the linear system of inequalities ( S ) : A x ≥ 0 , x ≥ 0has a solution x ∈ R n that is not zero x ≠ 0. Here A is a n × n square matrix with real entries.Of course I could combine the two inequalities in (S) into one and write it in the form of A ~ x ≥ 0 where A ~ is 2 n × n.

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the linear system of inequalities   (  S  )  :     A  x  ≥  0  ,     x  ≥  0has a solution   x  ∈        R    n   that is not zero   x  ≠  0. Here   A is a   n  ×  n square matrix with real entries.Of course I could combine the two inequalities in (S) into one and write it in the form of       A    ~    x  ≥  0 where       A    ~   is   2  n  ×  n.

Stevinivm · Accepted Answer

(S) is equivalent to  [  A     −      I    n    ]      [                            x                                      b                      ]    =  0  ,         [                            x                                      b                      ]    ≥  0         a    n    d           [                            x                                      b                      ]    ≠  0     (  ⇔     x  ≠  0  )According to Gordan’s theorem if (S) does not have solutions then there is   y  ∈        R    n   such that       [                                        A                      T                                                −                      I                      n                                ]    y  &amp;lt;  0 which comes to   y  &amp;gt;  0 and       A    T    y  &amp;lt;  0.

the linear system of inequalities ( S ) : <mtext> </mtext> A x &#x22

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