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Gordan - 2 A theorem of the alternative The separating...

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2 A theorem of the alternative The separating hyperplane theorem has a variety of applications. Amongst them is the very interesting result about existence of solutions to linear systems which we can use to determine conditions when arbitrages cannot exist. Suppose A is an n × m matrix and for z R k write z > 0 when z j > 0 for each j and z 0 when z j 0 for each j . Recall that ( Mz ) T = z T M T . Theorem 2.1 (Gordan’s theorem) Exactly one of the following systems has a solution: (1) y T A > 0 for some y R n ; (2) Ax = 0 , x 0 for some non-zero x R m . Proof: start by showing by contradiction that the two systems cannot both have solutions. If each has a solution then z T = y T A > 0 and hence 0 = y T ( Ax ) = z T x > 0 which is impossible. Next suppose that system (1) has no solution i.e. the two non-empty convex sets S 1 = { z R m : z = A T y, y R n } , S 2 = { z R m : z > 0 } are disjoint, that is S 1 S 2 = (though clearly z = 0 is on the boundary of both sets).
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