Condition number of a matrix bounded from below and above?

Question

Is condition number of an invertible matrix bounded from below? And is condition number always bounded from above for an invertible matrix?

Answer 1

The condition number of a matrix $A$ can be characterized by $$\kappa(A)=\frac{\lambda_{\max}}{\lambda_{\min}}$$ where $$\lambda_\min := \min \{|\lambda |: \lambda \in \sigma(A)\},\quad \lambda_\max := \max \{|\lambda |: \lambda \in \sigma(A)\},$$ and $\sigma(A)$ is the spectrum of $A$. So it is clear that $\kappa(A) \geq 1$ for every $A$ since $\lambda_\min\leq \lambda_\max$, however consider the matrix

$$A=\begin{pmatrix} \delta & 0 \\ 0 & 1 \end{pmatrix}$$ then for $\delta > 1$ we have $\lambda_\min=1$,and $\lambda_\max= \delta$ so that $$\kappa(A) = \delta \to \infty$$ as $\delta \to \infty$. So it is not bounded from above.