An invertible matrix is ill-conditioned when its condition number is large in the chosen norm. A determinant close to zero is not a scale-independent test: $A=\varepsilon I_n$ has determinant $\varepsilon^n$ but $\kappa_2(A)=1$.
In the 2-norm, the distance to the nearest singular matrix is $\sigma_{\min}(A)$ and the relative distance is $\sigma_{\min}(A)/\sigma_{\max}(A)=1/\kappa_2(A)$. Thus “nearly singular” should mean small relative distance, with scale specified.
Parents
Like how this is organised?
Build your own knowledge tree on knowen.md: notes, decisions and tasks in one connected markdown graph, shared with your team and your AI agents.