  Pivoting methods are of vital importance for linear programming, the simplex
method being the by far most well-known. In this paper, a primal-dual pair of
linear programs in canonical form is considered. We show that there exists a
sequence of pivots, whose length is bounded by the minimum dimension of the
constraint matrix, such that the pivot creates a nonsingular submatrix of the
constraint matrix which increases by one row and one column at each iteration.
Solving a pair of linear equations for each of these submatrices generates a
sequence of optimal solutions of a primal-dual pair of linear programs of
increasing dimensions, originating at the origin. The optimal solutions to the
original primal-dual pair of linear programs are obtained in the final step. It
is only an existence result, we have not been able to generate any rules based
on properties of the problem to generate the sequence. The result is obtained
by a decomposition of the final basis matrix.
