Two Proofs of the Farkas-Minkowski Theorem by Tandem Method and by Use of an Orthogonal Complement

Takao Fujimoto (), B.B.Upeksha P. Perera () and Giorgio Giorgi ()
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Takao Fujimoto: University of Kelaniya, Kelaniya
B.B.Upeksha P. Perera: University of Kelaniya, Kelaniya
Giorgio Giorgi: Department of Economics and Management, University of Pavia

No 129, DEM Working Papers Series from University of Pavia, Department of Economics and Management

Abstract: This note presents two proofs of the Farkas-Minkowski theorem. The first one is analytical, and this does not presuppose the closedness of a finitely generated cone. We do not employ separation theorems either. Even the concept of linear independence or invertibility of matrices is not necessary. Our new device consists in proving the Farkas-Minkowski theorem and the closedness of a finitely generated cone at the same time based upon mathematical induction. We make use of a distance minimization problem with an equality constraint. The second proof is algebraic, and a mixture of Gale’s and Ben-Israel’s methods. Our proof based on the orthogonal complement seems easy to understand in terms of geometrical images.

Pages: 12 pages
Date: 2016-11
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