Induced metric and matrix inequalities on unitary matrices

Recently, Chau (2011 Quantum Inform. Comput. 11 721) showed that one can define certain metrics and pseudo-metrics on U(n), the group of all n × 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 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).