  Given a normal matrix $A$ and an arbitrary square matrix $B$ (not necessarily
of the same size), what relationships between $A$ and $B$, if any, guarantee
that $B$ is also a normal matrix? We provide an answer to this question in
terms of pseudospectra and norm behavior. In doing so, we prove that a certain
distance formula, known to be a necessary condition for normality, is in fact
sufficient and demonstrates that the spectrum of a matrix can be used to
recover the spectral norm of its resolvent precisely when the matrix is normal.
These results lead to new normality criteria and other interesting
consequences.
