On the determinantal conjecture of Marcus and de Oliveira (Q1208265)

Let \(A\) and \(B\) be normal complex \(n\times n\) matrices with eigenvalues \(\lambda_ 1,\dots,\lambda_ n\) and \(\mu_ 1,\dots,\mu_ n\) respectively. \textit{M. Marcus} [Indiana Univ. Math. J. 22, 1137-1149 (1973; Zbl 0243.15025)] and \textit{G. N. de Oliveira} [Research problem: Normal matrices, Linear Multilinear Algebra 12, 153-154 (1982)] have conjectured that \(\text{det}(A + B) \in co\{\prod^ n_{j=1}(\lambda_ j + \mu_{\sigma(j)});\;\sigma \in S_ n\}\), where \(S_ n\) denotes the symmetric group on \(\{1,2,\dots,n\}\) and \(co\) denotes the convex hull in the complex plane. The authors provide a theorem and a counterexample relating to the determinantal conjecture of Marcus and de Oliveira.