Any example of condition number of matrix less than 1?

Question

Any example of the condition number of matrix less than 1? In our lectures, my professor defined the condition number under the matrix norms. i.e. $$K(A) = ||A^{-1}||_M||A||_M$$

Answer 1

The condition number $\kappa(A)$ of a complex square matrix $A$ is usually defined as $\|A\|\|A^{-1}\|$ for some induced matrix norm $\|\cdot\|$. Since the matrix norm is induced from a vector norm, it is submultiplicative and $\|I\|=1$. It follows that $\kappa(A)=\|A\|\|A^{-1}\|\ge\|AA^{-1}\|=\|I\|=1$, i.e. the condition number is always $\ge1$.

Answer 2

For non-square complex matrices, the easier way is to define the condition number as the ratio between the largest and smallest singular values. From this definition it is clear that $\kappa$ is always greater than or equal to 1.