# condition number and singular values

Let $$m \geq n$$ and $$A \in \mathbb{R}^{m\times n}$$. Let $$\sigma_1,\dots, \sigma_n \geq 0$$ be the singular values of $$A$$. We know that if $$r$$ is the rank of $$A$$, then $$\sigma_1,\dots, \sigma_r$$ are positive and the rest are $$0$$. If $$r = n,$$ then all singular values are positive, and we can define the condition number of $$A$$ to be \begin{align} \kappa(A) = \frac{\sigma_1}{\sigma_n} \end{align}

When $$1 \leq r < n,$$ some authors define $$\kappa(A) = \infty$$, but I've seen $$\kappa(A)$$ defined as \begin{align} \kappa(A) = \frac{\sigma_1}{\sigma_r} \end{align} where $$\sigma_r$$ is the smallest positive singular value. If a large condition number means the matrix is ill conditioned, the first definition says that less then full rank matrices are ill conditioned by definition. How can we see that less then full rank matrices are ill conditioned using the second definition? Is $$\frac{\sigma_1}{\sigma_r}$$ always large? What about when $$r=1,$$ and then $$\sigma_1 = \sigma_r,$$ and so actually the condition number is 1?

Short answer: these are two different definitions for two different things that are both called "condition number" (and there are many more), of course they don't agree.