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A Survey of the Marcus–de Oliveira Conjecture

  • Huajun HuangEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 277)

Abstract

We classify and survey the progress on the famous Marcus–de Oliveira determinantal conjecture (MOC) and related problems. The MOC claims that for two normal matrices A and B with eigenvalues \(a_1,\ldots , a_n\) and \(b_1,\ldots ,b_n\), respectively, the set \(\varDelta (A,B)=\{\det (A+UBU^*): U \text { is unitary}\}\) is in the convex hull of the set \(\{\prod _{i=1}^{n} (a_i+b_{\sigma (i)}):\sigma \in S_n\}\). We review the origin and the motivations of this conjecture from M. Fiedler’s work in the Hermitian case to Marcus and de Oliveira’s independent questions. Then, we survey the major positive cases of the MOC in terms of matrix degree, eigenvalues, and other things. We also describe some known properties of the set \(\varDelta (A,B)\), including compactness, connectivity, simply connectivity, convexity, star shapedness, and boundary and corner properties. Finally, we list some extended results that are related to the MOC. The main goal of this article is to provide a fairly comprehensive and brief overview of the progress of the MOC to interested readers and researchers for further explorations of the subject.

Keywords

Marcus-de Oliveira conjecture Normal matrix Determinant Eigenvalues Unitary matrix 

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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsAuburn UniversityAuburnUSA

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