)
Then L is onto, and L | ker L ⊥
is an isometry.
H di
.
Proof. If we assume the 〈·, ·〉 P n ,i norm on Tx(P ∗ n ), Lemma 8.4 implies
that the operator above is onto and L | ker L ⊥ is d −1/2
i times an
isometry.
For vectors, the relation between Fubini-Study and H di -induced
norm is
‖u‖ = √ 1 ‖u‖ i .
di
For covectors, it is therefore
‖ω‖ = √ d i ‖ω‖ i .
Hence, we deduce that L | ker L ⊥ is an isometry, when Fubini-Study
metric is assumed on P n .
Now we define
As before,
L x : F s → L(T x M, C s ),
⎡ ⎤
L 1x (f 1 )
(f 1 , . . . , f s ) ↦→
⎢
⎣ .
L sx (f s )
⎥
⎦ .