  Recently, Chau [Quant. Inform. & Comp. 11, 721 (2011)] showed that one can
define certain metrics and pseudo-metrics on U(n), the group of all $n\times n$
unitary matrices, based on the arguments of the eigenvalues of the unitary
matrices. More importantly, these metrics and pseudo-metrics have quantum
information theoretical meanings. So it is instructive to study this kind of
metrics and pseudo-metrics on U(n). Here we show that any symmetric norm on
${\mathbb R}^n$ induces a metric on U(n). Furthermore, using the same
technique, we prove an inequality concerning the eigenvalues of a product of
two unitary matrices which generalizes a few inequalities obtained earlier by
Chau [arXiv:1006.3614v1].
