I have a question about the effect of applying a linear transformation $M$ in $\mathbb{R}^{n \times n}$ to a vector $v \in \mathbb{R^n}$.

I know that if $M$ has p-norm $\|M\|_p = \lambda$, then by definition I can guarantee that for every vector $v \in \mathbb{R^n}$ $$\|Mv\|_p \leq \lambda \|v\|_p.$$

Is there a corresponding lower bound? In general the answer is no, (for example, choose $v \in Nullspace(M)$ if the matrix is not full rank. )But I am interested in a corresponding lower bound where

$v$ is chosen ``generically'', so that the probability of it being in a (fixed in advance) subspace is 0.

$\|v\|_p$ is sufficiently large.

In this case, is it true that $$\|Mv\|_p \geq C \|v\|_p$$ where $C$ is some positive constant greater than zero?