Matrices satisfying a conjecture of G. N. de Oliveira on determinants (Q1075404)

Let \(A\) and \(B\) be normal \(n\times n\) matrices with eigenvalues \(\alpha_ 1,...,\alpha_ n\) and \(\beta_ 1,...,\beta_ n\) respectively. Once the present reviewer [Normal matrices (research problem), Linear Multilinear Algebra 12, 153-154 (1982)] conjectured that if \(U\) is a unitary matrix then the determinant of \(A+UBU^*\) belongs to the convex hull of the \(n!\) points \(\prod^{n}_{j=1}(\alpha_ j+\beta_{\sigma(j)}\), \(\sigma\) running over the symmetric group of degree n. In the present paper the authors prove that the conjecture is true if \(A\) is positive definite Hermitian and \(B\) is a skew-Hermitian matrix. It should be pointed out that the proof of the result in this paper is quite complicated. The conjecture is known to be true also in the case \(A\) and \(B\) are Hermitian. In fact it was this result of \textit{M. Fiedler} [Proc. Amer. Math. Soc. 30, 27-31 (1971; Zbl 0277.15010)] that inspired the conjecture. The proof of Fiedler's result is difficult. So it seems that a complete answer to the conjecture will also be rather difficult.