Let $A \in \mathbb{R}^{m \times n}$ be a random matrix with i.i.d. entries (the distribution is not important), where $m < n$ (i.e. $A$ is a "wide" matrix). I would like a lower bound on $$ \phi(A) \triangleq \min_x \frac{\lVert Ax \rVert}{\lVert x \rVert} $$ that holds with high probability (apologies if the notation $\phi(A)$ conflicts with any established usage).

When $m \geq n$, evidently $\phi(A) = \sigma_{min}(A)$, the least singular value of $A$ (although I am not certain why this is true). Of course the distribution of the least singular value of a random matrix has been well-studied.

But when $m < n$, it seems that $\phi(A) \neq \sigma_{min}(A)$ in general. For example, if $m = 1$ and $n > 1$, then $\phi(A) = 0$ (just choose $x$ to be orthogonal to the vector $A$), but $\sigma_{min}(A)$ is the Euclidean norm of the vector $A$, which usually will not be $0$.

`$m\times n$`

matrix has`$\min(m,n)$`

singular values. $\endgroup$ – J. M. is not a mathematician Sep 3 '10 at 10:11