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Suppose $M$ is a square matrix (with elements that are continuous functions which are bounded above and below) and $v$ is a vector. I want a lower bound like $|Mv| \geq C|v|$ for constant $C$.

Do I have any luck here?

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    @RobertIsrael Doh, I actually do mean a lower bound.2012-11-16

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There can't be a better lower bound than $0$, because it is possible to have $Mv = 0$ with $v \ne 0$.

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    You might look at http://en.wikipedia.org/wiki/Min-max_theorem2012-11-18