Is the condition number of unitary matrix always equal to 1?

Question

I know that the 2-norm condition number $\kappa (\textbf U)={||\textbf U||_2}{||\textbf U^{-1}||_2}$ of a unitary matrix $\textbf U$ is always equal to 1. Is this true for all induced matrix norms, i.e. $\kappa (\textbf U)=||\textbf U||||\textbf U^{-1}||=1$?

Answer 1

The condition number with respect to the norm induced by vector norm $\|\cdot\|$ is equal to $1$ precisely when the image of the ball $B_1=\{x:\|x\|\le 1\}$ is another ball $B_r$ of some radius $r>0$.

For any vector norm that is not a constant multiple of the Euclidean one, there is a rotation of its unit ball that maps it into something other than a ball. For example, take the $\ell_\infty$ ball, i.e., a cube, and rotate it slightly. So the answer to your question is negative: up to scaling, the vector $2$-norm is the only norm for which all unitary matrices have condition number $1$.