{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Gordan

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

This preview shows page 1. Sign up to view the full content.

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).
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern