  We present several results associated to a holomorphic-interpolation problem
for the spectral unit ball \Omega_n, n\geq 2. We begin by showing that a known
necessary condition for the existence of a
$\mathcal{O}(D;\Omega_n)$-interpolant (D here being the unit disc in the
complex plane), given that the matricial data are non-derogatory, is not
sufficient. We provide next a new necessary condition for the solvability of
the two-point interpolation problem -- one which is not restricted only to
non-derogatory data, and which incorporates the Jordan structure of the
prescribed data. We then use some of the ideas used in deducing the latter
result to prove a Schwarz-type lemma for holomorphic self-maps of \Omega_n,
n\geq 2.
