I'm reading the following theorem from the book CONVEX STRUCTURES AND ECONOMIC THEORY by Hukukane Nikaido.
Here $P$-matrix refers to matrix whose principal minors are positive. I do not understand how such a $y$ can be chosen. If $x_{k}$ is increased to $y_{k}$ so that $(Ay)_{k}=0$, then there is a change in other co-ordinates of $Ay$. It is possible that for some $i \neq k$, $(Ay)_{i}$ becomes positive. Then again we need to increase some other co-ordinate of $x$ to make $Ay \leq 0$. It seems to be a cyclic process. Can someone explain how to choose $y$!
If $(Ay)_i$ became positive, it first became zero. So just stop when it becomes zero instead of continuing further and trying to make $(Ay)_k$ zero.
In other words, take any nonzero $v \geq 0$ whatsoever, and let $y = x + t v$, where $t$ is the smallest value which makes some coordinate of $Ay$ zero.