Efficient Computation of the Eigencomponents

The original scheme and its divided difference scheme are clearly closely linked, and an interesting question to ask is whether their eigenfactorisations are related in any transparent and useful way. 

Suppose that V is an eigencolumn with eigenvalue A, of a scheme S, of arity a, whose mask is M. 

When S is applied to V, the result is a copy of V, scaled by A. The first differences of V are therefore also scaled by A, and so the first divided differences are scaled by aX. 

Thus the first differences of an eigencolumn of S form an eigencolumn of S, and the corresponding eigenvalue is scaled up by a factor of a. 

Note that the unit eigencolumn vanishes in a puff of smoke, because its first differences are all zero. Yes, a column of zeroes is an eigenvector, but it is the trivial one, not to be considered beside the real ones. The number of eigencomponents of the divided difference scheme is therefore one less than the number in the original scheme. 

---