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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



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