  Consider an $n \times n$ matrix polynomial $P(\lambda)$. An upper bound for a
spectral norm distance from $P(\lambda)$ to the set of $n \times n$ matrix
polynomials that have a given scalar $\mu\in\mathbb{C}$ as a multiple
eigenvalue was recently obtained by Papathanasiou and Psarrakos (2008). This
paper concerns a refinement of this result for the case of weakly normal matrix
polynomials. A modification method is implemented and its efficiency is
verified by an illustrative example.
