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 6.9.1Let U be an open set in ℝ^{n}and let G : U → ℝ^{n}. Then G is calledprimitiveif it is of the 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 calleda flip if
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
Pm (x) ≡ (x1 x2 ⋅⋅⋅ xm 0 ⋅⋅⋅ 0 )T
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 6.9.2Let 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 primitive functionsG_{n},G_{n−1},
⋅⋅⋅
,G_{1}suchthat 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
( )T
h1(x) ≡ h (x) = α1(x) ⋅⋅⋅ αn (x)
( )T
Dh (0)e1 = α1,1 (0) ⋅⋅⋅ αn,1(0)
where α_{k,1} denotes
∂∂αxk
1
. 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