The question investigated here is: if two matrices $A$ and $B$ in $\CNN$ have identical behaviour in a unitarily invariant norm $\norm{\cdot}$, \ie,\ $\norm{p(A)} = \norm{p(B)}$ for every polynomial $p$ with complex coefficients, what properties do $A$ and $B$ have in common? After a preliminary result about eigenvalues, it is shown with a mildly restrictive assumption that if $A$ is unitarily reducible, so is $B$. A theorem is proved about Hadamard products of the form $H\circ\invt{H}$, where $H$ is Hermitian positive definite. Finally, an example is produced where $A$ and $B$ have identical behaviour in the Frobenius norm, but are not related to each other in any simple way.