Nonlinear Equations - UFRJ

124 [CH. 9: THE PSEUDO-NEWTON OPERATOR
Another convenient interpretation is the following: x = A † y is
the solution of the least-squares problem:
Minimize‖Ax − y‖ 2 with ‖x‖ 2 minimal.
If A is m×n of full rank, m ≤ n, then x is the vector with minimal
norm such that Ax = y.
Lemma 9.2 (Minimality property). Let A be a m × n matrix of
rank m, m ≤ n. Let Π be a m-dimensional space such that A |Π is
invertible. Then,
‖A † ‖ ≤ ‖(A |Π ) −1 ‖.
The same definition and results hold for linear operators between
inner product spaces.
In particular, when Let f ∈ H d and X ∈ C n+1 . Then,
Df(X) † = ( ) −1
Df(X) | ker Df(X) ⊥
whenever this derivative is invertible. In particular,
for any hyperplane Π.
‖Df(X) † ‖ ≤ ‖ ( Df(X) |Π
) −1
‖
While the minimality property is extremely convenient, we will
need later the following lower bound:
Lemma 9.3. Let A be a full rank, n×(n+1) real or complex matrix.
Assume that w = ‖A † ‖‖A−B‖ < 1. Let Π : ker A ⊥ → ker B ⊥ denote
orthogonal projection. Then for all x ∈ (ker A) ⊥ ,
In particular, for all y,
‖Πx‖ ≥ ‖x‖ √ 1 − w 2 .
‖B † Ay‖ ≥ ‖y‖
√
1 − w
2
1 + w .