Stiemke's Theorem: Only one of the following statements are true:
(a) $Ax\leq 0$ has a solution $x$.
(b) $A^Ty=0$, $y>0$ has a solutions $y$.
I'm trying to understand this theorem. Look at the following example:
$A=\left[\begin{matrix}1 & -2 \\ -1 & 2\end{matrix}\right]$, $x=\left[\begin{matrix}2\\1\end{matrix}\right]$ and $y=\left[\begin{matrix}1\\1\end{matrix}\right]$.
Then $Ax=0$ and $A^Ty=0$, which contradicts the theorem. What am I missing here?
Where did you get that version of the theorem from? Here and here, the theorem is stated as follows:
Only one of the following statements are true:
(a) $Ax > 0$ has a solution $x$.
(b) $A^Ty=0$, $y>0$ has a solution $y$.
In (a), there is a strict inequality.