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 19.3.1Let U be an open set in ℝ^{n}and let G : U → ℝ^{n}. Then G is called primitiveif it is ofthe form
( )T
G (x) = x1 ⋅⋅⋅ α(x) ⋅⋅⋅ xn .
Thus, G is primitive if it only changes one of the variables. A function F : ℝ^{n}→ ℝ^{n}is called a flipif
F (x1,⋅⋅⋅,xk,⋅⋅⋅,xl,⋅⋅⋅,xn) = (x1,⋅⋅⋅,xl,⋅⋅⋅,xk,⋅⋅⋅,xn)T .
Thus a function is a flipif it interchanges two coordinates. Also, for m = 1,2,
⋅⋅⋅
,n, define
P (x) ≡ ( )T
m x1 x2 ⋅⋅⋅ xm 0 ⋅⋅⋅ 0
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 19.3.2Let h : U → ℝ^{n}be a C^{1}function with h
(0)
= 0,Dh
(0)
^{−1}exists. Then there is an openset V ⊆ U containing 0, flips F_{1},
⋅⋅⋅
,F_{n−1}, and primitive functions G_{n},G_{n−1},
⋅⋅⋅
,G_{1}such that forx ∈ 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
( )T
h1(x) ≡ h (x) = α1 (x) ⋅⋅⋅ αn(x)
( )
Dh (0 )e = α (0 ) ⋅⋅⋅ α (0) T
1 1,1 n,1
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}
(U )
1
= 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