Given $a\in \mathbb{R}^N$ with at least one positive entry, and a positive definite $N\times N$ matrix $A$, I would like to prove the following set is non-empty:
$$S=\{ x\in \mathbb{R}^N : x\geq 0, Ax\geq a\}$$
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Is the idea here that $S$ depends on a choice of vector $a$? – hardmath Sep 25 '13 at 23:38
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@hardmath Yes I guess and for any $a$ (with the given condition), the set is non-empty. – Sudarsan Sep 25 '13 at 23:41
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Hint: Since $\mathbf{A}$ is positive definite, $\forall \mathbf{x} \in \mathbb{R}^n\text{ (non-zero)},\mathbf{x}^t\mathbf{A}\mathbf{x}>0$. Now following this, what can you say about our inequality $\mathbf{x}^T\mathbf{A}\mathbf{x}\geq \mathbf{x}^T\mathbf{a}$ given that at least one entry of $\mathbf{a} \in \mathbb{R}^n$ is positive?
Sudarsan
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