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
Definition B.3.1Let U be an open set in ℝnand let G : U → ℝn. 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 : ℝn→ ℝniscalled a flip if