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The validity of the Marcus-de Oliveira conjecture for essentially Hermitian matrices. (English) Zbl 0801.15005

Let \(x = (x_ 1, \dots, x_ n)\) and \(y = (y_ 1, \dots, y_ n)\) be \(n\)-tuples of complex numbers. A conjecture proposed by M. Marcus [Indiana Univ. Math. J. 22, 1137-1149 (1973; Zbl 0243.15025)] and G. N. de Oliveira [Research problem. Linear Multilinear Algebra 12, 153-154 (1982)] states that the set of possible values of \(\text{det} (X + Y)\), where \(X\) and \(Y\) run over the \(n \times n\) normal matrices with spectra \(x\) and \(y\), respectively, is contained in the convex hull of the numbers \(\Pi_ k (x_ k + y_{\sigma (k)})\), \(\sigma \in S_ n\).
The authors of this paper present a proof of the conjecture in the case \(x_ 1, \dots, x_ n\) are distinct positive real numbers and \(y_ 1, \dots, y_ n\) lie along a line through the origin. They also state its validity in the case \(x_ 1, \dots, x_ n\) are distinct complex numbers lying on a line \(\ell\) and \(y_ 1, \dots, y_ n\) lie along a line parallel to \(\ell\). These statements include as particular cases several previously known results.

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
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References:

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