If $Ax\geq b$, under what conditions on $K$ can I premultiply it and preserve the sign? i.e
What is a sufficient condition on $K$ for: $$KAx\geq Kb$$
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Suppose you have two vectors which satisfies the component-wise inequality (denoted $\ge_c$) $$\mathbb{x} \ge_c \mathbb{y}$$ If you want a matrix such that $$A\mathbb{x} \ge_c A\mathbb{y}$$ then that is equivalent to requiring $$\mathbb{a_i}\cdot \mathbb{x} \ge \mathbb{a_i}\cdot \mathbb{y}$$ for each row vector $\mathbb{a_i}$ of $A$. It is sufficient (and probably necessary) that you require $A$ to be non-negative. |
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