# Ill-conditioned matrices and their singular values

For ill-conditioned matrices, must it be that the smallest singular value is arbitrarily close to $$0$$? I know that $$K_2(A) = \frac{\sigma_{max}}{\sigma_{min}}$$ where $$\sigma$$ is a singular value of A. Can their be a case of $$\sigma_{max}$$ being extremely large and $$\sigma_{min}$$ being arbitrarily close to 1.

Of course. Think of the matrix $$A$$ defined as $$A = \mathrm{diag}(1, 2, \ldots, N),$$ where $$\mathrm{diag}$$ is the diagonal matrix. For any value $$N$$, the condition number of the matrix will be $$K_2(A) = N$$ and the smallest singular value $$\sigma_{\min} = 1$$. The matrix can also be small, for example $$A = \begin{pmatrix} 1 & 0 \\ 0 & N \end{pmatrix},$$ has the same spectral properties as the matrix above.