Symmetric Normal Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. Let ∆S be the restriction of ∆ to determinants of sums of symmetric normal matrices. This paper focuses on boundary matrices of ∆. We derive some properties of boundary matrices and boundary points. We conjecture that ∂∆ ⊆ ∂∆S. Speculations on how to prove this conjecture are given. We also present a second conjecture with regards to the form of normal matrices with magnitude symmetry. This paper builds on work in [1].

Comments: 15 Pages.

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Submission history

[v1] 2016-10-11 13:57:41
[v2] 2017-02-28 03:28:22
[v3] 2017-04-13 15:18:44
[v4] 2017-04-24 20:06:06
[v5] 2017-05-02 16:15:55

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