## A Simple Algebraic Proof Of Farkas's Lemma And Related Theorems (1998)

Citations: | 3 - 0 self |

### BibTeX

@MISC{Broyden98asimple,

author = {C. G. Broyden},

title = {A Simple Algebraic Proof Of Farkas's Lemma And Related Theorems},

year = {1998}

}

### OpenURL

### Abstract

this paper we have given an alternative proof of Farkas's lemma, a proof that is based on a theorem, the main theorem, that relates to the eigenvectors of certain orthogonal matrices. This theorem is believed to be new, and the author is not aware of any similar theorem concerning orthogonal matrices although he recently proved the weak form of the theorem using Tucker's theorem (see [5]). His proof of the theorem is "completely elementary" (a referee) and requires little more than a knowledge of matrix algebra for its understanding. Once the theorem is established, Tucker's theorem (via the Cayley transform), Farkas's lemma and many other theorems of the alternative follow trivially. Thus the paper establishes a connection between the eigenvectors of orthogonal matrices, duality in linear programming and theorems of the alternative that is not generally appreciated, and this may be of some theoretical interest.