In linear algebra it is shown that every invertible matrix can be written as a product of
elementary matrices, those matrices which are obtained from doing a row operation to
the identity matrix. Two of the row operations produce a matrix which will change
exactly one entry of a vector when it is multiplied by the elementary matrix. The other
row operation involves switching two rows and this has the effect of switching
two entries in a vector when multiplied on the left by the elementary matrix.
Thus, in terms of the effect on a vector, the mapping determined by the given
matrix can be considered as a composition of mappings which either flip two
entries of the vector or change exactly one. A similar local result is available for
nonlinear mappings. I found this interesting result in the advanced calculus book by
Rudin.
Definition B.3.1Let U be an open set in ℝ^{n}and let G : U → ℝ^{n}. Then G iscalled primitiveif it is of the form
( )
G (x) = x1 ⋅⋅⋅ α(x) ⋅⋅⋅ xn T .
Thus, G is primitive if it only changes one of the variables. A function F : ℝ^{n}→ ℝ^{n}iscalled a flip if
Thus a function is a flipif it interchanges two coordinates. Also, for m = 1,2,
⋅⋅⋅
,n,define
P (x) ≡ (x x ⋅⋅⋅ x 0 ⋅⋅⋅ 0 )T
m 1 2 m
It turns out that if h
(0)
= 0,Dh
(0)
^{−1} exists, and h is C^{1} on U, then h can be
written as a composition of primitive functions and flips. This is a very interesting
application of the inverse function theorem.
Theorem B.3.2Let h : U → ℝ^{n}be a C^{1}function with h
(0)
= 0,Dh
(0)
^{−1}exists. Then there is an open set V ⊆ U containing 0, flips F_{1},
⋅⋅⋅
,F_{n−1}, and primitivefunctions G_{n},G_{n−1},
⋅⋅⋅
,G_{1}such that for x ∈ V,
h (x) = F1 ∘ ⋅⋅⋅∘ Fn−1 ∘Gn ∘ Gn−1 ∘⋅⋅⋅∘G1 (x).
The primitive function G_{j}leaves x_{i}unchanged for i≠j.
Proof: Let
( )
h1 (x) ≡ h(x) = α1 (x ) ⋅⋅⋅ αn(x) T
( )T
Dh (0)e1 = α1,1(0) ⋅⋅⋅ αn,1 (0)
where α_{k,1} denotes
∂∂αxk1
. Since Dh
(0)
is one to one, the right side of this expression
cannot be zero. Hence there exists some k such that α_{k,1}
≠0. Therefore, by the inverse function theorem,
there exists an open set U_{1}, containing 0 and an open set V_{2} containing 0
such that G_{1}
(U1 )
= V_{2} and G_{1} is one to one and onto, such that it and its
inverse are both C^{1}. Let F_{1} denote the flip which interchanges x_{k} with x_{1}. Now
define