Small $\sigma_{\min}$ forces a column close to the span of the others: $\sigma_{\min}(A)\le s\Rightarrow \exists j,\ \operatorname{dist}(a_j,H_j)\le\sqrt n\, s$

Let $A$ be a real $N\times n$ matrix with $n\ge 1$, with columns $a_1,\dots,a_n\in\mathbb R^N$, and let

be its smallest singular value. For a column index $j$ let $H_j=\operatorname{span}\{a_i: i\ne j\}$, and write

If $\sigma_{\min}(A)\le s$ for a real number $s$, then there is a column index $j$ with

Indeed, for every unit vector $x$ choose $j$ with $|x_j|=\max_i|x_i|$, so that $|x_j|\ge n^{-1/2}$; then $x/x_j$ has $j$-th entry $1$ and $\operatorname{dist}(a_j,H_j)\le \|Ax\|/|x_j|\le\sqrt n\|Ax\|$. Hence $\min_j\operatorname{dist}(a_j,H_j)\le\sqrt n\,\sigma_{\min}(A)\le\sqrt n\,s$.

This deterministic "invertibility via distance" reduction reduces small-ball estimates for the smallest singular value of a random matrix to small-ball estimates for the distance of a single random column to a subspace spanned by the remaining (independent) columns.

Formalization Note. $\sigma_{\min}$ is GaussianMatrix.sMin, with $\|v\|_2$ written as Real.sqrt (v ⬝ᵥ v). The distance is expressed without a separate definition as the infimum of $\|Ax\|_2$ over vectors $x$ with $x_j=1$ (an infimum of a nonnegative family over a nonempty index set, so no sInf ∅ issue arises).