18
It follows that
Aj⅝' ≥ λicbii ⅛ λid - λ7c∣δn∙∣
= c(λiba — λj ∖bji I) + λid
> Xid
> 0.
Hence we have
On the other hand, it is not difficult to see that the procedure will terminate
within a finite number of iterations since each entry ‰7∙ is hnite. Hence the procedure
produces the matrix A = AU in standard form where U is a unimodular matrix.
We complete the proof. □
We remark that the origin of Rn is still contained in the interior of the convex
hull of the vectors ɑɪ,..., an+1. Moreover, the corresponding convex combination
coefficients remain unchanged. It is also easy to show that the volume of the simplex
does not change under unimodular transformation. For n = 2, we can construct
such unimodular matrices by adapting Scarf’s method in [10]. The procedure now
can be applied.
Let us give some examples.
Example 5. We are given
0 -1
1 1
-1 0
Then
1 1
-1 0