Some further notes on the Marcus-de Oliveira determinantal conjecture
Let \(A\) and \(B\) be normal matrices with eigenvalues \((a_ j)\) and \((b_ j)\), respectively. A famous conjecture claims that \(d=\text{det} (A+B)\) belongs to the convex hull of the products \(z_ \sigma = \prod (a_ j + b_{\sigma (j)})\), taken for all permutations \(\sigma\). The authors show that this is equivalent to the inequalities \(\min_ \sigma \text{Re} z_ \sigma \leq \text{Re} d \leq \max_ \sigma \text{Re} z_ \sigma \), and that this is true if all the \(z_ \sigma\)'s are collinear. They also show that the conjecture holds if among the \(a_ j\)'s (or, \(b_ j\)'s) there are at most two different values. The paper corrects an earlier paper of the same authors [Linear Algebra Appl. 121, 345-352 (1989; Zbl 0692.15003)].
determinant of the sum of two normal matrices
scientific article; zbMATH DE number 399303