How to get a lower bound on $|Mv|$ (matrix-vector product)? 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?
There can't be a better lower bound than $0$, because it is possible to have $Mv = 0$ with $v \ne 0$.
If $M$ is not zero, then there is a unique nonzero greatest lower bound.

If $M$ is nonsingular, then that best lower bound is $1 / | M^{-1}|$.

In general, see https://doi.org/10.1016/j.laa.2010.02.014

Joseph Grcar, "A matrix lower bound", Linear Algebra and Its Applications, Volume 433, Issue 1, 15 July 2010, Pages 203-220